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§3 such that Id§2 = 11 0 0. Since {x E (0, 1), hx = h} has positive measure, it follows that f == 0, that is 'IjJ = h . 0) started at any point y E aD, the first exit time inf { t > 0 Bt tI- D} is zero a.s. THEOREM 2.1. Let D be a bounded domain in IRd . A ssume that D is Dirichlet regular and let g be a nonnegative continuous function on aD. For every x E D, =
---->
==
deg('P) .
§2 and u : §3
---->
§2 be continuous maps. Then, it
H(w 0 u) = (deg w) 2 H(u) , whereas for every continuous 9 : §3 ----> §3 , then it can be deduced from the discussion above that H(u 0 g) = (deg g)H(u) . REMARK 8. It is worthwhile (in particular in view of the proof of case d) in Theorem 4 below) comparing the 3-Dirichlet energy of a map u : §3 ----> §2 with the 3-Dirichlet energy of a lifting 'P. In view of the integral definition of the degree d == H(u) = deg('P) = so that (11)
; 13 J'P d1i3 ,
1 31
1 38
FABRICE BETHUEL AND DAVID CHIRON
On the other hand, it is shown in [25] , using Remark 7, that for some smooth map Ud : §3 -+ §2 such that H (Ud ) = d and
r
iS3
d E Z, there exists
IV Udl 3 d'H3 < Gl dl � ,
so that, by (ll), the 3-energy of Ud becomes negligible compared9 to the 3-energy of any lifting 'Pd of Ud as d -+ +00. The previous argument can be adapted to prove for a given d E Z the existence of smooth maps Wd : B3 -+ §2 such that Wd = 0"1 §2 on 8B3, H(Wd ) = d and
E
83
IV'Wdl 3 dx < C1 dl ! .
Here, in the definition of the Hopf invariant H(Wd), we compactify the boundary 8B3 to a point. As above, one proves that for any liftingl O 'Pd of Wd on B3 , we have for some G 0
>
( 12) Choosing a good gauge. \Ve have seen in the case of coverings that the choice of
a lifting is very reduced. In particular, in many cases, the imposed regularity yields the uniqueness of the lifting, up to the action of an clement of 7fl (N) (see Lemma AA in the Appendix) . By contrast, when the fiber is a continuum, if the lifting exi�t�, then there is also a continuum l l of possible liftings, so that, if one imposes some regularity to the lifting, then one has to make a suitable choice, usually called a gauge fixing. In the case of the Hopf fibration, with abelian fiber § l = U (l) , we will describe next the Coulomb gauge. exp((J(x)O"l )g(X), so Let (J : BN --+ R be a scalar function, and set go(x) that U = IT 0 90 and explicit computations show that g;Idgo = g- I dg + (dB)O"I. In particular, _
We claim that there exi�ts Bo- such that the 1-form Al identify with the vector field AI , satisfies divAt = d*-Al = 0 (13) At n = (A d N = 0 -
.
Indeed, it suffices to choose
(Jo such that
= - div(Al) = -AI ' n
In particular, if A l the functional
gIn view of (9),
in BN , on 8BN.
E L2 (BN ), then (Jo can be obtained by minimizing on Hl (BN )
this means that the 3-energy of 'f'd is "concentrated" in A I , which is not seen
by IV'Udl .
lOThe lifting is not necessarily constant on 8B3: however, the image 'Pd(8B3) is of measure
zero in S3 .
llmuch larger than the fib er!
1 39
SOME QUESTIONS
The curvature equation. The term A l (g) is related to the projection by the following classical equation 12
(14)
dA 1 = 2u ( *
w
u = II 0 9
),
where w denotes the standard volume form on §2 , and u* (w) its pull-back on BN . In coordinates, " ." standing for the scalar product in su( 2 ) , u (w ) writes as
u
*
(w )
=
L I �J<j-::;'
N
aU U · (ax ' '
x
)
*
au dXi 1\ dX . j ax ·
J
Combining (13) with (14), we obtain an elliptic system for the Coulomb gauge A I . More precisely, by Poincare lemma, there exists a 2-form such that d = 0, T = 0 on aB N and
(15) Inserting (16)
(15) into (14), we are led to the elliptic equation 13 -� = u ( ) *
w
Lifting maps in W1,P ( BN, §2 ) . The lifting problem for the Hopf fibration II : SU(2) � §3 � §2 for s = 1 can be settled in a number of cases. (i)
THEOREM
If either
4. Let 1 :s; p :s; 00 and N 2 2 . (b) p 2 N 2 3,
(a) l :S; p < 2 :S; N,
then for every u E W 1 ,p (BN , §2 ) , there exists c.p E W1 ,p (BN , §3) such that u = IIoc.p. (ii) If either (c) 2 :S; p < 3 :S; N, then there exists a map u
W 1 ,P (BN , §3) .
This last Theorem (OQ 3) p = H l (B2 , §3 ) ?
N =
2,
E
W 1 ,P(BN , §2 )
(d) p = 3 < N, which does not admit any lifting c.p
E
4 leaves two cases open i. e.
does every map in H I (B2 , §2 ) admit a lifting c.p E
(OQ 4) Is the lifting property true for
3
Notice indeed that the case p > N = 2, which does not appear in Theorem can be handled using Proposition 5 or arguing as in the proof of case (b). As mentionned, we conjecture that the answer to (OQ 3) is positive.
4,
1 2This equation can be deduced by direct computation using the fact that dA + [A, A] = 0, which means that A = g - l dg has zero curvature as an 8U(2) connection (see e.g. [11] ) . 13To avoid the boundary conditions, one may consider the problem on § N by gluing together
two copies of EN as the north and the south hemispheres.
140
FABRICE BETHUEL AND DAVID CHiRON
PROOF .
The proofs for the different cases rely on arguments of rather different
nature.
Proof of case (a). The idea is base on Remark 6 as well as an averaging argument
of [15] . For 1 < p < 2, let U E Wl,P (EN , §2) and consider for e E §2 the map §3. We compute first by (10) and the chain rule 'Pc -
Integrating over
e
E
§2 and x E nN and using Fubini's theorem, we are led to
d1i2 ( e) is finite and independent of X dim §2. \Ve may therefore choose some E §2 such that since
i:']2 IX - e l -P
e
E
§2 for 1 <
P
< 2
=
1V''P e I P dx < Cp r lV'u l P dx. { .IBN lBN
Since 'Pc has the desired property, namely II 0 'Pe = u, this completes the proof of case (a) .
Proof of case (c). Since N > 3, we can consider the map u : EN (X l , X2 , X3 ) _ u (x ) =
->
§2 defined by
I (X l , X2 , X 3 ) 1 '
U(nN) for every 1 < P < 3, hence u E We have l V'ul (x) < I (X l , X2 , X3 )I-l Wl'P ( EN , §2) for 1 < p < 3. We are going to show that u can not be lifted to a map 'P E Wl,p(nN, §2) if 2 < P < 3. We argue by contradiction: assume that 'P E WI,p(nN, §2) satisfies u = II 0 'P. By the usual averaging argument we have used in the proof of Lemma 1, this yields a radius r E (0, 1 ) and a E EN-3 such that 'u ls� has a lifting 'P l s� E Wl,p(S; , §3), where S; is the sphere S; = {x = (X I , X2 , X3 , X') E nN, xI + x� + x§ = r2, x' = a}. Changing coordinates, this means that I d§2 has a lifting lP WI ,p (§2 , §3) . If p > 2, then lP is a continuous lifting of Ids2 , a contradiction with the property seen earlier. In the limiting case p = 2, wc argue by approximation. There exists a sequence 1/Jn E C = (§2 , §3) such that lPn -> lP in HI (§2, §3) . Therefore, II 0 lP", --> II 0 lP = Id§2 in HI (§2, §2) , which yields, by continuity of the degree in H I (§2 , §2) , deg(II 0 lPn) -> deg(Id§2 ) = 1 as n -> +00. However, since 'IjJn : §2 -> §3 is smooth and '7r2 (§3) = {O}, then §2 also, hence lPn : §2 -> §3 is homotopic to a constant, thus II 0 lPn : §2 deg(II 0 lPn) = 0, a contradiction.
E
E
-+
REMARK 9. An alternate proof of case ( c) relies on density properties of smooth maps in Sobolev spaces between manifolds. Recall that since 7r2 (§3) = {O}, then c= (nN , §3) is dense in Wl,p(nN, §3) for 2 < P < :� ::; N (see [1] ) . In particular, for any 'P E WI,P(EN, §3) , II 0 'P belongs to Coo (EN, §2 ) i- Wl,P(EN, §2 ) since 7r2 (§2) =I {O} (see [1] ) .
141
SOME QUESTIONS
This Remark raises the following question:
(OQ 5) For 2 ::; p < 3 ::; N, does any map in coo (B N , §2 ) has a lifting in W 1 ,P(B N , §3) ? In other words, do we have II(W 1 ,P(B N , §3 ) ) = coo (B N , §2 ) -=I W 1 ,P(B N , §2 ) ? REMARK 1 0 . The argument of the proof of case (c) shows that Id§2 has no lifting in VMO(§2 , §3 ) , and therefore, I d§2 has no lifting in WS,P (B2 , §3) for every ° < s < 00 and 1 < p < 00 such that sp 2 2 . We use in this case the continuity of the degree in VMO(§2 , §2 ) (see [7] ) .
Proof of case (d). I t suffices t o consider the case N = 4 (for N > 4, one adds suitably many dimensions) . The central argument in the proof is a dipole construction introduced in [4] . Consider, for A > 0, d E Z the function Wdip (A, d) : jR4 ----+ §2 defined for X l E jR, X' E jR3 by Wdip(A, d) (X 1 ' x, ) Wd ==
( 1 2 1 xAX'1 - 1 1 )
1 if X E B4 , and Wdip(A, d) is extended by 0'1 outside B4. Here, Wd is the map constructed in [25] and described in Remark 8. Notice that Wdip is locally lip schitz continuous away from (± � , 0, 0, 0) , where it has singularities of Hopf in variant ±d. Moreover, Wdip = 0'1 outside some neighborhood L: A of the segment L: oo == [(- � , O, O, O) , ( � , O, O, O)] which shrinks to L:oo as A ----+ +00. Finally,
r
W , d dx C d i . Jr.). IV dip(C W ::; 1 l The main point is that, as in Remark 8, any lifting of Wdip has a 3-energy larger than C 1 d l . We consider finally the scaled and translated versions of Wdip Wdip (A, C, d, a) WdiP ( ' d) � a , so that Wdip(A, C, d, a) = 0'1 outside L:i a a + CL:� and ==
,
==
� C ) £
( 1 7) Our counter-example writes
u L: Xr.n Wdi ( An , Cn , an , dn ) + NXB4\(Un24r. n ) , n 2:4 A n L:an ,<-n and for the choice an = ( 1n , 0, 0, 0) , dn = n , Cn --\ where L:n n and An diverges sufficiently fastly so that the sets L:n are mutually disjoint. By (17), we ==
==
infer
p
n
=
lB4 1Vul 3 dx ::; C nL:2:4 Cn ldn l i = C nL:2:4 n412
<
00,
cp of u, we have 3 dx 2 L: IVcpl 3 dx 2 C Cn l dn l = C � = 00, JB4 IVcp l n J r. n2: 4 n2:4 n n 2: 4
whereas for any lifting
r
r
L:
L:
142
FABRICE BETHUEL AND DAVID CHIRON
which establishes the result.
Proof of case (b). Recall that if p then coo(BN, §2 ) is dense in W1 ,P (BN , §2 ) (see, e.g. [26]) . Therefore, we will work first on smooth maps. Let v E CDC (BN , §2) . Since BN is contractible, v has a smooth lifting 9 C oo (BN, §3 ) . We claim that we may choose some smooth lifting 9 such that v = II 0 9 and
> N,
E
(18) In order to prove (18), the starting point are the inequalities (9), so that we have merely to choose 9 such that
(19) For that purpose, we choose the Coulomb gauge. Invoking (16), we deduce indeed by standard elliptic estimates that since � = � > � then
>1
1 11> l l w2 .!f < Cpl l u * (w) l l d < Cp l lV'u l l ip · Therefore, by (15) and Sobolev embedding, we have 1> E W1,p and (19) follows, that is 2 ' I I A l l b = l i d 1> I ILP < Cpl l V'uI ILP -
*
To complete thc proof, let 1L E W 1 ,P(BN, §2), and let Vn E coo (BN , §2 ) be such that Vn -+ 1L in W 1 'P (BN, §2 ) . Let 9n E COO(BN, §3) be the lift constructed above for v = vn . It follows from ( 18) that 9n is bounded in W1'P(BN, §3) . Therefore, up to a subsequence, we may assume gn ......l. g in W1'P(BN , §3) and check that 1L = 110g. D This finishes the proof of (a ) and thus of Theorem 4.
1 1 . The previous argument does not work for N = p = 2 since then u* (w) is only bounded in L l , and the elliptic theory does not allow to conclude. Actually, it turns out that inequality (18) does not hold for N = p = 2. More precisely, let P : rc -+ §2 \ { -ad be the stereographic projection and let, for A > 0, U\(z) = P(AZ) . Then, one has fB 2 1 V'u \ 12 < 1 §2 1 but REMARK
( inf \ -++00 lim
As a matter of fact, u\ A --> +00.
......l. - 0' 1
lV'g l 2 dx, 11 0 g = u\
)
in Hl (B2, §2) and u* (w)
=
+00.
......l.
1 §2 1 bo as measure as
However, this does not exclude the fact that ( 18) might hold for p = small energies, and this raises the open question
N
=
2 for
(OQ 6) Does there exists 6 > 0 and b > 0 such that if u E HI (B2, §2) verifies fIJ2 1V'u12 dx < 6, then there exists g E H1 (B2, §3) such that 1L = II 0 g
and
{ lV'g l 2 dx < b ? JB2
As in [5] , the case s i= 1 is also of interest . However, only the case (c) of Theorem 4 extends easily for 2 < sp < 3 in view of Remark 10.
1 43
SOME QUESTIONS
Appendix A A.I. Proof of Theorem 2. Proof of i). We use the same counter-example as in Lemma 1 , and check that the argument can be transposed. Indeed, for these s's and p ' s, s < 2 and 1 ::; p < 00, u E W 1 ,p and 8u (x) = 8Xl
�r I" (( l , O)
_
X1 (X�' X 2 ) , r
)
where r == (xi + x�) 1 /2 and 1" denotes tangential derivative for I' ( and a similar au au expression holds for aX2 ) , thus aX l E ws- 1 ,p for sp < 2. As we have seen during 0 the proof of Lemma 1 , u can not have a lifting in W 1 , 1 (0, f). For the proof of ii), we will use as in [5] the Gagliardo-Nirenberg inequality. We give here a general version (see e.g. [6] , Corollary 2) .
LEMMA A.L (Gagliardo-Nirenberg inequality) Let 0 < So, Sl < 00, , PO P1 ::; 00 and ° be a smooth bounded domain in JRN . For 0 < () < 1 , define 1 ())so and - == () + 1 - () So == ()S l + ( 1 Po Po P1 l l o o (0, JR) , then u E WS9 ,P9 (0, JR) and ,P (0, JR) n WS If u E ws ,p -
-
1 <
--
lul wS9 ,P 9 ::; Ciultvsl , P l luliv!o , po ,
where lul ws,p stands for the standard semi-norm in ws,p . In particular, if 0 < s < 00, 1 < p < 00 and u E WS,P (O, JR) n LOO (O, JR), then for every 0 < a < s, u E WC1, � (0, JR) and lul w<7 , "! ::; C (O, S ,p, a) lults,p I lull�-:a� . We will also use product estimates. Here, we will need a precise version, that we can also find in [6] (Lemma 6) . For s or ()s non integer, this result is not a consequence of the Gagliardo-Nirenberg inequality, and requires a "microscopic" improvement in the scale of Triebel-Lizorkin spaces. LEMMA A.2.
be such that
Let 0 < s < 00,
1 < p < 00, 1 < 1
()
-r + t
r < 00,
1 < t < 00
and 0 < () < 1
1
= -.
p
For every f E ws,t (O, JR) n L oo and 9 E WOs,p n LT , we have fg E WOS,P (O, JR) . In particular, if 1 < s < 00 and 1 < p < 00, then for any f, 9 E WS,P n L oo , f Dg E ws- 1 ,p. -
An easy consequence of the Gagliardo-Nirenberg inequality is the following LEMMA A.3. Let m E N* , 1 < p < (Xl and ° be a smooth bounded domain in If f E Cm (JR, JR) is such that f, 1', . . . , f (m) are uniformly bounded, then for every u E wm,p(O, JR) n w 1 ,mp (0, JR) , we have feu) E Wm,P(O, JR) .
JRN .
Proof. The proof of Lemma A.3 is easy, and relies on the combination of the Gagliardo-Nirenberg inequality and the Faa-di Bruno formula. For instance, since f' (u), f" (u) E Loo , 82 8d(u)
=
( 82 81 u)l' (u) + (82 u)(81 u)J"(u)
E LP
144
FABRICE BETHUEL AND DAVID CHIRON
in view of 02 OJ U E LP and 811J" Eh'u E L'/.P by hypothesis. For a third order derivative, 3) u) u) fJs02 od (u) = (i:h h8 l 1 !' (1/,) + (03U) (02U) (ot !( (u)+
((0201 U) (03U) + (83 82u) (lJ1u) + (ihEhu)(02U) ) !" (u) E LP
since ! '(u) , !,, (u) , f(:;) (u) E Loo , o30281 u E LP (first term) since U E W:{,p , Biu E L:Jp for i = 1 , 2, 3 (second term) since 'U E �V1,3p and for the third term, we use 83 u E L3p (since U E W1 .3p) and by Lemma A.I, u E W1 ,3p n W3,p C w2, "f , thus o201U E L ¥- and (lJ281 'u.) (03U) E LP by Holder. The general case follows by induction. The composition in fractional Sobolev spaces is much more delicate (see [6) and [22) ) . 0
Proof of ii) . Let. 'U E W8,P(l1 , N) . By the Gagliardo-Nirenberg inequality (cf. Lemma A.I), since N is compact, U E Loo and thus U E W1,SP(l1 , N) , Since sp > 2, then by Theorem 1 , there exists
' . •
We now consider an orthonormal frame (e1 (0 , . . , el (�) ) of Tt;f depending smoothly on � E f (£ is the dimension of N and f). Defining Oi(U) - [D\p1I') (ei (
(A. I )
V'P = L (VU, lli (u))ei('P), f
i=1
where (., .) stands for the scalar product in N, that is in lRq . The higher regularit.y of 'P will be a consequence of (A. 1 ) . 'vVe first consider the case oS integer. 'vVe prove the result by induction on the regularity exponent s , More precisely, the induction assumption at stage s is (Is ) For every 1 < p < 00 l>Ucil that sp > 2 and every u E WS,P(l1,N) and 'P E W),SP (rt, f) such that u = 11' 0 'P, then :p E W',P (rt, f).
In the Cal:i€ S = 1 , there is nothing to prove. Let us assume next that. (Is) holds for a given s E N" , and let 1 < P < 00 verifying (s + l)p > 2 , u E ws+1,P(rt, N) and 'P E W 1 ,(sH)P(rt, £) be such that lL = 11' 0 'P. By the induction assumption, we have 'P E W',P(rt, f) . We next claim that (A , 2)
'P E ws+l,P (rt, f),
Proof of (A.2) . We will repeatedly use the fact that N is compact, hence u E Loo . We have u E wsH,p n Loo , it follows from the Gagliardo-Nirenberg inequality that U E ws,p(1+ ! ) . Applying the induction hypothesis (Is) with lip" = p(I + ! ) E ( 1 , 00) verifying sp(l + !) = sp + s > sp > 2 , we obtain 'P E w s ,p( 1+ ; ) . Moreover, we have 'P E W 1 , ( s+ 1 ) p hy hypothesis, hence 'P E W·,p(l+ ; ) n WI ( H)p . ,
.•
1 45
SOME QUESTIONS
i
This implies, by Lemma A.3, that we have, for any 1 S; S; e, ei (rp) E ws,p(l+ � ) ( fl, lRq ) , since ei is a smooth map on E, with uniformly bounded derivatives of any order since N is compact. Using the Gagliardo-Nirenberg inequality (ei(rp) E Loo ) , one . s+1 infers that for any 0 S; j S; s, j E N, ei ( rp) E W S -J ,P s-j ( fl, lRq ) , so that in particular .
s+1
DS -J (ei(rp)) E £p s-j ,
with p��� = 00 (recall l ei l = 1 on E) if j = s. Next, since U E ws+ 1 ,p n L oo C ws+ 1 ,p n W 1 ,(s+l )p by Lemma A. 1 , we deduce lli (U) E ws+ 1 ,P ( fl, lRq ) by Lemma A.3. Applying Lemma A.2 with lli (U), U E ws+ 1 ,p n Loo , we infer 9i == (\7u, lli (U»)) E ws,p. We also have 9i E L(s+l )p since lli (U) E Loo and \7u E L(s+ l )p . Hence 9i E ws,p n L(s+l)p, from which we infer by Lemma A. 1 that, . s+1 for any 0 S; j S; s, j E N, 9i E WJ ,P HI , that is (A.3)
.
s+1
DJ (9i) E £P H I . For a = ( a 1 , " " a N ) E N N , we denote l a l a 1 + . . . + a N E N and al a i aa = - axr1 ax� 2 . . . ax(lr We' differentiate ( A. 1 ) s times: for any multi-index = ( a 1 , " " aN ) E N N such that l a l = s, the Leibniz formula yields (A.4)
==
-::--::-:--=--c:----=-=_
a
aa \7 rp =
L L 2= ENN , S R
a� (9i)aa - � (ei(rp)) , � �� � ( ( ) �� ) ( ) � a
... 1� where f3 S; a means f31 S; aI , f32 S; a 2 , . . . , f3N S; a N · For 0 S; 1f31 = j S; s, j E N, . s+1 s+l . we have by (A.3) and (A.4) , DS-J (ei(rp)) E LP s-j and DJ (9i ) E £P HI , with
s-j j+1 p( s + 1) + p( s + 1 )
1 p'
thus a�(9i)aa-�(ei (rp) E £P by Holder and then Ds+ 1 rp E £P, that is rp E ws+ 1 ,P ( fl, E) as claimed. This finishes the proof by induction for integer s. We turn now to the case s rf N. Let m = [s] E N* , so that m < s < m + 1. First, as for the case s E N, U E ws,p n UXJ c W 1 ,sp implies lli (U) E ws,p n Loo . Since U E ws,p n L oo , Lemma A.2 yields gi E ws- 1 ,p. Moreover, 9i E LSP since lli (U) E Loo and \7u E Up. Hence,
9i E w s - 1 ,p n Up. Furthermore, by Lemma A . 1 , U E ws,p n Loo , thus ( m < s) U E wm, � n L oo . The result being already established for s integer, we infer rp E wm, � . Moreover, rp E W 1 ,sp by construction, thus rp E W 1 ,sp n wm, � and Lemma A.3 implies (A.6) fi ei (rp) E wm, � n L oo . From (A.5) and (A.6), we are now in position to apply Lemma A.2 with "(f, g, s, t, r, p, O)" = (A.5)
==
(fi, gi , m, � , Sp,p, s�l ), so that "Os" = s - 1 and the relation 1 0 1 s - l = 1 -r + -t = sp + -m;: p is satisfied, which yields fi 9i E ws- 1 ,p for any 1 S; i S; e. \7 rp E ws- 1 ,p, that is rp
Therefore, (A. 1 ) gives 0
E WS,P, and the proof is finished for non-integer s.
146
FABRICE BETHUEL AND DAVID CHIRON
A.2. Proof of Theorem 3. Proof of i). The case
1 < sp < 2 < N follows
once more from the map introduced in Lemma 1 . Indeed, we have U E W 1 ,p n LOO for any 1 :::; p < so that by the Gagliardo-Nirenberg inequality, u E ws ,p for every 0 < s < 1 and 1 < p < 00 verifying sp < As we have already seen by a slicing argument, U can not have a lifting in W1 /q,q for 1 < q < 00, for otherwise, I would have a lifting in VMO (St , £). 0
2,
2.
Sinee N is a compact manifold without boundary, there exists firstly a constant C > 0 such that for any u, v E N,
Iu - vi
>
:::;
d./lf(u, v) < Clu
-
i
v ,
and secondly a constant Ii 0 such that the nearest point projection p : No -+ N from the Ii-neighborhood No of N in ]Rq onto N is well-defined and smooth. Finally, since N is compact and 11' is a covering, there exists another constant T) > 0 having the following property: if B is a geodesic ball in N of radius < T), then 1T-1 (B) is a union of disjoint geodesic balls in c diffeomorphic and isometric (by 11' ) to B. In particular, inf{ds (cp, 'Ij! ) , IT ( cp) = 7r('Ij!) , 1/J =j:. cp}
(A.7)
Proof of ii) : the case
8p
> 2T) > O.
< 1 . We proceed essentially as in [5) (Appendix A).
The construction is based on a approximation of maps in WS'P, sp < 1 by maps which are piecewise constant on cubes. First, we extend u by an arbitrary constant a E N outside 0 and still have a map in W·,p, which allows to reduce to the case o = (0, l)N. We define Pj the dyadic partition of 0 into 2j N cubes of size 2-j , and let Xj be the set of maps in L l ( O , lRq ) which are constant on each cube of Pj ' For u E L1 (0, N) and j E N, we set
Ej (u)(x)
1
=
u(y) dy,
I Qj (x ) 1 Qj (x) where Qj (x) E Pj is the cube defined by x E Qj (x). Fix a E N and let, for j > 0, if Ej (u) E No, Ej (u)) p ( UJ (x) otherwise. a We have Uj E Xj and Uj -+ u a.e. as j -+ +00, since Ej (u) To define cp, we will make use of the following claim.
Claim 1 . There exists C = C (c, N) exists cp E E satisfying 7r (cp) = u and
->
u a.e. as j
->
+00.
> 0 such that for any 1/J E E and u E N, there
ds (cp, 1/J) < C l u - 7r('Ij!) I .
Proof of claim 1 . Since N is compact, there exists K c E compact such that 7r(K.) = N (cover N by finitely many closed balls trivializing the covering 7r ) . Let be given u and 'Ij!. There exists h E 7rl (N) such that14 h · 'Ij! E K. If lu - 1T(1/J) 1 > 1], then let cp' E K. be such that 7r(cp') = u, and define cp h- 1 . cp' E c. Then _
IT(cp) = 1l'(cp' )
= U,
and since h acts isometrically and h . 1/J, cp' belong to K.,
de (cp, 1/J) = ds (h - 1 cp', 1/J) = ds (cp' h 1/J) < diam(K.) :::; CT) :::; Cl u .
l1h
.
1jJ denotes the action of h
E
1l']
(.�
,
on
.
1jJ
E E.
..
-
1l'(1/J ) I
147
SOME QUESTIONS
provided C 2': diam(K)1] - I . If Iu - 1f ( 1,b) I < 1], then 1f is an isometry from Be: ('P, 1]) onto BN(U, 1]), of inverse a , and it suffices to take 'P = a(u), so that
and the proof of the claim is finished. Let 'Po E E be such that 1f( 'Po) = Uo. We then define 'Pj E Xj by induction. Assume 'Pj E Xj is constructed, and consider Qj E Pj and Qj + ! E Pj + l , Qj + ! C Qj . Claim 1 applied with 1,b = 'Pj ( Qj) (so that 1f(1,b) = 1f ('Pj ( Qj)) = Uj ( Qj)) and u = Uj + l ( Qj + d gives us a 'P E E, and we set 'Pj + ! ( Qj + l) = 'P. Then, we have 'Pj + l E Xj + ! and (A.8) Claim 2. There exists ( O, I ) N ,
C = C(E , N)
> 0 such that for any j E N* and a.e.
m
IUj - Uj - 1 1 :::; C(lu - Ej (u) 1 + lu - Ej- 1 (u)l). Proof of claim 2. If Ej (u) � No (or Ej_ 1 (u) � No) , then l u - Ej (u) 1 2': 8 (or lu - Ej_ 1 (u) 1 2': 8) , hence IUj - Uj - 1 1 :::; diam(N) :::; Clu - Ej (u) 1 (or :::; Clu Ej - 1 (u) 1) provided C 2': diam(N)8 - 1 . If Ej(u), Ej_ 1 (u) E No, then we have, since
p
is lipschitzian,
so the proof of the claim is complete. As a consequence, one has by (A.8), claim 2 and taking the 00
LP norm 1 5
00
j L 2 SP dfp ( [l , e:) ('Pj , 'Pj - l) :::; C L 2spj I IEj (u) - ull fp · j= 1 j =O From Theorem A . I in Appendix A of [5] , we have 00
L 2spj I I Ej (u) - u W �,p :::; Clul fvs,p , j =O thus I:� 1 2Spj df p ([l e: ) ('Pj , 'Pj - d converges and we infer that ('Pj) is Cauchy , LP(O, E) , so that there exists 'P E LP(O, £) such that 'Pj 'P in LP(O, £) j +00. As in [5] , we now prove that 'P E WS ,P(O, E) and (A.9)
----+
in as
----+
(A. IO)
l (n lPW5,p r
=
1 1[lx[l
de: ('P (x), 'P(Y))P dxdy < C � 2Spj l E - (u) - uI P < CluiP . l I LP Ws,p - L I x Y I sp+ N j =1 _
J
We propose a direct proof of (A.IO), based on the following claim.
15dLP (f!,e) denotes the natural distance in LP(O, £) defined by dLP(f!,e) (ip, 'IjJ)P If! de (ip(x), 'IjJ(x))P dx.
_
148
FABRICE BETHUEL AND DAVID CHIRON
Claim 3. There exists GI, G2 E Pj ,
C=
C ( s, p, N) such that for every j E N and every cubes
(A . 1 1 )
Assuming claim 3 for a moment, we complete the proof of ii) when sp < 1 . Summing (A. Il) over all cubes C] , C2 E Pj gives, using t:laim Z for the last inequality,
-- 1 £
de (CPj + I (X ) , CPj +l (Y» P d d I CPJ. +l IPW··P X Y I I ."L - Y N+.p . . fl x ll de ( cpj (x) , cPj ( Y» P dxdy + 2C2spj r I U x) - U x) IP dx < r ( .( + + N sp J In J l Jflxn Ix - yl < l'Pj lfv,.p + G'Zspj I l lu - Ej(u) IP + I u - Ej +I(U) IP dx,
J
i
where the com;tant C' depends only on N, p and infer for k E N ,
k+ l ICPk l�v" p < G L 2Spj I IEj (u) - uW;',p j =O
Since
<
s.
By induction, m;ing (A.9) , we
00
C L zSpj I IEj (u) - ul l t p j=O
< G lulf,.,·" p ·
CPk ---> cP in LP (O, E) k +00, we then deduce from Fatou's lemma that inf I CP k ltv,.p < Glulf-v '.v , 1'Plfv',p < lim k-+oo as
-+
which concludes the proof in the case
claim
Proof of 3. �otice that CPk E Q] , Q 2 elements of PH I , we have
sp
< 1 . \Ve turn now to the proof of claim 3.
Xk for every k E N.
Let
CI, G2 E Pj '
Denoting
dxd y r (A.IZ) d ) e(CPj+I( Q 1), 'Pj +l (Q 2) P L y s x lN+ Q I 1 p J XQ2 Q , CCI ,Q2 CC2 , Q , # Q 2 The supremum of all these last integrals for Q I t= Q2 E Pi+l is achieved when Q1 and Q� have a face in common, thus it suffices to estimate this integraJ when Ql = (0, 2-j- 1 ) N - l (0, 2 -j - 1 ) and Q 2 = (0, 2- j -1)N - 1 ( 2 -j-\ 2-j ) , for which
we
infer by scaling
J
Q I X Q2
( A 13) .
X
X
dxdy I x - y lN+ sp
dxd y rJ (O,1)N X « O,I) -I X(I, 2» Ix - yl N+sp j ) p N < < ( ), -s CTj(N-sp G2 I I O": :: X :C N< P w 1 " p «O , 1 ) N - I X (0 ,2» -
= C2 - 2Nj 2j ( N+sp)
J
...
149
SOME QUESTIONS
where C is a constant depending on N, s and p, XO :S: xN :S: 1 stands for the char acteristic function. The last inequality follows from the fact that XO <- xN <- l E WS ,P ((O, I )N - 1 x (0, 2)) (for sp < 1 ) . Moreover, by (A.8),
de (
J
Since the cubes
J L Q1 CC1 ,Q2 CC2 ,Q1 #Q2
Q l , Q2 have a volume 2 -(j + 1) N, the last sum is j r � C2 N IUj + 1 (X) - Uj (x) IP dx, ic1uC2
and this concludes the proof of claim 3.
D
Proof of ii) : the case sp � N. If sp > N, then maps in ws ,p are Holder continuous, and have therefore a continuous lifting. In the limiting case sp = N, maps in ws,p need not be continuous but belong to the space VMO. We follow the construction in [5] (Section 2 ) of liftings, and notice first that a map U in WS ,p(n, N) has an extension
Ul E ws + i ,p(n x (0, t d , N) for some tl > ° depending on Obviously, U 1 has no reason in general to take values into N. However, in the limiting case ws ,p c VMO, the usual extension u u.
of u given by
u(x, t)
==
1 )1 I Bt (X
r u(y) dy iB, (x)
belongs to w s+ i ,p(n x (0, t d , lR.q ) and has range "close" to N, say in Nfl for t 1 small enough, thus p(u) E ws + i ,p(n x (0, t 1 ) , lR.q) and has values into N and extend u. For k E N* such that s + � < 1 , we iterate this construction to define a map
Uk+ 1 E W s+ � ,p(n x (0, h ) x (0, t2 )
x . . . x (0, t k+ 1 ) , N) extending U k . If n is the first integer for which s + � :::: 1, we have then constructed a map
E N*
Un E Ws+ 'g' ,p (n x (0, h ) x (0, t 2 ) x . . . x (0, tn ),N). By Theorem 1 (if s + � = 1) or Theorem 2 (if + � > 1 ) , Un has a lifting
x
x
1 50
FABRICE BETHUEL AND DAVID CHIRON
(0, tn-2), and finally define cP = CPo to be the trace of CP 1 on O. It is clear that cP E W',P(O, £) is a lifting of u. In the case £ is not compact, we use the geodesic distance to define W',P(O, £) instead of the euclidean distance: from [8] , Proposition 6, we know that a map in WC7,Q(W, £) (0 < (7 < 1 , (7q > 1) has a trace 1 in W"- q ,Q(8w,£). For the first trace, we may have 17 = s + ; 1 , and we use then Remark 14 in [8]. 0 . . . X
•
•
•
>
A.3. Uniqueness of the lifting for
sp
> 1. The question of the uniqueness
of the lifting is treated in the following Lemma. We can obviously not have such a result for sp < 1.
< 00 and 1 < p, q < 00 be such that sp > 1 and (7q > 1 , and let 0 be a smooth bounded connected domain in RN . If cP E W',P(O, £) and ¢ E w",q(O, £) satisfy 11" 0 cP = 11" 0 ¢ , then cP and ¢ differ from the action of an element of 7f1 (N) , i. e. there exists h E 7f1 (N) such that ¢ = h . cP in O. In particular, cp, ¢ E w s ,p n w",q(n, £) . LEMMA
A.4. Le t ° <
S, 17
In order to prove Lemma A.4, it suffices, by standard arguments, to treat the case where 0 is the cube Q = (O, I ) N (cp and ¢ are extended by reflection on ( - 1 , 2) N). We argue by induction on N, and assume first N = 1 . Since 7f 0 cp = 7f 0 ¢, we deduce that for almost every x E (0, 1), there exists hx E 7f1 (N) such that ¢ (x) = hx . cp (x) . Moreover, 7f1 (N) is at most countable, hence one may choose some h E 11"1 (N) such that {x E (0, 1 ) , hx = h} has positive measure. Next, we set PROOF.
x ...... f (x) = dd¢ (x) , h ·
If(y)
-
f(z) 1 < I de(1j; (y), h · cp(z)) - de(1j;(z), h ·
x 1 l +, -
since
2.0
deC1j;(y), 1j; (z») + dc; (h ·
=
dc;(i/J(y) , 1j; (z)) + dc; (
+ j+
acts isometrically on E. Therefore, for every x E (0, 1 ) , x x 1 x +< 1 f (y) - J( z) dz dy < € 2 IJ(y) - f (z) l dydz < I+II,
h
X-e
where I
3
1
(2£) 2
2€
+ ["'+< 1"' <
}x - <
x-£
,
4
X-g
d£ (V' (y), i/J(z) dydz
x-<
C
and
-< x
e
c
lJ -
(2£1 ) 2
"+' , x 1 1+ x-<
"' _ <
de(
° (and II 0 symmetrically) as € We claim that I ° uniformly for x E (0, 1), which will prove that f E VMO « O, I» . If s = p = 1 or sp > 1, then 'P E CO([O, 1] , E), thus 1 0 as c -> ° uniformly for x E (0, 1 ) . Therefore, we are left with the case ° < s < 1 sp < p, for which we are in a VMO type1 6 embedding w s ,p c VMO. In I, we bave ly - z ls+ ! « 2c) � since sp = l . With -->
-->
->
->
=
s+
F (y , z) = dd¢ (y) , ¢ (z)) E P « O " 1 ) 2 ) I Y Z I 1.I> -
16However, the definition of VMO((O, 1 ) , £) should be precised, since [; is not endowed with the euclidean distance.
• ·
• •
• • ·
,
SOME QUESTIONS
we then infer by Holder inequality (with p' = OS
(2€) �
IS ( 2€) 2
s
l l x+c
X-c
x+c
X-c
(2€)2( � -1) (2€) :r
151
� E (1, 00) such that � + ? = 1)
F( y , z) dydz
(1�:c 1�:€ FP(y, z) dYdZ) � = (1�:€ 1�:€ FP (y, z) dYdZ) �
as € � ° uniformly for x E (0, 1) since FP E L 1 . Therefore, f E VMO((O, 1), JR) and it follows from [7] (Section 1.5) that the essential range f((O, 1)) of f is connected. Since 7l" 0
F. BETHUEL, The approximation problem for Soboley maps between two manifolds. Acta Math. 167, no. 3-4 (J991), 1 53-206. F. BETHUEL AND F. DEMENGEL, Extensions for Soboley mappings between manifolds. Calc. Var. Partial Differential Equations 3, no. 4 (1995), 4 75-491. F. BETHUEL AND X . M. ZHENG, Density of smooth functions between two manifolds in Soboley spaces. J. Funct. Anal. 80, no. 1 (1988), 60-75. H. BREZIS, J-M. CORON AND E. H . LIEB, Harmonic maps with defects. Comm. Math. Phys. 107 (1 986), 649-705. J . BOURGAIN, H . BREZIS AND P . MIRONESCU, Lifting in Soboley spaces. J. Anal. Math. 80 (2000), 37-86. H. BREZIS AND P. MIRONESCU, Gagliardo-Nirenberg, composition and products in fractional Soboley spaces. J. Evol. Equ. I , no. 4 (2001), 387-404. H. BREZIS AND L. NIRENBERG, Degree theory and BMO I. Compact manifolds without bound aries. Selecta Math. (N.S.) I, no. 2 (J995), 197-263. D. CHIRON, On the definitions of Soboley and BV spaces into singular spaces and the trace problem. To appear in Commun. Contemp. Math. . D . CHIRON, Regularity o f minimizers o f a Ginzburg-Landau type energy with metric cone target space. Preprint. J. DAVILA AND R. IGNAT, Lifting of BV functions with values in §1 . C. R. Math. Acad. Sci. Paris 337, no. 3 (2003), 159-164. B.A. DUBROVIN, A.T. FOMENKO AND S . P . NOVIKOV, Modern geometry - methods and ap plications. Part II. The geometry and topology of manifolds. Translated from the Russian by Robert G. Burns. Graduate Texts in Mathematics, 104. Springer-Verlag, New York, 1985. P-G. DE GENNES AND J. PROST, The physics of liquid crystals ( second edition) . Oxford Science Publications, Oxford (J 993). R. HARDT, D. KINDERLEHRER AND F-H. LIN, Stable defects of minimizers of constrained variational principles. Ann. Inst. Henri Poincare, Anal. Non Lineaire 5, no. 4 (J988), 297322 R. HARDT AND F-H. LIN, Harmonic maps into round cones and singularities of nematic liquid crystals. Math. Z. 213, no. 4 (1993), 575-593. R. HARDT AND F-H. LIN, Mappings minimizing the LP norm of the gradient. Comm. Pure Appl. Math. 40 (J 987), 555-588. T.IsOBE, Obstructions to the extension problem of Soboley mappings. Topol. Methods Non linear Anal. 21, no. 2 (2003) 345-368. J. JOST, Equilibrium maps between metric spaces. Calc. Var. Partial Differential Equations 2, no. 2 (J994), 1 73-204.
� O
1 52
FABRICE BETHUEL AND DAVID CHIRON
[18J M . KLEMAN, Points, lines and walls. In liquid crystals, magnetic systems and various orcleroo media.
Wiley-Tnter.. r.ience P.ublication. John Wiley
f1
Sons, Inc., New York, 1 983.
t argets. Cummun. Anal. Geom. 1 , no. 4. (1993), 561-659. F.R. L IN On nem at ic liquid crystals with variable degree of orientation. Comm. Pure Appl. Math. 44, no. 4 (1991), 453-468. C . R. LUCKHURST, A missing phase found at last ? in Nature, vol. 4 30, july 2004. V. MAZ 'YA AND T. SHAPOSH>lIKOVA, An element ary proof of the Brezis and Mironescu theo rem on the composition operator in fractional Sobolcv spaces. J. Evol. Equ. 2, no. 1, (2002),
[19J N. J. KOREVAAR AN D R. M. SCHOEN, Sobolev spaces and h armonic maps for metric space [20J [21J [22J
,
113-1 25. [23J N . V . MERMIN, The topological theory of defects in ordered media.
Rev. Mod. Phys. 5 1
(1 979), 591-648.
[24J P . OSWALlJ AND P . PIERANSKI, Les Cristaux Liquides . Con cepts et proprietes physiques
( French) Gordon and Breach Sr.ienr.e Publishers, (2000).
ilIustn;es par des experiences. Tome I
Smectiques et phases colonnaires
:
Mesophases avec .
un
ordre orientationnel. Tome II :
[25J T . RrVIF:R.B, M i ni mizing fibrations and p-harmonic maps in homotopy classes from
§3 into
§2 . Comm. Anal. Geom. 6, no. 3 ( 998), 427-483. [26J R. SCHOEN AND K . UHLENBECK, A regularity theory for harmonic maps. J. Differential Geom. 17 (1982), 307-335. Correction Tbid. 18, no. 2 (1988), 82.9. LADORATOIItE JACQUES-LOUIS LIONS, UNIVERSITE PIERRE ET MARIE CURIE PARIS VI, 4, PLACE JUSSIEU Be 187, 75252 PARIS, FRANCE
E-mail address:
bethuell§ann . j us s i eu . f r
LABORATOIRE J-A. DIEt:DONNE, UNIVERSITE DE NICE
E-mail
-
SOPHIA ANTIPOLIS
address: chironlllmath . uni c e . f r
-
Contemporary Mathematics Volume 446, 2007
Normal forms and the nonlinear Schrodinger equation
J. Bourgain
Dedicated to H. Brezis
1. Introduction We consider nonlinear Schrodinger equations (NLS ) with periodic boundary conditions. Our main focus is the behavior of classical solutions. Consider for instance in 1D the defocusing quintic NLS
(1.1) or in 2D the defocusing cubic NLS
(1.2) It is well-known that classical solutions exist for all time. More precisely, let u(O) E H S (1f) for ( 1 . 1) (resp. u(O) E H S (1f2 ) for (1.2)) with s ;::: 1 . The corresponding Cauchy problem is globally well-posed and u(t) E H S for all time (see for instance [1] for results on the NLS Cauchy problem with periodic boundary conditions. The case of equation ( 1.2) is due to H. Brezis, and T. Gallouet ( [9] ) . Conservation of the Hamiltonian provides an apriori bound on the H 1 _ norm, thus sup Ilu(t) II Hl < C. t
A first main issue is the possible growth of higher Sobolev norms of smooth solutions, which may be referred to as 'Hamiltonian' or 'weak turbulence'. PROBLEM 1.1. Assume u a solution of smooth. Is it possible that for some s > 1
( 1.3) Iiso, how fast may
( 1 . 1) or (1 .2) and
u(O),
hence
u(t)
Ilu(t) II Hs = 00 7 tlim ->oo Ilu(t) II H s
grow when t
--t
00 7
2000 Mathematics Subject Classification. Primary: 35B65, Secondary: 35B15. ©2007 American Mathematical Society
153
J. BOURGAIN
154
<
<
As pointed out earlier, SUP l t l
t
( 1 .4)
for equations ( 1 . 1 ) , ( 1 .2) and more generally any defocusing NLS in D � 3 which are subcritical with respect to the Hi-norm (and with sufficiently smooth nonlinearity) . Notice t.hat the Rubcriticality only imposes a condition for D > 3. In particular in 3D, the equation ( 1 .5 ) is sub critical for p < 6. The main ingredient in the proof of ( 1 .4) are refined versions of Strichartz' theory for the linear Schrodinger group eitf', (which are of course also essential to the Cauchy problem) . Unlike the Zakharov-Shabat ID cubic NLS
( 1.6) equations ( 1 . 1 ) , ( 1 .2) are non-integrable and it is generally believed that weak tur bulence may occur (one may in fact come up with examples of NLS with smooth nonlinearity that do exhibit the phenomenon, but no example with a local polyno mial nonlinearity seems known) . An issue closely related to weak turbulence is the question of abundance of time quasi-periodic and almost-periodic solutions in phase space. Again, since ( 1 . 1 ) , (1.2) are non-integrable, we certainly do not expect every classical solution of (1.1) or ( 1 .2) to be almost periodic. Recall that if u(t, x ) is a solution and u : lR � H8 (']fd ) is almost periodic as a function of time, then surely SUPt I lu(t) II H' < 00. Over the last decade, lots of work has been done on the construction of quasi-periodic solutions for NLS both in I D and D > 2. A quasi-periodic solution corresponds to a finite-dimensional invariant torus in phase space (its dimension is the number of frequencies) . Two methods have been used to study the problem (involving an infinite dimensional phase space). The first is an adaptation of the classical KAM theory to the PDE-setting (see in particular [10]). The second is a direct applica tion of a Nash-Moser type scheme with a control on the linearized (non-diagonal) equation through more sophisticated small divisors analysis (which is reminiscent to the theory of localization for quasi-periodic lattice Schrodinger equations, cf. [2] ) . This second method applies also beyond the Hamiltonian context and has the appearance of more general implicit function theorems involving small divisors. Among the several expository and reference works, we mention [10], [11] , [8],
[2] , [13] .
Much less has been done an almost periodic solutions ( infinite dimensional invariant tori) for Hamiltonian PDE's (putting the integrable cases such as KdV, KPII or I D cubic NLS aside) . The papers of J. Poschel (see [15]) and the author (see [3) ) , present constructions for ID NLS and I D NLW ( nonlinear wave equation) for a full set of frequencies. Both arguments are based on the idea of consecutive perturbations of finite dimem,ionaJ tori, eventually producing an invariant torus of
NORMAL FORMS AND THE NONLINEAR SCHRODINGER EQUATION
1 55
full dimension, but with very strong compactness properties (in fact the decay of the action variables is hardly explicit ) . These constructions surely do not address in any way the issue of 'abundance', even not in the real algebraic category. In recent work [4] , the author succeeded in carrying out the KAM sheme in the context of a ID NLS
(1.7) where P(t) is an arbitrary polynomial and M is a bounded Fourier multiplier pro viding an extra source of parameters. Thus
(1. 8 )
M¢(x)
=
¢ (n)Mn e27rinX L n EZ
and the Mn are parameters. In this setting, invariant tori of full dimension are constructed, which are 'abun dant' in the real analytic or even Gevrey category. We follow the 'standard' normal forms procedure as worked out in [14J for short range interactions (notice that the nonlinearity in (1.7), expressed in Fourier modes, is not short range) but the analysis here is significantly different. It involves the fine features of the model, including the behavior of the spectral asymptotics {n 2 l n E Z} and the form of the nonlinearity. We do not know at present how to carry out a similar argument in 2D. A key issue is of an arithmetical nature and may be explained as follows. The system of equations
{ nnil -- nn�2
= a i=- 0 =b
n l , n 2 in ID (i.e. a, b, n l , n2 E Z) but not in 2D. PROBLEM 1.2. Construct almost periodic solutions for the analogue of (1.7) in
determines
2D.
The problem may be more than just technical. Of course, the other issue is to dismiss the Fourier multiplier M. This forces us to extract parameters from the nonlinearity by amplitude-frequency modulation. In lD with nonlinearity of the form iUt + Uxx + mu + u l u l 2 + ( higher order) = O. (1.9) this was achieved in [12J in the construction of finite dimensional tori. In the present problem where an infinite dimensional invariant torus is pro duced, the major difficulty is that on one hand the iterative scheme requires a suffi ciently fast decay of the action variables In = I qn 1 2 (qn = ,;;(t) ( n)) ; but if the action variables are small, so is the frequency modulation and the small divisor problems become worse. PROBLEM 1.3. Construct invariant tori of full dimension for (1.9) . The paper [5J illustrates the difficulty brought up above, where it is to some extent resolved in the more modest attempt of establishing Nekhoroshev (=long time) stability properties for small solutions of (1.9) . It is shown in [5J that if Ilu (O) II H s
< C
J. BOURGAIN
156
(s large) and u(O) is 'typical' in some sense, then in particular lI u(t) II H' < 1 for I t I < To C )A, where A A(s) .�';" 00 . =
=
In the more recent papers [6] , [7] we started exploring the application of normal forms to NLS for coarser topologies. A first task here is to understand the precise mapping properties on HS-spaces of the sympleetie transformations involved in the Birkhoff normal forms reduction of the nonlinearity to a 'resonant part' and a 'small' remainder term (recall that the symplectic space of NLS expressed in Oarboux coordinates is L2) . The description 'resonanL' refers Lo the ull-IIlodulated frequencies {n2 ln E Z} (in 10). Thus a monomial q"l qn2 . . . q"2. - 1 q"2. ' where nl n2 + . + n2s - 1 n 2. = 0 is called resonant if -
.
-
.
( 1 . 10)
(removal of such terms requires modulation of the n2-frequencies, for instance us ing a Fourier-multiplier M as in (1 .7) and leads to the 'small-divi::;Uf::;' analysis encountered in the construction of 'true' invariant tori) . We cite two POE results that came out of this analysis. Both related to equation (1.1). •
THEOREM
1 .4. �6]). Let
s > 1 and u the solution of the Cauchy problem =
i.ut + Uxx u l ul4 u(O) E W (1J'). -
Then
,-,; - 1
l I u ( t ) I I H' « t · for t
The second is THEOREM
is
a
-+
00.
low regularity result.
1 .5. ([7]).
Cauchy problem (1.11)
u (x, t) = u (x + 1 , t )
0
There is
a
Sobolev exponent 0 < So <
i,1J.t + nxx 1/,17),14 = 0 u(O) E W (1r) , 8 > 80 -
globally wellpo8ed and u(t)
E
1 2
such that the
n(:c, t) = n (x + 1 , t)
HS for all Lime.
Recall that ( 1 . 1 1 ) is locally wellposed for all 8 > 0 (see [1] ) . The claim is thus that this local solution extends to a global one for 8 > 80 (it is unknown if we may take 8 0 = 0) . The interest of the second theorem is that it provides a direct proof of the uniquely defined invariant dynamics on the support of the (formally invariant) Gibbs measure introduced in the work of Lebowitz, Rose and Speer (1.12)
where ( 1 . 1 3)
H (4))
=
. 11'
(1\74>12 + p2 1cPI2 + 1 4>16)
(see again [1] and relevant reference::;).
i ,
,
,
•
NORMAL FORMS AND THE NONLINEAR SCHRODINGER EQUATION
157
The measure ( 1 . 12) is indeed absolutely continuous with respect to the 'free' measure induced by the Gaussian process ( 1 . 14) where
g e21Cinx L Jnn(w) + p2 2 n EZ
{gn l n E Z} are independent normalized complex Gaussian random variables. References
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
J. Bourgain, Nonlinear Schrodinger equations, in Hyperbolic equations and frequency in teractions, ( Park City, UT, 1995) , 3-157, lAS/Park City Math. Ser. , 5, Amer. Math. Soc., Providence, RI, 1999. J. Bourgain, Green's function estimates for lattice Schrodinger operators and applications, Annals of Mathematics Studies, 158, Princeton University Press, 2005. J. Bourgain, Construction of approximative and almost periodic solutions of perturbed linear Schrodinger and wave equations, Geom. Funct. Ana!. 6 (1996), no 2, 201-230. J. Bourgain, On invariant tori of full dimension for lD NLS, J. of Funct. Ana!., to appear. J. Bourgain, On diffusion in high-dimensional Hamiltonian systems and PDE, J. Ana!. Math,
80 (2000), 1-35.
J. Bourgain, Remarks on stability and diffusion in high-dimensional Hamiltonian systems and partial differential equations, Ergodic Theory Dynam. Systems 24 (2004), no. 5, 1331-
1357.
J. Bourgain, A remark on normal forms and the "I-method" for periodic NLS, J. Ana!. Math.
94 (2004), 125-157.
W. Craig, Problemes de petits diviseurs dans les equations aux derivees partielles, Panoramas et Syntheses, 9, Societe Mathematique de France, Paris, 2000. H. Brezis, T. Gallouet, Nonlinear Schrodinger evolution equations, Nonlinear Ana!. 4 ( 1980) , no 4, 677-681. S. Kuksin, Nearly integrable infinite-dimensional Hamiltonian systems, Lecture Notes in Mathematics, 1556, Springer-Verlag, Berlin, 1993. S. Kuksin, Fifteen years of KAM for PDE. Geometry, topology, and mathematical physics, Amer. Math. Soc. Trans!. Ser 2, 212, Amer. Math. Soc, Providence, RI, 2004. S. Kuksin, J. Poschel, Invariant Cantor manifolds of quasi-periodic oscillations for a non linear Schrodinger equation, Ann. of Math. (2) 143 ( 1996), no 1, 149-179. T. Kappeler, J. Poschel, KAM and Kd V, Springer, 2003. J. Poschel, Small divisors with spatial structure in infinite-dimensional Hamiltonian systems, Comm. Math. Phys. 127 ( 1990) , no 2, 351-393. J. Poschel, On the construction of almost periodic solutions for a nonlinear Schrodinger equation, Ergodic Theory Dynam. Systems 22 (2002), no 5, 1537-1549. INSTITUTE FOR ADVANCED STUDY, PRINCETON, NJ E-mail address: bourgain
08540
Contemporary Mathematics Volume 446, 2007
Extremal Solutions and Instantaneous Complete Blow-up for Elliptic and Parabolic Problems Xavier Cabre
Dedicated to Haim Brezis on the occasion of his sixtieth birthday. ABSTRACT. We consider some reaction-diffusion problems that depend on a parameter ).. In the elliptic case they admit a minimal solution for each ), :-:::: ). * , where ). * is a certain extremal parameter, and admit no solution for every ). > ). * . In both the elliptic and parabolic settings, we describe recent results concerning the existence, uniqueness, regularity, and stability of solutions for the extremal problem at ).* . Two phenomena are studied in detail: an apparent "failure" of the Implicit Function Theorem at the extremal problem, and the complete blow-up (instantaneous complete blow-up in the parabolic case) of approximate solutions for )' > ). * .
1 . Introduction This article is concerned with several semilinear elliptic problems of the form g>. (x, u) in 0 (1 . 1 ) o on 80, that admit a minimal (or smallest ) solution for each A � A * , where A * is a certain extremal parameter, and that have no solution for every A > A* . The minimal solution u* corresponding to A = A* is called the extremal solution. An example of this situation is given by Gelfand's problem
{
-D.u u
Aeu 0
in 0 on 80,
and more generally by (1 . 2 )
Ag(u) o
in 0 on 80,
2000 Mathematics Subject Classification. Primary 35J60, 35K55, 35B35, 35B40, 35R25, 35D05. Key words and phrases. Nonlinear elliptic and parabolic problems, extremal solutions, regu larity of solutions, non-existence of very weak solutions, instantaneous complete blow-up. Supported by MEC Spanish grant MTM2005-07660-C02-0l , the RTN Program Front Singularities HPRN-CT-2002-00274, and by the European Science Foundation PESC Programme "Global" . ©2007 American Mathematical Society
159
XAVIER CABRE
160
whenever 9 > 0 is increasing and convex 011 ]R and superlinear at + 00 in the sense (2.2) stated helow. Under these assumptions, the extremal parameter of (1.2) satisfies 0 < ,\* < 00. Here and throughout the paper, !1 is a smooth bounded domain of ]RN . For each 0 < ,\ < >. * , the minimal solution of (1. 2) is classical. Its limit as >. r ,\* is the extremal solution u' , which may be singular (i.e., unbounded) for some dimensions, nonlinearitiet>, and domains. \Vhen g(u) = e", it is known that u · E LOO(n) if N < 9 (for every !1), while u* (x) = log(1/lxI 2 ) if ::::: 1 0 and !1 = B1 • Brezis and Vazquez [BV] raised the question of determining the regularity of u' , depending on the dimen sion N , for general nonlinearities 9 as above. Below we explain recent results of Nedev [NI , N2] and of the author and Capella [CCa2] on this delicate regularity issue. While nonexistence of elassical solutions for ,\ > ,\* follows by the definition of >" , Brc�is, Cazenave, Martel, and Ramiandrisoa [BCMR] proved that, in addition, no weak solution of (1 .2) exists for >. > >.* . L ater Brezis and Vazquez [BV] characterized singular extremal solutions of ( 1 .2) by means of a stability condition. FOT some equations in dimension N > 1 1 , they also show that an app uent "failure" of the Implicit Function Theorem occurs at ( u* , >. * ) , since the linearized problem at · u turns out to be formally invertible (for some problems) and, however, no weak solution exists for ,\ > >" . This article is a survey of several results and methods from [BC, BCMR, BV, CCa2, CMal, CMa2, Mal, NI, N2, PV] concerning the previous issues, both in the elliptic and the parabolic settings. Special attention will he made on the regularity of the extremal solution u· , as well as on how nonexistence develops for >. > >" . We will see that this last question is related to a phenomenon of complete blow-up of approximate solutions, and of instantaneous complete blow-up in the parabolic setting. Sections 2 and a are devoted to existence, nonexistence, uniqueness, stability, and boundedness of solutions of ( 1 .2). We describe some re�ults of [BCMR, BV, CCa2, CMal, Mal , NI, N2] ahout weak solutions of ( 1.2). We are particularly interested on the regularity of the extremal solution u ' and on the linearized prob lem of ( 1 .2) at the extremal solution and its connections with the Implicit Function Theorem. Section 4 is concerned with [Be], where Brezis and the author make a detailed analysis of how nonexistence develops for ,\ > ,\. , and also of the apparent "failure" of the Implicit Function Theorem, for the simple model
N
,
,
(1 .3)
-6.71,
u
U
2
Ixl 2
o
+ ,\
'
in !1 on 8!1,
in all dimensions N > 2. Here 0 E !1, u· = 0, and >. * = O. Indeed, it. is proved in [Be] that, for every >. > 0, (1 .3) has no solution in a very weak sense. Moreover, approximate solutions of ( 1 .3) blow-up everywhere in !1, i.e., there is complete blow-up in the sense established by Baras and Cohen [BaCo] for Home parabolic problems. In Section 5 we present, related nonexistence and blow-up results obtained in [Be] for the parabolic version of (1.3) . They extend, in particular, a result. of Peral
-
161
EXTREMAL SOLUTIONS AND BLOW-UP
{
and Vazquez [PV] . Namely, the problem
Ut flu u u(O) log (1/ l xj 2 )
(1.4)
-
= =
2(N - 2)
e
0
U
Uo and Uo =t u, =
in (0, T) x B1 on (0, T) x aB1 on B1 ,
with Uo 2': u := has no weak solution u(t, x) 2': u(x) even for small time: instantaneous complete blow-up occurs. To prove this result we will consider the equation satisfied by u u. It leads to the parabolic version of problem (1 . 3 ) . The potential I x l - 2 also appears when linearizing problem (1.4) at u. Hence, the study of the linear problem Ut flu = >"lxl - 2 u is also of interest. In Section 6 we discuss the classical work of Baras and Goldstein [BaG] on this equation, as well as results of the author and Martel [CMa2] on linear parabolic equations with singular potentials: -
-
{
(1.5)
Ut flu u u(O) -
a (x)u 0
= =
Uo
=
in (0, T) x D on (0, T) x aD on D,
with a 2': 0 and Uo 2': 0 in Lfoc (D) . We give conditions, based on the validity of a Hardy type inequality with weight a (x) , that ensure either the existence of global weak solutions or the instantaneous complete blow-up of approximate solutions of (1.5). Replacing a (x) by >..a (x) in (1.5), the extremal parameter >..* can be explicitly computed for some critical potentials a (x) This work extends the result proved by Baras and Goldstein [BaG] in 1984. Namely, if a (x) = >" lxl - 2 , 0 E D, and N 2': 3, then >.. * = (N - 2) 2 /4. More precisely, (1.5) has a global weak solution for every Uo E L 2 (D) and every 0 :::; >.. :::; >..* = (N - 2) 2 /4. Instead, if >.. > >.. * = (N - 2) 2 /4 then (1.5) has no positive weak solution for every T > 0 and every Uo E Lfoc(D) with Uo 2': 0 and Uo =t O. .
The paper is organized as follows: 1. Introduction. 2. The extremal solution. 3. An apparent "failure" of the Implicit Function Theorem. 4. Nonlinear elliptic problems: very weak solutions. 5. Nonlinear parabolic problems. 6. Linear parabolic problems with singular potentials. 2. The extremal solution We start considering the problem
(2.1>J
{ -fluu
= =
>..g (u) o
in D on aD,
where >.. 2': 0 is a parameter, and D is a smooth bounded domain of JRN , N 2': The nonlinearity 9 is a C 1 , increasing and convex function on [0, (0 ) such that
g(u) +00. U Since 9 > 0, we consider nonnegative solutions of (2.1>.). The cases g(u) g(u) (1 + u)P, with p > 1, are classical examples of such nonlinearities. (2.2)
g(O) > 0
=
and
lim
u-->+oo
2.
=
= e
U
and
16 2
XAVIER CABRE
These reaction-diffusion problems appear in numerous models in physics, chem
istry, and biology. The case of an exponential reaction term
g(u) = eU is a very Since the work of Gelfand [G] in the sixties,
simplified model in combustion theory.
problem (2.1,), ) has heen extensively studied.
2.1. Minimal stable solutions. It is well known that there exists a param eter A* with 0 < ..\* < 00, called the extremal parameter, such that if 0 < A < "\" then (2.1,),) has a minimal classical solution u,),. Here, by "minimal solution" we mean that u,), is smaller than every other solut.ion or supersolution of (2. 1,), ). On the other hand, if ..\ > ..\ * then (2.1,),) has no classical solution. The set { u,), ; 0 < A < A' } forms a branch of solutions increasing in A. More over, every u,), is a stable solution, in the sense that the first Dirichlet eigenvalue of the linearized problem at u,), is positive: 11'1 {-.6. - >.g' (u,),}; O} > o. In particular, for the quadratic form associated to the linearized problem we have
(2.3)
Qu,, (r.p} :=
10 1\7r.p1 2 - 10 ..\g' (u,), )r.p2 > 0
for every r.p
This condition, that we refer as the "semi-stability of
u,),
E HJ (O).
, is equivalent to /1 1
where /11 is the first eigenvalue of the linearized problem above. Recall that the first Dirichlet eigenvalue in
-.6. - a (x )
0
is defined by A /1 1 {' -u
(2.4)
-
a(
x ) ,. H "}
-
_
.
mf
O;;E
11
> 0,
of a linear operator of the form
In I \7r.p I
2
2 a(x}r.p In In r.p2 -
.
Later, we will use this expression as definition of generalized first eigenvalue when
E
L�oc (O), - 00.
a
a
> 0 a.e"
and a is singular. In this case, /11 {-.6.
- a(x}j O}
could be
{ u,), ; 0 :S ..\ < ..\*} can be proved using the Implicit FUnction Theorem (starting from ..\ = 0). The solution u,), may also be obtained by the monotone iteration procedure, with 0 < >. < >" fixed, starting from u = 0 (note that u = 0 is a strict subsolution of the problem ) . These results, among others concerning (2.1,),), are proved in [CrR] and [MP ] . The existence of the branch
2.2. Weak solutions.
The increasing limit of
tremal solution U · . It is proved in
..\ =
>. *
in the following sense:
DEFINITION U E
for all
We say that
2.1.
LI (O),
( E C2 (0)
with
to the boundary of
[BCMR]
=
0
on
ao,
as ..\ i >. * is called the ex
is a weak solution of
u*
is a weak solution of
u
g(u)8 E Ll(O), (
that
u,),
u.6.( =
and
where
(2. 1 >J
8(x)
n
= dist (x,
80)
n
(2. 1,),)
for
if
>.g(u)(
denotes the distance
0.
More generally, the definition of weak solution for problem
one: we now assume
in the above integral.
g,). (x, u(x))8(x)
E
Ll (O)
The nonexistence of classical solution for
Cazenave, Martel, and Ramiandrisoa
and we
A > A* [BCMR] :
( 1 . 1 ) is the analogous replace >.g(u)( by g,),(az, u)(
has been improved by Brezis,
EXTREMAL SOLUTIONS AND BLOW-UP
THEOREM 2.2 ( [BCMR] ) .
For every A > A* ,
(2.1>.)
1 63
admits no weak solution.
In case g(u) = eU, a similar result had been obtained by Gallouet, Mignot, and Puel [GMP] . The proof of Theorem 2.2 given in [BCMR] uses a new and interesting method. One assumes that (2.1>.) has a weak solution for some A > A*. One then considers the equation satisfied by 1>( u), where 1> is a positive, bounded, and concave function which is chosen appropriately depending on the nonlinearity g. It is possible to construct in this way a bounded supersolution (and hence a classical solution) of (6( 1 - 0)>')' for each 0 < E < 1 . But this is a contradiction if E is small enough to guarantee ( 1 - E)A > A * . This procedure could be called the "method of generalized truncations" 1>(u) of u. It is also used in [BCMR] to study the global existence and the blow-up in finite time of solutions to the evolution problem Ut - flu = Ag(U ) , and in [Ma2] to prove results on complete blow-up of solutions. A refined version of the method is also used by Martel [Mal] to show the following property of extremal solutions: THEOREM 2.3 ( [Mal] ) .
u* is the unique weak solution of (2.1;. . ) .
2.3. Regularity of the extremal solution. The extremal solution u * may be classical or singular depending on each problem. For instance, for Gelfand's problem we have: THEOREM 2.4 ( [CrR, MP, JL] ) . Let u* be the extremal solution of (2. 1>.) . (i) If g(u) = eU and N :S 9, then u* E L oo (0.) (i. e., u* is classical for all 0.) . (ii) If 0. = B1 (the unit ball), g( u) = eU, and N � 10, then u* = log ( 1/ lxI 2 ) and A* = 2(N - 2) . There is an analogous result for g(u) = ( 1 + u)P. In this case, the explicit radial solution is given by Ixl - 2 / (p- 1 ) 1 and coincides with the extremal solution u* when 0. = Bl for large values of N and p (see [BV, CCa2] for more details). Part (i) of Theorem 2.4 was proven by Crandall and Rabinowitz [CrR] and by Mignot and Puel [MP] . The proof uses the stability of the minimal solutions, as follows. One takes cp = eC>U>' - 1 in the semi-stability condition (2.3) . One then uses that u>. is a solution of (2. 1>.). This leads to an L 1 (0.) bound for ef3u>. , uniform in A, for some exponents (3. That is, -flu>. is uniformly bounded in Lf3 (0.) and, hence, u>. is uniformly bounded in W 2 ,f3(0.). If N :S 9, the exponent (3 turns out to be bigger that N/2 and, therefore, we have a uniform L oo (0.) bound for u>.. Part (ii) of the theorem had been proved by Joseph and Lundgren [JL] in their exhaustive study of the radial case using phase plane analysis. More recently, Brezis and Vazquez [BV] have introduced a simple approach to this question, based on the following characterization of singular extremal solutions by their semi-stability property: -
THEOREM 2.5 ( [BV] ) . Let u solution of (2.1>.) for some A > O.
HJ (0.), u rt L oo (0.), be an unbounded weak Then, the following are equivalent :
E
(a) The solution u satisfies (2.5) Ag'(u)cp2 � 0 IV'cpl 2
In
- In
(b) A = A* and u = u* .
for every cp E HJ (0.).
XAVIER CABRE
164
That (a) holds for u' follows immediately from the semi-stability (2.3) of miIl imal solutions, by letting .\ r '>" . The key idea behind the other implication of the theorem is that, for each .>. > 0, (2.h) has at most one semi-stable solution -a consequence uf the convexity of g. Following [BV] we can deduce part (ii) of Theorem 2.4 from Theorem 2.5. Indeed, let 0 = B 1 and u = log(1/lxI2). A direct computation shows that this function is a solution of (2.1>.) for .>. = ,\ = 2(N 2). The linearized operator at u i� given by 2(N 2) £r.p = t:J.r.p 2(N - 2) e r.p = - t:J.r.p r.p. 2 I xl H N > 10 then the first eigenvalue of L in B1 "atisfies P1 { L; Bd > O. This is a consequence of Hardy 's ine.q1tality: ,
-
-
(N - 2) 2
1
B,
u
-
r.p2 < Ix l 2 -
1V'r.p12
-
-
for every r.p
H3 (B1) ,
B, and the fact that (N - 2)2/4 > 2(N 2) if N > 10. Applying Theorem 2.5 to (u, .>.) we deduce part (ii) of Theorem 2.4. Before proceeding we recall that, by scale invariance, Hardy's inequality also holds in every domain 0 C jRN with N > �t Namely, (N - 2)2 r.p2 2 for every r.p E Hci (O). (2.6) < 1V'r.p1 4 in fl Ixl 2 Moreover, if 0 E 0 then the optimal constant is (N 2)2/4 and it is not attained in JIJ (O); see [BV] . Theorem 2.4 is a precise result on the boundedness of u' when g( u) = e U • A related regularity result from [BV] states that if the nonlinearity satisfies in addition lim inf7l->oo ug'(u)/g(u) 1 , then u· E HJ (O) for all 0 and N. However, to establish reglilarity of u' without additional assumptions on 9 a question raised by Brezis and Vazquez [BV]- is a much harder task. The best known results for general domains and nonlinearitiet:> (within the hypothese made at the beginning of Section 2) are due to Nedev [NI, N2]: 4
E
-
r
-
>
THEOREM 2.6 ( [NI, N2] ) . Let u* be the e:r,tremal solution of (2 . h ) . (i) If N < 3, then u· E £= (0) (for every 0) . (ii) If N < 5, then u· E HJ (0) (for every 0). (iii) For every dimension N , if 0 is strictly wrwex then u* E H6 (O) .
The proof of this t.heorem uses a very refined version of the method of [erR, MP] described above in relation to Theorem 2. 4 . There are still many question;; to be answered in the case of general 0 and g. For instance, it is not known if an extremal solution may be singular in dimensions 4 S N < 9, for some domain and nonlinearity. However, the radial case 0 = Bl has been recently settled by the author and Capella [CCa2] . This work does not use phase plane analysis, but instead PDE techniques. The result establishes optimal regularity results for general g. To state it, we define exponents qk fur k E {O, 1 , 2, 3} by 1
-
(2.7)
=
qk qk
=
1
2
-
-
+ 00
.IN - l k - 2 +� N N.
for N > 10 for N < 9.
-
EXTREMAL SOLUTIONS AND BLOW-UP
165
Note that all the exponents are well defined and satisfy 2 < qk :'::: +00. As in Theorem 2.6, the following result holds for every nonlinearity 9 satisfying the as sumptions at the beginning of Section 2. THEOREM 2.7 ( [CCa2] ) . Let n = B 1 and u* be the extremal solution of (2.1).,). (i) If N :'::: 9, then u* E L= (Bd . (ii) If N = 10, then u* (lxl) :'::: C log(I/lxl) in B1 for some constant C. (iii) If N ;::::: 1 1 , then
u* ( l xl) :'::: C ! X I- Nj 2+ v'N - 1 +2 Jlog(I/lx l ) in B1 for some constant C. In particular, u* E Lq(Bd for every q < % . Moreover, for every N ;::::: 11 there exists PN > 1 such that u* tI- Lqo (B1) when g( u) = (1 + U) PN (iv) u* E W k ,q(Bd for every k E { I , 2, 3} and q < qk ' In particular, u* E H3 (Bd for every N . Moreover, for all N ;::::: 10 and k E { I , 2, 3 } , 1 8k u * (lxl) 1 :'::: C !xl- Nj 2 + v'N - 1 +2 -k (1 + Jlog(I/lxl)) in B1 for some constant C. Here 8k u* denotes the k-th derivative of the radial function u* . Theorem 2.4, which deals with g( u) = eU, shows the optimality of Theo rem 2.7(i) (ii) , including the logarithmic pointwise bound of part (ii) . The Lq regu larity stated in part (iii) (q < qo ) is also optimal. This is shown considering 9 ( u) = (1 + U)PN (for an explicit PN ), in which case u* (lxl) = I x l - Nj2+ v'N - 1 +2 - 1 . This function differs from the pointwise power bound (2.8) for the factor Jlog(I/lxl). It is an open problem t o know i f this logarithmic factor i n (2.8) can b e removed. (2.8)
•
The exponents qk in the Sobolev estimates of part (iv) are optimal. This follows immediately from the optimality of qo and the fact that all qk are related among them by optimal Sobolev embeddings. The proof of Theorem 2.7 was inspired by the proof of Simons theorem on the nonexistence of singular minimal cones in ]RN for N :'::: 7. The key idea is to take cp = (r - /3 - 1 )ur , where r = l xi, and compute Qu * ((r - /3 - 1)ur) in the semi-stability property (2.3) satisfied by u*. The nonnegativeness of Qu* leads to an L 2 bound for urr - a, with 0: depending on the dimension N. This is the key point in the proof. A similar method was employed in [CCal] to study stability properties of radial solutions in all of ]R N . The method originates from certain relations between semilinear equations and minimal surfaces studied recently in connection with a conjecture of De Giorgi (see [CCa2] for more details) . Theorem 2.7 has been extended in [CCaS] to equations involving the p-Lapla cian and having general semilinear reaction terms. Previously, Boccardo, Escobedo, and Peral [BEP] had extended Theorem 2.4 on the exponential nonlinearity to the p-Laplacian case in general domains. See [CS] and references therein for recent extensions of Theorem 2.4 to more general nonlinearities in the p-Laplacian case and in general domains. 3. An apparent "failure" of the Implicit Function Theorem 3.1. Hardy'S inequality and the Implicit Function Theorem. As a con sequence of (2.3) , we know that the extremal solution is semi-stable, i.e., it satisfies ILl {-�-A*g' (u*); n} ;::::: O. If u* is classical then necessarily ILl { - � -A * g'(u*); n} =
166
XAVIER CABRE
O. This is an immediate consequence of the Implicit Function Theorem and the
>
nonexistence of classical solutions of (2.1,),) for each A A*' Brezis and Vazquez [BVI point out that the property jjd-� - A*g' (U' ) i Q} = 0 fails for some problems in which the extremal solution u' is singular. They also show that jjd -� - Ag'(u,), ) ; Q} may decrease to a positive number as A r A* . An example of this situation is given by -�u u
(3.1)
= =
Ae" 0
in Bl on OBI ,
in every dimension N > 1 1 . Indeed, we know that for this problem 'U' = log(1/lxI2) and A* = 2(N - 2). The linearized operator at u' is given by L = -� 2(N 2) l x l -2 , which has positive first eigenvalue in Bl if N > 1 1 , by Hardy's inequal ity (2.6). We conclude that, for N 1 1 , the linearized operator of (3. 1 ) at u * is bijec l tive, for example between HJ (Br) and H- (Bl ). However, the Implicit Function Theorem can not be applied to problem (3.1) at (u' , A'), since there are no weak solutions of (3. 1 ) for A > A', by Theorem 2.2. Similarly, the Inverse Function Theorem can not be applied neither, since it is proved in [BCMRI that, for each A 0, problem -�u = 2(N - 2)eU + A in BI , U = 0 on OBI admits no weak solution if N > 10. This apparent contradiction is explained by the absence of appropriate func tional spaces in which one could apply the Imp li cit or Inverse Function Theorems. -
>
>
,
3.2. Weak eigenfunctions. Motivated by this apparent "failure" of the Im plicit Function Theorem, the author and Martel [CMal] have shown that the linearized problem of (2.1,),") at 'U" always admits a positive weak eigenfunction belonging to Ll (Q) with eigenvalue 0, even for the problems in which this operator has positive first eigenvalue in HJ (Q). The fact that 0 is a weak eigenvalue is one explanation of the apparent "failure" of the Implicit Function Theorem at u· . THEOREM
3.1 ([CMal]). There exists a f'Unction 'P > 0 in Q such that in n on aQ,
-�'P = A' g' (u* )'P 'P = 0
(3.2)
in the weak sense of Definition 2.1, i.e., and
In
'P�(
=
n
A*g' (U* )'P(
for all ( C2 (Q) with ( = 0 on aQ. We recall that 8(x) = dist(x, (0) . Moreover, 'P E U(O) for every 1 < q < N/(N - 2).
E
-
We also prove the existence of a positive weak eigenfunction of the operator -� - A"l(u') with eigenvalue f.l, for every f.l such that o < f.l < lim" Il-l{ -� - Ag' (U,), ) ; O}: A T>-
Hence, there is a phenomenon of continuum spectrum if this limit is positive and, in particular, if jjl { -� - A"l (u" ) ; O} O . To prove Theorem 3.1, we approximate 9 by an increasing sequence of convex functions gn for which the extremal solution u�, is classicaL For this, it is enough to take gn with subcritical growth, by standard regularity theory. We know that f.ll{-� - A�g�(U�)i O} = 0, wherc A;, is the extremal parameter corresponding
>
1 67
EXTREMAL SOLUTIONS AND BLOW-UP
gn' We consider the first eigenfunction 'Pn , which has eigenvalue 0, of -� A�g�(U�), normalized so that II 'Pn ll £l(O) = 1. We show that, up to subsequences, u� and 'Pn converge in LT(O), for every 1 :S r < N/(N - 1), respectively to u* and 'P, with 'P > 0 a weak solution of (3.2) as in the theorem. The key step of passing to the limit as n ---t 00 in the approximate equations is accomplished through an equi integrability result. A similar method was employed by Baras and Cohen [BaCo] to
-
for some nonlinear parabolic problems. Next, we make a detailed study of all weak eigenfunctions of (3.2) whenever o = B1 is the unit ball, g(u) = eU or g(u) = (1 + u)P, with p > 1, and u* is singular. In these cases, we know that u* can be written explicitly, and that the corresponding linearized operator is -� - c lxl 2 , with 0 < c :S (N 2 ) 2 / 4. We express all weak eigenfunctions and eigenvalues of
{
-
- �'P 'P
-
=
=
in terms of harmonic polynomials and Bessel functions. Looking at which of these eigenfunctions belong to HJ (B1 ), we obtain the following improved Hardy inequal ity due to Brezis and Vazquez [BV] : THEOREM 3.2
([BV]). If N 2: 2 then for every 'P E H6 (0),
where H2 > 0 is the first eigenvalue of the Laplacian in the unit ball of �2 , and WN is the measure of the unit ball of �N . 4. Nonlinear elliptic problems: very weak solutions Brezis and the author [Be] have studied in detail how nonexistence develops for A > A * , and also the phenomenon described above in relation with the Implicit Function Theorem, for the following model problem. Let 0 be a smooth bounded domain of �N , N 2: 2, such that 0 E O. For A E �, consider the problem
{
( 4.1)
-�u u
= =
u2 A Ixl2 +
o
in 0 on
aO.
A formal analysis suggests the existence of a small solution of (4.1) for A small, since the linearized operator at the solution (u, A) = (0, 0) is -�, which is bijective. However, we show that for every A > 0 (no matter how small) , (4. 1 ) has no solution in various weak senses. On the other hand, the results of [BS] show that (4. 1 ) has a minimal negative solution for every constant A < O. Hence, for this problem we have u* = 0 and A* = O. 4.1. Non-existence. The nonexistence of generalized solutions of (4. 1 ) for
A > 0 follows from the following result: THEOREM 4.1 ([BC]) . If 0 E 0, u E Lfoc (O \ {O}), u 2: 0 a. e., and -lxl 2 �u 2: u2 in V' (0 \ {O}), then u O. Here V'(O \ {O}) denotes the space of distributions in 0 \ {O} . ==
168
XAVIER CABRE
Theorem 4.1 is proved using appropriate powers of test functions -a method due to Baras and Pierre [BaPJ . An immediate consequence of the theorem is the nonexistence of generalized solutions ( �o called very weak solutions in [BC] ) for the following problem : THEOREM 4.2 ( [BC] ) . If 0 E n , f > 0 a. e., and f =j 0, then - tl.u u
in n on an,
-
has no solution u E Lroc(n \ {O}) in the sense of distributions V'(n \ {O} ) .
Note Lhat, in the previous results, the test functions vanish in a neighborhood of 0, and no assumption is made on the behavior of 'U near O. In particular, these results imply nonexistence of weak solutions in the sense of Definition 2 . 1 in Section 2. A second corollary of Theorem 4.1 is that u = 0 is the only solution of (4. 1) when ).. = 0 (compare this with the uniqueness property of u ' stated in Theo rem 2.3) . A last consequence of Theorem 4.1 is the nonexistence of local solutions (i.e., nonexistence in every neighborhood of D, wiLhout imposing any boundary condition) for a vcry simple nonlinear equation: THEOREM 4.3 ( [BC)) . If 0 E n and )" > 0, then there is no function 'U Lroc (n \ {OJ) such that
E
in V' (n \ {O}).
4.2. Complete blow-up. Next , we study a natural approximation technique for problem (4.1). Consider, for example, the equation -tl. u
with )" > O. For every
n,
min{ u2 n } + ).. in n IxI 2 + ( 1/11,) o on an, ,
Lhere exists a minimal solution
Un.
Vlfe
then have:
THEOREM 4.4 ( [BC] ) . If ).. > 0 then
( )
un x
b(x)
--»
+ 00
uniformly -in x E n, as 11,
-+ 00 .
Hence, approximate solutions blow-up everywhere in n, i.e., there is complete blow-up. This confirms that there is no reasonable notion of weak solution for (4. 1) when ).. > O. The proof of Theorem 4.1 uses the nonexistence result for (4. 1), the mono tonicity property u" < Un+ l , as well as an appropriate lower bound for the Green's function of the Laplacian ill n.
EXTREMAL SOLUTIONS AND BLOW-UP
169
4.3. More general problems. The previous nonexistence and blow-up re sults are extended in [Be] to the problem
{
-�
� : � (x)g(u) + f(x)
in 0 on a�,
g ?: 0 on lR, g is continuous and increasing on [0, 00) , ds < 00, g(s) with a E L fo c (O) , a ?: 0, a(x) dx = 00 x I B,, (O) IN-2 for some 'rJ > 0 small enough, and f E Lfoc (O) , f ?: 0, and f =I'- O. assuming only
1
J
oo
For this problem, we prove nonexistence of solutions in the weak sense of Sec tion 2, using a variant of the method of generalized truncations. Note that here we do not assume g to be convex. 4.4. The existence criterium of Kalton and Verbitsky. We now consider the previous problem in the case of power nonlinearities g, that is
{
= a(x)uP + f(x) in 0 on a�, u = 0 with p > 1, a and f in L foc (O), a ?: 0, a =I'- 0, f ?: 0, and f =I'- 0 in O. (4.2)
-�u
Kalton and Verbitsky [KV] have found an interesting necessary condition for the existence of a weak solution of (4.2) : THEOREM 4.5 ( [KV] ; see also [BC] ) . If (4.2) has a weak solution u ?: 0 (in the sense of Definition 2 . 1 ) , then aG(J)Po E £1 (0) and G(aG(J)P) < _1_ in O . ( 4.3 ) G(J) - p - 1 Here G = (_�) - 1 with zero Dirichlet boundary condition. The explicit constant 1/(p- 1) was found in [Be] . Note that Theorem 4.5 easily
implies some of the previous nonexistence results. For instance, it gives Theorem 4.2 (for weak solutions instead of very weak solutions) whenever f E LOO (O) , since in this case G(J) o. The interest of Theorem 4.5 also lies in the fact that no assumption is made on the set of singularities of the potential a. In particular, a could be singular near some parts of a�. This is in contrast with previous problems considered in this paper, where a had an isolated singularity in an interior point. In [Be] we give a simple proof of condition (4.3 ) using a refinement of the method of generalized truncations described in Section 2. More precisely, we consider the equation satisfied by v1jJ(u/V), where u is a weak solution of (4.2) , v = G(J), and 1jJ is a bounded concave function appropriately chosen. Next, we replace f(x) by Af(x) in (4.2 ) and we study the problem of existence of solution depending on the value of A. We then have: rv
THEOREM 4.6 ( [BC] ) .
Assume that G(aG(J)P) E LOO (O) . G(J)
XAVIER CABRE
170
For A � 0, consider the problem a (x)uP + Af(x)
-�u u
(4.4)
o
in n on an.
Then there exists A' E (0, 00) such that : (i) If 0 < A < A*, then (4.4) has a weak solution U;,. satisfying A<
G�»
< C(A)
in n
for some constant C(A) depending on A. (ii) If A = A' , then (4.4) has a weak solution. (iii) If A > A' , then (4.4) has no weak solution. Moreover. •
p- 1 P 1 * ,, I G(aG(f)") < (A ) p p 1 G (f)
I < -----, P- 1 LOO (fI)
-
-
-
5. Nonlinear Parabolic Problems The paper [BC] also contains parabolic versions of Theorems 4.1, 4.2, and 4.4, for very weak solutions of the equation Vt
(5. 1 )
-
v2
�v
I x l2 0 vo
v v(O)
in (0, T) x 0
on (0, T) x an on n.
We show that for every initial condition Vo E £,X' (O) , Vo > 0, Vu 'f. 0, approximate solutions Vn of (5.1) blow-up everywhere in (0, T) x 0 (complete blow-up) and that this happens for all T > 0 (instantaneous blow-up) . More precisely, for all 0 < T < T, we have
vn (t, x) o(x)
---»
c in ( T, T) +00 unhormly
x
" r. ,
as
n --> 00.
This extends the following result of Peral and Vazquez [PV1 : THEOREM 5 . 1 ( [PV] ) . Let N > 3 . For every Uo > u and for every T > 0, the problem -
(5.2)
Ut - �u u u(O)
2 (N - 2) eU
o
Uo
:=
log (1/IxI2), Uo 'f. u,
in (0, T) x B1 on (0, T) X OBI on B1
has no weak solution u such that u(t, x) > u(x) in (0, T) x BI .
In [BC] we give a simple proof of this theorem by applying the nonexistence results for (5.1) to v : = 'U - U > 0, with u as in Theorem 5. 1 . Indeed, it is easy to check that v satisfies Vt - �v > (N - 2) l x l - 2v2 . We obtain nonexistence of solution of (5 .2) in the distributional sense V' CCO, T) x (B1 \ {O} » . The assumptions on Uo and u in Theorem 5.1 have been improved and studied in detail by Vazquez [Vl .
171
EXTREMAL SOLUTIONS AND BLOW-UP
6. Linear parabolic problems with singular potentials
{
We consider the linear heat equation with potential (6. 1)
Ut - D.. u u u (O)
= a (x) u = 0 = Uo
in (0, T) x 0 on (O, T) x 80 on 0,
0 is a smooth bounded domain of ]RN . We assume that a E Lfoc(O), Uo E Lfoc(O), and that a 2: 0 and Uo 2: 0 a.e. in O. Note that a is time independent.
where
We only consider nonnegative solutions of (6. 1 ) . Note that problem (6. 1 ) with a (x) = 2 ( N - 2) l xl - 2 i s the linearization of (5.2) at the stationary solution u. In 1984, Baras and Goldstein [BaG] considered equation (6. 1 ) with a (x) = )' l x l - 2 , 0 E 0 C ]RN , and N 2: 3. For this problem they prove that if 0 :::; ), :::; (N - 2) 2 /4 =: ),* then there exists a global weak solution of (6.1 ) for every Uo E L 2 (0). Instead, if ), > (N - 2 ) 2 /4 = ),* then, for each T > 0 and each Uo E Lfoc(O) with Uo 2: 0 and Uo :t= 0, (6.1) has no positive weak solution; moreover, in this case there is instantaneous complete blow-up of approximate solutions of (6. 1 ) . Their proof of nonexistence uses Moser's iteration technique, as well as a weighted Sobolev inequality. The critical constant ),* = (N - 2) 2 /4 is related to Hardy's inequality (2.6) . In [CMa2] , the author and Martel consider the case of general potentials a (x ) , and establish relations between the existence of solutions and the validity of a Hardy type inequality with weight a (x ) . In particular, a new and simple proof of the nonexistence result of Baras and Goldstein is given. In order to state the precise results, let 8 (x ) = dist (x, (0) for x E 0, and let LHO) = L 1 (0, 8 ( x ) dx). For 0 < T :::; 00 and Uo E LHO), we say that u 2: 0 is a weak solution of (6. 1 ) if, for each 0 < S < T, we have that u E L 1 ( (0, S) x 0), au8 E L 1 ((0, S) x 0), and
loS in u ( -(t - D..() - in Uo ( (0) loS in a u ( =
for all ( E C2 ([0, 8] x 0) with ( (8) == 0 on 0 and ( = 0 on [0, 8] x 80. If T = 00, we say that u is a global weak solution. Recall the definition of generalized first eigenvalue /1 1 { -D.. - a(x); O} given by (2.4) . The following result states that /1 1 { -D.. - a(x); O} > -00 is a necessary and sufficient condition for the existence of global weak solutions of (6. 1 ) with (at most ) exponential growth. THEOREM 6. 1 ( [CMa2] ) .
(i) Suppose that, for some Uo E LHO) and some constants C and M, there exists a global weak solution u 2: 0 of (6.1) such that Il u(t)81Iu (!1 ) :::; CeMt for all t 2: O. Then /1 1 { -D.. - a (x) ; O} > (ii) Suppose that /1 d -D.. - a(x); O} > -00. Then, for each Uo E L 2 (0) with Uo 2: 0, there exists a global weak solution u E C ( [O (0) ; L 2 (0)) of (6. 1) such that Ilu(t) II £ 2 (!1) :::; Iluo ll £2 (!1) e - I-'l t for all t 2: 0, (6.2) where /1 1 = /1 d -D.. - a(x); O}. The next result states that condition /1 1 { -D.. - a(x); O} > is "almost - 00 .
,
- 00
necessary" for the local existence of weak solutions. More precisely, we have:
172
XAVIER CABRE
6.2 ( [CMa2]) . Snppose that f.L1 { -� (1 - c)a(x); O} = - 00 for some constant c > O. Then, for every T > a and every Uo E L� (O) with Uo > a and Uo =f= 0, there is no weak solution u > 0 of (6.1). Moreover, there is instantaneo1/.S complete blow-up for (6. 1 ) , in the following sense : FaT eveTY 'II > 1 , set an(x) = min(a(x ) , n), uon(X) = min( uo (x) 'II ) , and let Un be the unique global solution of (6.1) with a and uQ replaced, Tcspcctively, hy an and UOn . Then, for all a < 'T < T, T HBOREM
-
,
---> ,
+ 00 Il.1l.iformly ,in ('T, T)
x
0, as n
->
00 ,
The following inequality is the key ingredient in the proof of the necessary conditions in these theorems. Suppose that (6.1) has a positive local solution , and let u be the minimal solution (i.Il., the one obtained as increasing limit of 1tpproximaLe solutions) . Then, for all 0 < t1 < t2 < T, we have
o
a( x)cp2
-
lV'cpl2 ::; r Jo
1 r log t2 - h Jo
�tt2�
1!. u
for every 'f E C� (O).
L cp
l This inequality is provlld IIlultiplying the approximate problems of (6.1) by cp2 /un , then integrating by parts in space, integrating in time, and finally letting 'll 00. We apply the previous theorelllS to some specific equa.tions with "critical" po tentia.ls. First, our method provides a new and quite elementary proof of the result of Baras and Goldstein [BaG] mentioned above. Namely, suppose that 0 E n e ]RN , N > 3, and a(x) = ..\lxl- 2 . Let ..\* = (N - 2)2/4. For this potential, [BaG] establishes the following: (a) If 0 < ..\ < ..\ * , then (6. 1 ) h as a global weak sulut,ion for every Uo E L2 (0), Ito > O. (b) If ). > ).* , then (6. 1 ) has no positive weak solution for every T > 0 and cvery Uo E L H n) with Uo > 0 and Uo =f= O. Moreover, there is insLanLaneous complete blow-np of approximate solutions. We obtain (a) and (b) through direct applications of Theorem 6,1 (ii) and The orem 6.2, respectively. Here the e:;8eIltial role is played by (2.6), i.e., Hardy's inequality with best constant. In case a < ..\ < ..\* = (N - 2? /4, we also obtain that solutions decay exponen tially. More precisely, )'Ie have --->
Ilu (t ) II U (f1) < I l uo Ii U(f1) e-J1.t
for all t >
0,
where f.L = H2 (WN /101)2/N > () This is a consequence of estimate (6.2) and the improved Hardy inequality (Theorem 3.2) . New improved Hardy inequalities and a detailed description of solutions of (6. 1 ) when a(x) = ).1.7: 1 -2 and 0 < ). ::; (N - 2) 2 /4 have been ohtained by Vazquez and Zuazua [VZ] . In [GP] , Garcia Azorero and Peral have extended the results of Baras and Goldstein to the p-Laplace operator. A second application of Theorems 6.1 and 6.2 is the c&;e when 0 c jRN is of elass C2 , N > 2, and a(x) = ..\S ( X)-2 . Here ). 0 is a constant and 5(x) = dist (x, (0) . For th is prohlem, we obtain state ments ( a) a.nd (b) above with ..\* = 1/1. Here we use some Hardy type inequalities proved in [BM] and [D] . ,
>
.
EXTREMAL SOLUTIONS AND BLOW-UP
173
Finally, recall that the results for the potential Alxl- 2 required N 2 3. In dimension N = 2, we prove that the potential a(x) = Alxl- 2 (1 - log Ixl)- 2 is "critical" , in the sense that the dichotomy ( a) - (b) occurs with A * = 1/4 for all domains n with 0 E n e Bl (0) c ]R.2 . References [BaCo]
Baras, P. , Cohen, L . ,
heat equation, [BaG]
Baras, P. , Goldstein, Math. Soc . ,
[BaP]
284,
[BEP]
121-139 (1984).
Baras, P . , Pierre, M., Inst. Fourier,
Complete blow-up after Trnax for the solution of a semilinear 71, 142-174 (1987). J . A . , The heat equation with a singular potential, Trans. Amer.
J. Funct. Anal.,
34,
Singularites eliminables pour des equations semi-lineaires,
185-206 (1984).
Ann.
A Dirichlet problem involving critical exponents, & Appl. , 24, 1639-1648 ( 1995). X., Some simple nonlinear PDE's without solutions, Boll. Unione
Boccardo, L., Escobedo, M., Peral, I., Nonlinear Anal . , Theory, Meth.
[BC]
Brezis, H . , Cabre,
[BCMR]
Brezis, H . , Cazenave,
Mat. Ital . ,
revisited, [BM]
25,
217-237 (1997).
Brezis, H . , Strauss, W.A., Soc. Japan,
[BV]
T . , Martel, Y. , Ramiandrisoa, A . , Blow up for U t - L!.u = g(u) 1 , 73-90 (1996). M . , Hardy 's inequalities revisited, Ann. Scuola Norm. S up . Pisa
Ad. Dilf. Eq.,
Brezis, H . , Marcus Cl. ScL ,
[BS]
I-B, 223-262 (1998).
25,
Semi-linear second-order elliptic equations in L 1 ,
565-590 (1973).
J. Math.
Blow-up solutions of some nonlinear elliptic problems, Rev. 10, 443-469 (1997). Cabre, X. , Capella, A., On the stability of radial solutions of semilinear elliptic equa tions in all of JRn , C. R. Math. Acad. Sci. Paris, 338, 769-774 (2004) . Cabre, X . , Capella, A . , Regularity of radial minimizers and extremal solutions of semilinear elliptic equations, Journal of Functional Analysis, 238, 709-733 (2006). Cabre, X., Capella, A., S anch6n, M . , Regularity of radial minimizers and semi-stable solutions of semilinear problems involving the p-Laplacian, preprint, 2006. Cabre, X., Martel, Y., Weak eigenfunctions for the linearization of extremal elliptic problems, J. Funct. Anal. , 156, 30-56 (1998). Cabre, X., Martel, Y., Existence versus instantaneous blowup for linear heat equations with singular potentials, C. R. Acad. Sci. Paris Ser. I Math. , 329, 973-978 (1999). Cabre, X., Sanch6n, M., Semi-stable and extremal solutions of reaction equations in volving the p-Laplacian, to appear in Comm. Pure Applied Analysis, 2006. Crandall, M .G. , Rabinowitz, P. H . , Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rat. Mech. Anal. , Brezis, H . , Vazquez, J . L . ,
Mat. Univ. Compl. Madrid,
[CCa1] [CCa2] [CCaS] [CMa1] [CMa2] [CS] [CrR]
58, 207-218 (1975).
The Hardy constant,
[D]
Davies, E . B . ,
[GMP]
Gallouet, T., Mignot, F., Puel, J.-P. ,
[G] [JL] [KV] [Mal] [Ma2] [MP]
=
307, Serie I, 289-292 ( 1988). Garcia Azorero, J . , Peral, I. , Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations, 144, 441-476 (1998). Gelfand, I.M., Some problems in the theory of quasilinear equations, Amer. Math. Soc. Trans! . , 29, 295-381 (1963). Joseph, D . D . , Lundgren, T . S . , Quasilinear Dirichlet problems driven by positive sources, Arch. Rat. Mech. Anal . , 49, 241-269 (1973) . Kalton, N . J . , Verbitsky, I . E . , Nonlinear equations and weighted norm inequalities, Trans. Amer. Math. Soc . , 351, 3441-3497 (1999) . Martel, Y., Uniqueness of weak extremal solutions of nonlinear elliptic problems, Houston J. Math. , 23, 161-168 (1997). Martel, Y., Complete blow up and global behavior of solutions of Ut - L!.u = g(u) , Ann. Inst. H. Poincare Anal. Non Lineaire. , 15, 687-723 ( 1998). Mignot, F . , Puel, J.-P. , Sur une classe de problemes non lineaires avec nonlinearite positive, croissante, convexe, Comm. P.D. E . , 5, 791-836 ( 1980). C. R. Acad. Sci. Paris,
[GP]
4 6 , 417-431 ( 1995). Quelques resultats sur Ie probleme -L!.u >.eu ,
Quart. J . Math. Oxford,
174
XAVIER CABRE Nedcv, G . , Regularity of the extremal solution of semilinear elliptic equations, C. R. Acad. Sci. Paris Ser. I Math . , 330, 997-1002 (2000) . Nedev, G . , Extremal solutions of semilincar elliptic equations, preprint, 2001. Peral, L , Vazquez, J.L., On the stability or instability of the singular solution of the semilinear heat equation with exponential reacti on term, Arch. Rat. Mech. Anal., 129, 201-224 (1995) . Vazquez, J.L., D omain of e.:r:istence and blow-up for the exponential reaction-diffusion equation, Indiana Univ. Math. J., 48, 677-709 (1999). Vazquez, J.L., Zuazua, K, The Hardy inequality and the asymptotic beha1liour of the heat equation with an inverse-square potential., J. Funet. Anal., 173, 103-153 (2000) .
INl] IN2] IPV]
IV] IVZj
POLITECNICA DE CATALUNYA, 1 , Av. DIACONAL 647, 08028 BARCELONA, SPAIN
ICREA AND UNIVERSl'T'A'T'
A PLIC ADA
E-mail address:
,
xavier . cabrelDupc . edu
DEPARTAMENT DE MATEMATICA
Contemporary Mathematics Volume 446, 2007
Capillary Drops on an Inhomogeneous Surface L . A . Caffarelli and A . Mellet
To Haim with admiration and our best wishes. ABSTRACT. This paper is concerned with equilibrium liquid drops lying on a horizontal plane when small periodic perturbations arise in the properties of the support plane (due for example to chemical contamination or roughness). We first establish the existence of a minimizer for the energy and study its regularity. We then show that the free interface stays in a small neighbor hood of a portion of sphere (corresponding to an equilibrium drop lying on a homogeneous plane) , thus showing the existence of sphere-like capillary drops.
1. Introduction The energy of a drop described by the set E c plane (z = 0) is given by (1.1)
a
Jrir
z>o
I D'PE I
-
a
1
z=O
]Rn + 1 and resting on a horizontal
(3 'PE(X, 0) dx +
1
z>O
p f 'PE dx dz
where 'PE is the characteristic function of E, a is the surface tension, (3 is the relative adhesion coefficient between the fluid and the solid, f is the gravitational energy and p is the local density of the fluid. In this paper, we neglect the effect of gravity and assume f = O. The Euler-Lagrange equation associated to the minimization of ( 1 . 1 ) under a vol ume constrain gives rise to a mean-curvature equation, together with a contact angle condition (see [Fl ) . This last condition, known as the Young-Laplace equa tion reads: cos ,",! = (3, where ,",! denotes the angle between the free surface of the drop 8E and the hori zontal plane {z = O} along the contact line 8( E n {z = O}) (measured within the 1991 Mathematics Subject Classification. Primary 49F22; Secondary 74Q05. Key words and phrases. Capillary surfaces, Calculus of variation, Isoperimetric inequality, Homogenization. L. Caffarelli is partially supported by NSF grant DMS-0140338. A. Mellet is partially supported by NSF grant DMS-0456647.
1 75
©2007 American Mathematical Society
1 76
L. A. CAFFARELLI A)lD A. MELLET
fluid) . The coefficient (3 is determined experimentally and depends on the proper ties of the materialt; (solid and liquid) . It is usually assumed to be constant, but it is very sensitive to small perturbations in the properties of solid plane (chemical contamination or roughness) . Thp.',e inhomogeneities are responsible for many in teresting phenomena such as contact angle hysteresis and sticking drop on inclined surfaces (see [JG] , [LJ]). In [HM] , C. Huh and S. G . Mason investigate the effect of roughness of the solid surface on the equilibrium shape of the drop by solving approximately the Young-Laplace equation, for some particular type of periodic roughness (radially symmetric). The purpuse of
this paper is to investigate the properties of equilibrium drops in the case of general periodic inhomogeneities, that. is when the relative adhesion coefficient satisfies (3 (3(x/=:) , with y f-7 /3(11) Zn-periodic. The existence of equilibrium drops (minimizing the energy functional) will be shown, within the class of �d.s of locally finite perimeter. Then, we will investigate the properties of that minimizer, showing in particular that the contact liTle has finite Hausdorff measure. Finally, we will prove that the equilibrium drop converges uniformly to a spherical cap when the size of the inhomogeneities (c) goes to zero (homogeni
Du
div
,
u
·v
in {u > O}
= Ii
=
(J (x /c)
on EJ{u > OJ.
In [eM] , we investigate further consequences of t hose results, justifying in particular the two phcnomena Lhat we mentioned earlier: Contact angle hysteresis and 8ticking drop on an inclined plane (for non-vanishing gravity) .
2. Notations, main results and organization of the paper 2.1. Sets of finite perimeter. We recall here the main facts about sets of
finite perimetcr and BY functions. The standard reference for BY theory is Giusti [Gi] . Let l1 be an open subset of Rn+ l ; B V (l1) denotes the set of all functions in £ 1 (l1) with bounded variation:
BV(l1) = where
ID f l = sup
f E £1 (l1)
:
o
IDf l < + 00
f (x)div g(x )dx : 9 E [C6 (nW + 1 , I gi < 1
•
If E is a Borel set, and l1 is an open set in Rn+1, we recall that the perimetel" of E in l1 is defined bv o
"
o
P(E, l1)
=
!!
I Drp£ i .
1 77
CAPILLARY DROPS ON AN INHOMOGENEOUS SURFACE
A Caccioppoli set is a Borel set E that has locally finite perimeter ( i.e. P(E, B) < 00 for every bounded open subset B of 0). Note that sets of finite perimeter are defined only up to sets of measure O. We shall henceforth normalize E ( as in [Gil) so that 0 < I E n B(x, p) 1 < I B(x, p) 1 for all x E 8E and all p > O. Furthermore, it is well known that if the boundary 80 of 0 is locally Lipschitz, then each function f E BV(O) has a trace f + in £ 1 (80) ( see Giusti [Gil ) . l.From now on, we denote by
0 the upper half space: O = ]Rn x (0, +00), and we denote by (x, z) an arbitrary point in n, with x E ]Rn and z E [0, +00) . We denote by
V > O:
c&'(V) the class of closed Caccioppoli sets in 0 with total volume
{
c&'(V) = E c O :
10 IDipEI < +00, l E I = V } ,
where l E I = In 'PE demotes the Lebesgue measure of E. Since Caccioppoli sets have a trace on 80 = ]Rn x {z = O}, we can define the following functional for every E E c&'(V):
(2 . 1 )
/ (E)
J 1> 0 ID'PEI - 1=0 /3(X) 'PE (X, O)dx P(E, 0) - r }En{z=O} /3(x) dyen(x).
In this framework, equilibrium liquid drops are solutions of the minimization prob lem: (2.2)
inf / (F) / (E) = FEe'(V)
2.2. Constant adhesion coefficient. When /3 = /30 is constant, the ex istence of a minimizer was established by E. Gonzalez [GJ . The corresponding functional reads (2.3)
/o (E) =
10 I D"'E 1 - /30 J 'PE (X, 0) dx.
When /30 = - 1 (hydrophobic surface) , the absolute minimizers are the spheres of volume V in {z > O}. In particular, the equilibrium drop does not touch the support plane. On the contrary, if /30 > - 1 , it is easy to see that, though a sphere of volume V is still a local (degenerate) minimizer of the functional / , the absolute minimizer must touch the solid support. An important tool, when the adhesion coefficient is constant, is the Schwarz symmetrization ( see [Gl ) : For every E E c&'(V), the set (2.4)
E8 = {(x, z) E 0 j I x l < p(z)} ,
is a Caccioppoli set with volume
(
where p(z) = W� 1
V satisfying
/0 (E8 ) ::; /o (E)
J 'PE (X , Z)dX)
1
n-
178
L. A. CAFFARELLI AND A. MELLET
with equality if and only if E was already symmetric. This clearly implies that any minimizer should have axial symmetry. Actually, it can be shown that the minimizers are spherical caps; that is the intersection of a ball Bpo (O, Zo} in ]Rn+1 with the upper-half space O. We denote by Btu (za)
=
Bpa (O, zO } n {z > O}
such a spherical cap. Our main result is a stability/uniqueness result for the minimization problem with constant coefficient: THEOREM 2 . 1 . Let E be such that
(2.5) (2.6)
E E C(V), E lies in a bounded subset BR of 0, 'rIF E C(V) . :38 > 0 s.t. /a (E) < /0 (F) + 8
Then there exists a universal a > 0 and a constant G (depending on R) such that IE6.Bta I :S G8a,
where Bta is such that
lB.;;;' I = V
and the cosine of the contact angle is {3a ·
Note that we recover in particular the fact that the gravity-free equilibrium drop is a constant mean-curvature surface satisfying the Young-Laplace condition. If moreover E satisfies some non-degeneracy conditions, then Theorem 2.1 im plies the uniform stability in the following sense: For any 1) 0, there exists .50 such that if (2.6) holds with .5 < 80 , then
>
B�_1))p C E
c
B�+1))p '
In other words, the free surface aEn {z > O} stays between aB�+1)) p and aB� _1))p ' REMARK 2.2. When {3a = - 1 (so / (E) is the perimeter of E in ]Rn+1 ) , Theorem 2.1 is nothing but a quantitative version of the isoperimetric inequality, also known as Bonnesen inequality. In dimension 2 it is well known (see R.R. Hall [H] ) that we have (with Po such that lEI = I Bpa l ) : I E 6.Bpa l :::; G IEI3/4 (p( E)
-
P(Bpo ) ) �
thus corresponding to a = 1/2 in our theorem. In higher dimension, a similar result was established by Hall with a = 1/4, though it was conjectured that the inequality should hold with a = 1/2 in any dimension. In the last section of this article, we will prove Theorem 2.1 with a = 1/3 in dimension 2. The question of whether the result holds with a = 1 /2 (in dimension 2 at least) is still open.
2.3. Periodic adhesion coefficient. When the relative adhesion coefficient
f3 depends on x , the Schwarz symmetrization (2.4) could increase the wetting energy
J {3(x)'PE(x, O)dx.
However, if {3 = {3(X/E) is periodic with small period E, the wetting energy should not increase by more that GE after symmetrization. If we can prove that fact, Theorem 2.1 will allow us to show that the minimizer associated with a periodic adhpBion coefficient is almost a spherical cap (in L1 and LOO).
CAPILLARY DROPS ON AN INHOMOGENEOUS SURFACE
179
We now make our framework precise: We consider the following energy func tional: (2.7) where and
{3 satisfies
We denote by
/ (E) =
J 1>0 I D
-1
<
(3(x) < 1 ,
X
(3(x)
({3) the average of {3: f-+
for all
x E IRn
Zn-periodic
r
i [o , l ]n {3(x) dx, Bta ( zo ) = Bpo (0, zo ) n {z > O} the minimizer of the ({3)
and by coefficient functional
/o E ) =
Then we prove:
(
=
constant adhesion
J 1>0 I D
(i) For all V and c > 0, there exists E minimizer of / in g(V) . Moreover, up to a translation, we can always assume that 1 E c { x l ::; Ro V n+1 1 , z E [0, To V n + 1 ] } with Ro and To universal constants. (ii) The contact line 8( E n {z = O}) has finite (n - 1) Hausdorff measure in IRn . (iii) There exists a constant C such that if c ::; C(V)ry (n+1) /o:, then + ) · B(+l -'l/)po c E C B(H'I/ po In other words, the free surface 8E n {z > O} lies between 8B(;. + 'I/ ) PO and 8B(;._ 'l/) PO for small enough. THEOREM 2.3.
I
c
2.4. Organization of the paper. The proof of Theorem 2.3 (i) is very clas sical and will be developed in Sections 3 and 4; It relies on the properties of BV functions and some monotonicity formulas for the minimizers of / . The regularity of the contact line (Theorem 2.3 (ii)) is established in Section 5 and allows us to show that the energy of the minimizer only differs by from the energy correspond ing to the wetting coefficient ((3). The last point will then follow as a corollary of Theorem 2 . 1 , the proof of which is presented in Section 6.
c
3. Constrained minimizer The lower semi-continuity of the functional (2. 1 ) in the £1 (0) topology is easy to establish. Moreover, it is a classical result that bounded sets of BV are pre compact in Lfoc (O) . Those two facts give the convergence of a minimizing sequence to a minimizer if 1 0 1 < +00. Since we are considering drops lying in the upper half space, we can only prove the existence of minimizers if we first restrict ourself to Caccioppoli sets lying in a bounded subset of 0 (constrained minimizers) ; this is the object of this section. In Section 4, we will show that this minimizer is actually an unconstrained minimizer
180
L. A. CAFFARELLI AND A. MELLET
To guarantee the compactness of minimizing sequences, we first look for mini mizers that stay within a bounded subset of ]Rn+ l . Let
rR,T
=
E
{ (x, z)
and
En+l ; Ixl < R, z E [O, T] },
{E E g(V) ; E C rR,T}. We call constrained minimizer a set E E G'R,T(V) satisfying / (E) = min / (F). (3.1 ) gR ,T (V)
=
F E$R,T(V)
We have the following proposition: PROPOSITION
-
3.1. If R and T are such that there exists a ball B with volume.
V in rR,T, then there exists a minimizer ER,T for (3. 1). Moreover, we have and £,n(ER,T n {z = O}) < CV ,,+l . (3 . 2) "
The proof is very classical. The key is to establish the lower continuity of the functional in a topology that makes minimizing sequences pre-compact. We recall the following classical result for functions with bounded variation (see [Gil): LEMM A
3.2. Let 0 C En be an open set, and (lj ) a sequence. of functions in
BV(O) which converges in L�oc{O) to a function f. Then rz
It follows: LEMMA
ID fl < lim inf
IDh l·
J - OC
,
3.3. The functional / is lower continuous with respect to the L l topol-
ogy: If (Ej) is a sequence of Caccioppoli sets such that Ej / (E) < lim inf / (Ej)
•
E in L l then
}-=
Proof: Assume on the contrary that there exists 6 > ° such that or z>
ID'PEj l < o
%>0
ID'P E I +
We recall the following estimate (see [Gil):
1=0
I'PE - 'PEj I dx <
J
rR"
1
%=0
ID'PE I + +C(t)
{3('PE - 'PEj)dx - 6
J tR.t ID'PEj I z> o
I'PE - 'PEj Idxdz.
The last term clearly goes to zero as j goes to infinity, so lim inf
1
n\rR ,t.
I DCPEj I
::;
J1
z >O
ID'PE I +
1
rX , t
IDcpE I - 6,
from whidl we derive a contradiction, taking into account the lower continuity of the perimeter (Lemma 3.2) and letting t goes to zero. 0 Next, we recall the following compactness result for functions of bounded vari ations:
1 81
CAPILLARY DROPS ON AN INHOMOGENEOUS SURFACE
Let 0 be a bounded open set in IRn with Lipschitz boundary. Then sets of functions uniformly bounded in BV -norm are relatively compact in L 1 (0) . LEMMA 3.4.
The existence of a constrained minimizer is therefore a consequence of the following lemma: LEMMA 3.5 (A priori estimates ) .
/ (E)
�
for all E E 6"(V) with
1 - f3max 1z>o 2
-1
If
f3 < 1, then ID
1-
1z=o
2
f3max = sup f3 ( x) . x
Note that Lemma 3.5 and the fact that B lies in rR,T give (3.2), since
where
/ (E) :::; / (B) = JL;;� l V n+
l
JLn+ 1 denotes the isoperimetric constant in IRn+ 1 .
Proof: Note that if g(x) is a non-negative function, then
1>0 g(x) ID
Hence
and therefore
r ( 1 - f3(x) f3(x))
2
or
�
+
2
-
+
2
2
o
Lemma 3.3, 3.4 and 3.5, together with the fact that 6"R,T(V) i=- 0 gives Propo sition 3. l . We now have t o prove that i f R and are large enough, ER,T i s i n fact an unconstrained minimizer.
T
4. Unconstrained minimizer In this section we prove the following result: PROPOSITION 4 . 1 .
There exists
To and
Ro
such that if
T To Vn+l , 1
and � then ER,T E 6"Ro,To . In particular, ER,T is an unconstrained minimizer. After rescaling, we only have to prove the result when l E I = V =
1.
182
L. A. CAFFARELLI AND A. MELLET 4.1.
Bounded above. First, we prove that for T large enough, the minimizer
for ,/ in gR,T(V) is actually a minimizer in gR(V) where
gR(V) = gR,oo (V).
This follows from the following result;
There exists Tl such that for all T > T1 , there exists a minimizer E of ,/ in gR,T that satisfies PROPOSITION 4.2.
E E gR,T, (V).
The proof relies on a slight modification of an argument first presented by E. Barozzi in [B] (see also E. Barozzi and E. Gonzalez, [BG] ) . The key lemma is the foHowing: LEMMA 4.3. Let R and To be such that there exists a ball B of volume V lying in r R,To ' Then there exists Tl � To such that for any T > Tl and E minimizer of ,/ in gR,T(V), there exists t, To < t < Tl with -
.7F (E n {z = t}) = 0.
Let us recall here that the coarea formula gives I EI = 1+oc .}t"n (E n {z = t}) dt,
which in particular implies that a.e. t E R
Proof of Proposition 4.2: Let E be a minimizer of ,/ in gR,T(V), and let
t be as in Lemma 4.3. Then, the part of E that lies above z perimeter of E by at least 1 -- I E n {z > t } 1 n+1 , !J,n+ l where !J,n+l is the isoperimetric constant. Thus, if we define
=
t contribute to the
n
Eo =
E
o
in {O < z < t} \ B in {z > t} in B -
E U pB
-
-
with p � 1 such that IEol = l E I , we have and Eo
E gR,T, (V).
,/ (Eo) < ,/ (E) =
FE6'R.T(V)
min
,
,/ (F),
Letting T go to infinity, the compactness property of the minimizing sequence gives the existence of a set E E gR,T, (V) such that ,/ (E) =
FE0"R(V)
min
,/ ( F) .
In other words, E is a vertically-unconstrained minimizer.
Proof of Lemma 4.3: Let t l , t 2 and t 3 be three positive numbers with
0 < h < h < t3 < T
o
CAPILLARY DROPS ON AN INHOMOGENEOUS SURFACE
183
and such that B e {z < td.
VI = IE n {tl < z < t2 } 1 V2 = IE n {t2 < z < t3}1
We denote
and
The isoperimetric inequality yields:
n v t: 1 v2n + 1
J.Ln+1(2m + 81 ) :::; J.Ln + 1 (2m + 82) , :::;
where 8i i s the surface of the lateral boundary:
8i =
1{ti
In particular, we have:
(4. 1)
n n vt+ 1 + V2n+ 1
:::;
JLn+1 (4m + 8)
with 8 = 81 + 82 . Let us now define the set F by
F=
{
E
in fR ,T \ {h < z < t3 } \ B in {tl < z < t3} in B
0
E U pB
where we recall that B is a ball with volume I F I = l E I· Then,
f (F) :::; f (E)
-
8 + 2m +
Since E is a minimizer, we deduce:
V lying in fR,T ,
1 (VI + V2) n + l . J.Ln + 1
--
(4.2) Thus, inequalities (4.1) and (4. 2 ) implies
and therefore
(4. 3) Let ao = To and bo = Tl with To such that B e {z < To} .
Given a k and bk , we define
and p is such that
n
L. A. CAFFARELLJ AND A. MELLET
184
Then, for
hk
=
�, ak�bk + �)
bk�a.
it is possible to find h and t3 E (bk - hk , bk ) such that
.n'Jn(E n {z = ti}) Setting
<
h;,
(ak , ak
E
hk) '
t2
E (a.�b.
-
for i = 1, 2, 3
Vie
vf = IE n {t; < z < tHdl, Vk+l = min(v�, v�)
we choose
+
i = 1, 2
(tI . t2 ) if v f :::; v� (t2 ' t3) if v� < vf
Let
It follows from (4.3) that
ntl
Vie + ! < Cm le Vie mk < - h ie
(and thus
hk+ 1 �
mle+l
and therefore, with the motation a = < -
!jf).
utln
C h
It follows that
ie mle
nt1 ,
C1+0+o ' + . , , +0'
Q
",k m o
h Ie h'" k - l ' " h0 C",-
<
Since hk >
<
n
"
+
.. + 1
h'"k " ho-k+1 k-l
bQ�kaQ, we deduce
•
•
•
Ic + l
h0
mo
•
•
Thus, choosing Tl large enough so that <1 we have
mk
)
0
which conclude the proof of Lemma 4.3.
as
k -+
00,
o
CAPILLARY DROPS ON AN INHOMOGENEOUS SURFACE
185
4.2. Monotonicity Formula and Non-degeneracy. The proof of Propo sition 4. 1 now relies on some monotonicity formula that we derive in this section. We start with a simple observation: 1
,/ :::; 0"R (min ,/ + CVo- n+ l 6V. ,/ ::; 0"Rmin Vo+8 V) 0"R ( VO ) ( VO )
( 4.4)
min
To establish this fact, take
E minimizer of ,/ in 0"R(VO).
The vertical dilatation of
Et = { (x, z) E lRn + 1 ; (x, (1 + t) - l z) E E},
E:
satisfies
and
P( Et , D) :::; (1 + t)P(E, D),
but the wetting energy is left unchanged. Thus
,/ ( Et ) :::; ,/ (E) + tP(E, D). So if we take t = 6VIVo, we get and
,/ ( Et )
< <
,/ (E) + tP(E, D) min ,/ + C( Vo) 6V,
0"R (VO )
C(Vo) = Jo P(E, D) , thus giving the second inequality in (4.4) fact that P(E, D) ::; C l Vo l n+ l ) . Similarly, if F is a minimizer of ,/ in 0"R (VO + 6V) , then the set Pt { (y, z) E lRn + 1 ; (y, (1 + t)z) E E}, with t = 6VIVo satisfies and P(Pt , D) :::; P(F, D) where
(using the
=
and therefore proving (4.4) . The next lemma establishes the non-degeneracy of E near a point of the contact line. It will allow us to show that E lies in a bounded set of D. Its proof relies on a monotonicity type argument. Let rr (XO) denote the cylinder rr (XO) =
B�(xo) x lR = { (x, z) E lRn + 1 ; Ix - xo l < r},
then we have the following:
There exists C1 , Co > 0 such that for any minimizer E of ,/ in in the projection of E onto {z O} (i.e. there exists Zo such that (xo, zo) E E) then I E n rr (xo) 1 > corn+ 1 , for all such that I E n rr (XO) I ::; cl l EI . LEMMA 4.4. 0"R (V), if Xo lies
r
=
186
L. A. CAPPARELLI AND
A
MELLET
Proof: We denote U(r) = IE n rr (xo ) l , S(r) = .non(E n 8rr(xo) n {z > O})
Note that U'(r) = S(r). We also introduce
A1(r) = p(E , rr n {z > O} ) the area of the free surface in rr and A2 (r) =
'PE(x, O)dx
B;:(xo) the wetted area in rr ' Since E is bounded above, if x E E n {z = O} , going from the slice {z = O} to the slice {z = T}, we must cross 8E. Therefore we have A2 < Al. The isoperimetric inequality then gives: U(rt/ (n+1 ) < ,un +l (2A1 + S(r » . Consider now the set F = E \ fr (xo). It satisfies:
,/ (F) < ,/ (E) - A l + ,SmaxA2 + S, and IFI = IEI - U(r) , and using (4.4), with IFI = Vo and OV = U, we get ,/ (E) = min ,/ = min ,/ < ,/ (F) + C CIEI )U(r). 0"( IEI) S(Vo + U) It follows that
min ( I , 1 - 'smax)A l < A l - 'smaxA2 < S + C(/FI) U(r), with C(F) < �o peE, n) < CCIEj) as long as U < IEI/2. Thus
U (r t/(n+ l )
and if U < c- ( n+1) /2, we deduce
:s;
CU'(r) + CU(r) ,
U(r),,/(n+ l ) < CU' (r),
o
and Gronwall's Lemma gives the result. Using similar arguments, we can also establish the following lemma:
LEMMA 4.5. Let (xo , zo) E BE with Zo > O. There exists c, universal constant, such that for all r < Zo we have IBr(x o, zo) n EI > crn+ 1 I Br (xo, zo) \ EI > crn+ 1
Proof. For
r
< zo, we define
U1 (r) = IBr (xo, zo) n EI U2 (r) = I Br (xo, zo) \ E I
S1 (r) = .non (8Br (Xo , Zo) n E) S2(r) = .non CBBr (xo, zo) \ E)
As in the previous proof, the minimality of E and the fact that Br lies entirely in n for r < Zo give (by estimating ,/ (E \ Br) and ,/ (E U Br) respectively) : pe E, Brexo , zo» < Sl (r) + CUI (r) pe E, Br(xo, ZO» < S2 (r) ,
CAPILLARY DROPS ON AN INHOMOGENEOUS SURFACE
1 87
which together. with the isoperimetric formula yields U1 (r) n+l ::; U2 (r) n+l ::;
Since U: (r)
=
2Iln+ l (Sl (r) + CUI (r)) 2Iln+ S2 (r) , 1
S ( r) , Gronwall's Lemma gives the result.
o
i
4.3. Unconstrained minimizers. We now complete the proof of Proposi tion 4. 1 : Let G denote the projection of E onto {z = O}. As a consequence of Lemma 4.4, there exists Po = OIEI n�l such that if x E G, then
IE n rpo (x) 1 2: Cp�+ l Consider the familly {Bpo (x) I x E G}. We can extract a subfamilly Bpo (xj ) finite overlapping still covering G. In particular, j > C(n) � )E n rpo (xj ) 1 j > C(n) L P�+ l . j This implies that the subfamilly contains at most IEIIpo n +1 odicity of /3, we deduce that G has at most radius
with
balls. Using the peri
1Lpo CIEI n�l . Cp�+ l =
So the proof of Proposition 4.1 is complete.
o
5. Properties of the minimizers In this section, we investigate the regularity of the minimizer E. The regularity of the free surface 8E n n is a consequence of classical regularity results for minimal surface. We then determine the Hausdorff dimension of the contact line 8E n {z = O}. 5.1. Regularity of the free surface. Note that if then E also minimizes the functional
E is a solution of (2.2),
among the Caccioppoli subsets of n satisfying I FI = lEI. We can therefore apply the classical regularity results for minimal surfaces (see E. Gonzalez et al. [GMT] ) : THEOREM 5 . 1 . If E minimizes the functional ,/ in 6"(V), then 8* En n is an analytic {n-l}-manifold and HS [(8E \ 8* E) n n] 0 for all s > n - S. In particular, if n + 1 ::; 7, the singular set is empty. =
HIS
L. A. CAFFARELLI AND A.
MELLET
5.2. Hausdorff measure of the contact line. We now establish the follow ing proposition: 5.2. The contact line aCE n {z Hausdorff me.asure in Rn and PROPOSITION
=
O}) in JRn has finite n
-
1
Yr- l (8( E n {z = O})) < CV ::.:t .
The proof relies on the monotonicity formula and a couple of lemma. We start with the following:
LEMMA 5.3. Let X o be a pO'int in Rn. There exists a critical So > 0 such that if
B;: (xo O ) \ {z < Sor} ,
then
c
E,
B0 2 (XO, 0) c E
Proof, Let us renormalize and take r = 1 . The proof relies on a De Giorgi type argument: Consider the vertical cylinders rk
and
=
B;!k (xo, 0)
x
JR,
and r�
B;'k (xo, 0)
=
X
(0, 05)
with rk
=
consider
� + 2-k ,
Vk = IE" n (r � \ r f+ 1 ) 1 where EC = !l \ E is the complimentary set of the drop. Since
we deduce that there is a cylinder r between r k and r k + 1 such that Ji"n(8rr n EC n {O < z < o}) < 2k+ 1 Vk , r
Next, we consider F
=
E u r� (where r�
r r n {O < z < S}), we have IFI > l E I · =
and
< ",f (E ) - peE, r� ) + ,7t"n(8rr n EC n {o < z < S} ) -fJrninJi"n(Ec n fr n {z = O}I) < ",f (E ) - p( E , r�) + Ji"71 (8rr n EC n {O < z < oJ) + max(O, -fJmin)P(E, r�) < ",f ( E) - min(l, 1 + fJrnin)P(E, r�) + Ji"n (8rr n ec n {O < z < S}) where fJmin = inf fJ (x). In particular, inequality (4.4) and the minimality of E yield k min(l, 1 + fJmin)P(E, f�) < £"' (arr n EC n {O < z < o}) < 2 +l Vk . ",f (F)
-
By the isoperimetric inequality, we get
" < I Ec n rr l J.Ln+ 1 (2P( E, f�) + £"'(8rT n EC n {O < z < S})) �l
n�l k +l V +l < C(2 Vk) . k 0 Therefore, 2k+1 Vk 0 if Vo is small enough (i.e. if 05 is small enough) . This Lemma, together with the monotonicity formula (Lemma 4.5) allows us to control the perimeter of E in the neighborhood on the contact line: and thus
'
CAPILLARY DROPS ON AN INHOMOGENEOUS SURFACE
189
5.4. If (xo, 0) E 8E, then for every r, P(E, B;:(xo, 0)) � Crn where C > 0 is a universal constant. Proof. Let V1 (r) = IE n Br(xo, 0) 1, 81 (r) = £n (E n 8Br(xo, 0)) V2 (r) = IBr(xo, 0) \ EI 82 (r) = £n (8Br(xo, 0) \ E). By the previous lemma, either B: (xo, 0) \ {z < tSor} c Ee, or there exists (Yo, zo) E 8E n Br /2 (xo, 0), with Zo � tSor /2. In the first case, we clearly have V2 (r) � crn+ 1 COROLLARY
and
V1 (r)
rv
IE n fr(xo, O) 1 � crn+ 1
by Lemma 4.4. In the second case Lemma 4.5 gives: and
c(tSor) n+ l
VI
�
IE n Boor(YO , zo) 1
V2
�
IBoor (Yo, zo) \ E I � c(tSor) n + 1 .
�
In either case, we deduce
Vi(r) � crn+ 1
(5.1)
i = 1, 2.
Moreover, the isoperimetric inequality gives n v1n: l � /-Ln +1 (81 + P(E, Br(xo, 0))) v2n +l � /1n+ l (82 + P(E, Br (xo, 0))) and It follows that
�
1
�
v1n+ l + v2n+l
-
T n+ l
n (VI + V2) n+l
which yields the result thanks to (5. 1).
�
2/-Ln+ l P(E, B: (xo, 0)),
D
Proposition 5.2 will now be a consequence of the following Lemma:
There exists a constant C such that P(E, {O < z < t}) � CV nn+ ll t Proof of Proposition 5.2. Let Uj Bo (xj ) be a covering of 8{E n {z = O}} with
LEMMA 5.5.
finite overlapping. Then by Corollary 5.4, we have
P(E, Bo (xj )) � CtSn .
But thanks to the finite overlapping property,
L P(E, Bo (xj )) � CP(E, {O < z < tS}) � CV;:+� tS, -l and therefore the number of balls is less than CV nn+1 tS 1 - n , hence the result. Proof of Lemma 5. 5. Let F the set obtained by cutting E at level t: F = {(x, z) E jRn+ 1 ; (X, Z + t) E E} n { Z > O}.
D
190
L. A. CAFFARELLI AND A. MELLET
Then
IE I - IE n {O < z < t} 1 > lEI wnRnt (where Wn denotes the volwne of the unit ball in JRn) since by Proposition 4.1 we have E c rR with R = cv1/(n+ 1) . Thanks to (4.4) we deduce I FI
=
-
/ (E) < f (F) + CV- n+l Rni n < / (F) + CV- n H V n+' i. 1 1
Moreover
/ (E) - / (F)
=
p eE , {O < z < t })
-
J (3(x/r::) [ipE (X, 0)
-
ipE(X, t) ] dx,
but if x belongs to the symmetric difference of E n {z = O} and E n {z = t}, then, going from the slice {z = O} to the slice {z = t}, we must cross BE, and therefore
lipE ( X, O) - ipe (x, t) ldx < peE, {O < z < i}) . We deduce
( 1 sup( I (3I» P (E, {O < which completes the proof. -
z
n- l < t}) < CV n+l t
5.3. Sphere-like minimizers. We can now prove the following result:
o
PROPOSITION 5.6. There exists a spherical cap Bta such that n -l IEt.Bta l < C( V n+I E)a .
Moreover', I Btu I
=
V and the cosine of the contact angle is «(3) .
Proof. We observe that
(5.2)
« (3) - (3(X/r:: » ipE (X, O) dx ::; L
c,
«(3max
-
(3m in ) dx ,
where we sum on all the cells Ci of r::zn that intersect the contact line. Thanks to the n - ll r:: 1 results of the previous section, the number of such cells cannot exceed V n+ n. Since the area of each cell is En , we deduce: n- 1 (3 (X/E) ipE (X, O)dx + CV ..+ , E. Thus, if we introduce the energy functionnal in which the adhesion coefficient (3(x/E) is replaced by its average «(3) :
/o(F) = (J" we have
z>o
z=o
/o (E) ::; / (E) + CV n + 1 E. n-l
Moreover, if Eo denotes the minimizer of /0 , we also have n -I / (Eo ) < /o(Eo) + CV n +' E, hence (using the fact that / (E) ::; / (Eo» (5 .3 )
n-I
/o(E) ::; /o(Eo) + CV n +1 E,
CAPILLARY DROPS ON AN INHOMOGENEOUS SURFACE
and so
E satisfies
(2.6) with b =
191
-
CV nn+ll 10, and Theorem 2.1 gives the proposition.
o
5.4. Proof of Theorem 2.3- (iii) . We conclude this section by proving that this implies Theorem 2.3-(iii): It is a consequence of the following nondegeneracy result: LEMMA 5.7. Let 0 < ry < 1/2, then (i) If there exists (x, z) E E \ B (H'7) p then
I E \ B: I ::::: C(ryp) n+ l . (ii) If there exists (x, z) E B0 -'7) p \ E then
IB: \ EI ::::: C(ryp) n + l . Proof of Theorem 2.3-(iii). Let 0 < ry < 1/2. = b cv:-::;:i 10 � C(pry) (n+ 1 ) /<> then Thus, for 10 �
Theorem 2.1 yields that if
C(V)ry (n+ l ) /<>, Lemma 5.7 implies E \ B0+ '7) p = 0 , and B0 '7) p \ E = 0, -
which gives the last part of Theorem 2.3.
Proof of Lemma
distinguish the case z Lemma 4.5 yields
0
5. 7. We only give the detailed proof of (i). We have to ::::: ryp and z � ryp. When z ::::: ryp, the monotonicity formula
IB'7P (x, z) n EI ::::: C(ryp) n + l , and since B'7P (x, z) n B: = 0 , we have When z
::::: ryp, then the monotonicity formula Lemma 4.4 gives
A simple geometric argument shows that if ry
<
and the lemma follows similarly.
1/2, then
o 6. Stability
We now turn to the proof of Theorem 2.1. Throughout this section, we assume that E is such that (2.5) and (2.6) hold.
1 92
L. A. CAFFARELLI AND A. MELLET
6.1. Schwarz syrnrnetrisation. The first step is to prove that E has almost axial symmetry. More precisely, we prove:
6.1. If E is such that (2. 5) and (2. 6) hold, then there exists a constant C (depending only on R, f3max, l E I and n) and a > 0 (depending only on the dimension n) such that P ROPOSITION
Moreover, E3 also satisfies (2. 6).
Note that (2.6) and Lemma 3.5 implies that peE) < C for some constant depending only on ,6max and lEI. We will u�c this fact throughout this section. Also, thanks to Giusti [Gil , we know that E can be approximated by a sequence of Coo sets Ej such that --> DO .
as j
In particular, proceeding as in Lemma 3.3, we can show that
Thus Ej satisfies (2.5)-(2.6) with Vj = I Ej I instead of V (where Vj V) and OJ instead of 0 (where OJ --> 0) . It is t.hus enough to establish Theorem 2 . 1 for smooth sets E. For the sake of simplicity, we restrict ourself to the 3-dimensional case ( n = 2) which is the most relevant ease from the point of view of applications. All the arguments are however valid in higher dimension, but with sometime different con stants. In particular, the coefficient a is related to the exponent p in the Bonnesen inequalities (see below) and will thus depend on the dimension. -->
We recall that ES denotes the Schwarz symmetrization of E; E" = { (x, z) ; Ixl < PE (Z)} with (6. 1 )
PE (Z )
(6.2)
AE(Z)
with the notation
(71"-1 AE(zW I
ipe (x, z) dx = .Yt'2 (Ez ) ,
Ft = F n {z = t}.
We recall that ES is a Cacciopolli set satisfying I E s l sation preserves the volume) and so (2.6) yields
= lEI
(the Schwarz symmetri
fo (E S ) < /o (E ) ::; /o (ES) + o. Since we deduce (6.3)
130
J ips (x , 0) dx = f30
ipE" (x, 0) dx,
P ( E", O) < P (E O ) < P(ES , O) + 0, ,
To deduce something regarding the symmetry of the drop, we establish the following proposition;
CAPILLARY DROPS ON AN INHOMOGENEOUS SURFACE
1 93
P ROPOSITION 6 . 2 . Let F be a subset in n, such that F c rR, T, and let F* be the set obtained by replacing each horizontal slices of F by a disk with same area and same center of grovity:
with
\ J
ap(z) = Ap z) x'Pp(x, z) dx Then there exists a constant C(R, T, P(F)) such that I F�F* I :::; C(P(F) p ( Fs ) ) 1 /3 . -
Note that by an approximation argument, it is once again enough to establish this proposition for smooth set F. The proof relies on the following Bonnesen type inequality: The Fraenkel asym metry of a set G is defined by
IG n B(a, p) 1 = inf IG�B(a, p) 1 2IB(a, p) 1 IB(a, p) 1 where p is such that l E I = I B(a, p) l. Then R. R. Hall [H] (see also Osserman dimension 2) showed that P(G) ::::: P(B(a, p)) ( 1 + /1 1 A* (G) P ) with p = 2 in dimension 2 and p = 4 in higher dimension. In dimension 2, it follows that there exists a constant /1 1 such that A* (G) = I
_
sup
a
a
[?] in
(6.4) with p = pp, ,\* = ,\} and p(z) = P(Fz, IRn). We also define the following quantity for each horizontal slices Fz of F:
(6.5) where
a(z) is the center of gravity of Fz .
It is readily seen that
Ap(z) :::; A* (FJ +
:(�) A*(Fz).
In order t o estimate F�F* = J Ap(z)H2(Fz) dz, we thus need t o control the isoperimetric default p(z) - 21l"p(z), which will be done using the following lemma (the proof of which is postponed to the appendix) : LEMMA 6.3. Let F be a smooth set in n, let p(z) denotes the (n - I)-perimeter of the slice Fz . Let p(z) be defined by {6. 1} with E = F, and let F8 be the Schwarz symmetrization of F. Then
P(F) ::::: and
J Vp2
+
(21l"pp' )2 dz
1 94
L. A. CAFFARELLI AND A. MELLET
Proof of Proposition 6.2. Lemma 6.3, implies
P(F) - PcPS) >
(p - 2rrp)
VpZ
2rrp dz. + (2rrpp') 2
For any borel set Ao in �+ , we have Ao
1 /2 < So if Ao is the set Ao =
CKTl /2
Ao
(p - 2rrp ) p dz
•
{ z ; Vp2 + (2rrpp')2 > (P(F) - P(F8» - 1 /3 } ,
I Aol < P(F) (P(F) - P(F8» 1 /3, and 1-{2 ( Fz�D (a( z) , p(z » ) dz = r >'F (Z)1-{2 (Fz) dz Iii!.\ Ao R\Ao l Z S / { P F ) P F) ( 1/ < - CKT 2 (P(F) - P(F·» 1/3 :5 CRT 1 /2 (p {F) - p(F8» 1 /3 .
we have
It follows that
l /2 )(P (F) p( F8 » 1 /3 < ) RT R2 F + F (F C( P I I � * _
which gives Proposition 6.2.
o
Proof of Proposition 6. 1. If we tried to apply Proposition 6.2 directly to E, we would still have to determine how ES differs from E*. This means that we need to control the variations of the center of gravity a{z), which appears to be a delicate task. Instead, we make use of a different approach: Let H be an hyperplane in jRn+ 1 , perpendicular to {z = a}, and let H+ and H- be the two half space defined by H. By sliding H in the normal direction, we can ensure that H cuts the set E into two sets E+ = E n H+ and E- = E n H with same volume V/2. If we denote E1 the set formed by adjoining to E+ its reflexion with respect to H, and Ez the set form by adjoining to E- its reflexion with respect to H, we obtain two set El and Ez such that Repeating the same operation with El and Ez, with respect to an hyperplane H' perpendicular to H and {z = a}, we obtained four sets satisfying:
I El l = I Ez l = I E3 1 = I E41 = V , p eEl , 0) + P(E2 ' 0) + P(E3 , 0) + P{E4 , 0) = 4P{E, 0).
CAPILLARY DROPS ON AN INHOMOGENEOUS SURFACE
1 95
Moreover, each of those set is symmetric with respect to the axis H n H' , which we assume to be given by x = 0, and therefore
(6.6)
Now, if we denote
S( E) = we have
r
J{z=O}
IPE (X, 0) dx,
S(E1 ) + S (E2 ) + S(E3 ) + S (E4 ) = 4S (E) ,
(2.6) , we deduce: /o (Ed + /0 (E2 ) + /0 (E3 ) + /0( E4 ) = 4 /0 (E) � 4 /0 (Eo) + 8, for any Eo E 6"(V). If we choose Eo to be a minimizer for /0 , we also /o (Eo) � /O (Ei ) for each i = 1 · · · 4, and thus i = 1 · . · 4. /O (Ei ) � /o(Eo) + 8, But P(Ef ) � P(Ei ) and S(Ef ) = S(Ei); therefore i = 1 · . · 4. P(Ei ) - P(Ef ) � 8, Proposition 6.2, together with (6.6) implies (6. 7) i = 1 · · · 4. I Ei�Ei l � C81/3 and using
have
In order to conclude, we now reconstruct the set
ES = (Ef n H+ n H'+ ) U (E� n H+ n H'- ) u (E3 n H- n H'+ ) U (EX n H- n H'- ) , We clearly have IE�Es l � C81 /3, so we only have t o check that IEs �Es I � C81/3 . To that purpose, we need to show that for a given z, the slices Ei z have almost the same radii for i = 1 · . . 4. This is a consequence of the strict convexity of the square function: We note that the sum of the area of the slices Ei z is equal to four times the area of Ez, and the same is true for the perimeter. In other words: 2:Pi(Z) 4p(z ) 2: Pi(Z) 2 = 4p(z) 2 , where Pi and Pi denote respectively the perimeter of Eiz and the radius of Eiz . =
Thus, if we denote by
J-Li = Pi - 27rPi
Eiz, we have: 4 4 1 1 27rp � P = 4 L Pi � 4 L (27rPi + J-Li ) . i=l i= l
the defect in the isoperimetric inequality for the slice
It follows that
196
L . A. CAFFARELLT AND A . MELLET
and a direct computation shows that:
L I Pi - Pi 1 2 i<j
4 <
L ,u: + 2 L /-liPj ;= 1 i<j 4
<
L /-IT + 2 L /-liPi + 2 L /-li (Pj - Pi) i<j i=1 i<j 4
<
L /-lT + 2 L ,uiPi + 2 L /-l7 + � L I pj - p; j 2 i= 1 i<j i<j i<j
and so
4
4
L IPi - pj l 2 $ C L /-l7 + C L /-li Pi. i <j i=1 Next, lemma 6.3 implies
and so
r /-liPi dz $ C(P(Ei ) - p(Et) )2/3 JI!\Ao with Ao = { z ; Jp2 + (21rpp') 2 > (P(Ei) - P(Et))-1/3}. Since /12 -< /I . p ' ,-, '- '
,
and
/I. p ' < /I . p '
1"" '"
1.
_
,-'1. '1
we deduce IlIt\Ao
IPi - Pj 12dz < C82/3,
'i <j R\Ao
I PT - p21dz < C81/3.
i<j which implies that Since
I Aol
<
C81/3, the result follows.
o
6.2. Sphere-like minimizer. Proposition 6 . 1 shows that E has almost axial symmetry. We now complete the proof of Theorem 2.1 by showing that a symmetric set satisfying (2.6) cannot differ too much from a spherical cap: PROPOSITION 6.4. There exi.5ts a ball Bpo (xo, zo) in IRn +1 such that
IE�( Bpo (xo, zo) n O)1 < C8Ot.
Moreover, Bpo (xo, zo) n o is the minimizer associated to the averaged functional /0 and is fully determined by the conditions
I BI'D n 121
=
V,
cos 'Y = /30 '
197
CAPILLARY DROPS ON AN INHOMOGENEOUS SURFACE
Proof.
For any given ball Bp(x, z), we denote
B; = Bp(x, z) n O
and
B;; = Bp(x, z) \ O .
In view of the result of the previous section, we only have to show that
I Es �B:a 1 � C8"
(6.8)
where ES denotes the Schwarz symmetrisation of assume that V = l. Let L; denotes the wetting surface
E.
Again, after rescaling, we
L; = ES n {z = O}. Note that L; is a disk, so there exists a ball B = Bp(xo, zo) in ]Rn satisfying B n {z = O} = L;, IB n {z > O}I = V. To establish inequality (6.8), we compare the perimeter of B and that of the set obtained by adjoining B;; = Bp(xo, zo) n {z < O} to E8: G = ES U B;; . /o(ES) + yen (L;)(Jo +
/o(B;) + yen (L;)(Jo + Moreover, (2.6) implies and since
r IDipB-p 1 r IDip I . J{ z
J{ z
P
/o(E) � /o(B; ) + 8
/o(ES) � /o(E), we deduce
r
J/itn+l
r
ID ipa l � J/itn+l I DipBp I + 8
or (6.9)
which measure the quantitative loss in the isoperimetric inequality. We recall that if we introduce
' *(F) = 1
A
with p such that IBp(a) 1 constant J.L 1 such that
= IFI,
_
�
s p
IF n Bp(a) 1 IBp(a) 1
then R.R. Hall [H] proved that there exists a
P(F) � P(Bp)(l + J.L1 .A * (F) 4 ).
Moreover, the exponent 4 can b e replaced by 2 i n the case of axially symmetric domains (which is the case of G). Therefore, we have
.A * (G) -< C
P(G) - P(Bp) P(Bp)
Finally, since V = 1, the measure of the wetting surface £n (L;) is bounded, and so the radius p (and therefore P(Bp)) is bounded. It follows that
.A * (G) � C8 1 / 2
L. A. CAFFARELLI AND A. MELLET
198
which gives (6.8) with P instead of Po and a 1/2. =
To complete the proof of Proposition 6.4, it remains to see that I Bt � B:O I < C8, where Bt" is a spherical cap with volume V such that the cosine of the contact angle is /30, First, we notice that Bt satisfies (2.6), and in particular /o(Bt ) < /o(B:O) + C6. (6. 10) So the result will follow from a simple computation: If B: is a spherical cap with radius and contact angle , ( Figure 1), the volume conditions yields r
see
, ,,
," y � . , .' "
,
, ,,
,
, ,,
,
,
r
..
'
-' •
FIGURE 1. r=
(6.11)
which implies /o(R;) = .J(COSi) =
We deduce:
B+ r
1/3 3V 7r (2 + cos,)(cos, - 1)2 7r 1 /3 (3V)2/3 (/30(1 cos i) + - 2) , 3 3 1 (cos, + 2)2/ ( COSi - 1) /
1/3 (3V) 2/3 2n ) /3 ( c ( ' 0 OSi 5/ + 4 .J COS i) (cos , 1) /3 (COS i 2) 3 In particular, the only minimum is reached when cos, /30 , and since 27r l /3 (3V)2/3 > 0 .J (/30) (/30 1)4/3 (/30 + 2)5/3 we have that for every /30 E (-1, 1 ) , there exists a constant C such that /o (B:) - /o (B:O) > CI cos , - /301 2 and so (6,10) gives that the contact angle i of B: satisfies 1 cos , - /30 1 < C6 1 /2 , I
=
_
=
/I
=
_
199
CAPILLARY DROPS ON AN INHOMOGENEOUS SURFACE
any
Finally, (6. 11) implies that P is a Lipschitz function of cos l' on (-1, 1 - 'T/) for 'T/ > O . Since /30 < 1 we deduce that
I p - Po l � C (/3o) I cos l' - /30 1 , if cos l' � (1 + (30 ) /2. When cos l' � (1 + (30 ) /2, then I cos l' - /301
yields
o�
� (1 - (30 ) /2 which
C(/3o ) .
In either case, using (6. 10), we deduce the existence of constants
IB+ !:l.B+ I P Po
<
-
C such that
C01/2 ,
so the proof of Proposition 6.4 is complete.
o
Appendix A. Proof of Lemma 6.3. We recall that F is a smooth set. Let n(x, z) be an extension of the unit outward normal vector to oFz in IRn , and take
lu(zW + Iv(zW
� 1
for all z E R
Then
1 'Ppdivx z
where
1 1 'Ppdivx (n(x)) dxu(z) dz + 1 1 'Pp dx ozv(z) 1 p(z)u(z) 1 A' (z)v(z) dz -
-
A(z)
=
Taking
1 'Pp(x, z) dx. and
v (z)
-
A' (z)
-�====::::;:;;:
Vp2 + A,2 '
we check that lu(zW + Iv(zW � 1 and therefore
1 ID'Ppl 1 Vp2 + (27fpP')2 dz . �
We now prove that there is in fact equality in the case of the Schwarz sym metrisation FS : Let 'P(x, z) be a test function, then:
1 'Pps O'P dx dz 11 !l
uZ
=
D(O,p)
O'P
!l
uZ
dx dz
200
L. A. CAFFARELLI AND A. MEL LET
and if we denote v(B)
=
(cos B, sin B) the unit normal vector, we have
a
a az o
P12..
211"
o
D(O,p)
+
Therefore
-
Thus,
if <1> =
p'
2..
a
-
o
alP dx az
o
p(p')
divx f/J dx dz D(O,p)
- J 102" f/J(pv, z) · vp dfjdz
(f/J,
-
2"
o
p(p')
f/J(pv, z) . vp dB dz p pp'
•
(pv, z) dB dz
and so
o References [B] E. Barozzi, It Problema di Plateau in Domini lllimitati, Rend. Sem. Mat. Univ. Padova, 70 (1983). [BO) E . Barozzi, E. Gonzalez, Least Area Problems with a Volume Constraint, Soc. Math. de France, Aterisque 118 (1984), pp. 33-53. [CM] L. A. Caffarelli, A. Mellet, Capillary drops on an inhomogeneous surface: Contact angle hysteresis and Sticking drop, Calc. [F] R Finn, Equilibrium capillary surfaces. Springer-Verlag, New York, 1986. [Gi] E. Giusti, Minimal surfaces and fUllctions of bounded variation. Birkhuser Verlag, Basel, 1984. [G] E . Gonzalez, SuI problema della goccia appoggiata, Rend. Sem. Mat. Univ. Padova, 58 (1977). [GMT] E. GOllzalez, U. Massari, 1. Tamaruni, On the regularity of boundaries of sets minimizing perimeter with volume constraint, Indiana Ulliv. Math. J . 32 (1983) , 25-37. [H] RR Hall, A quantitative isoperimetric inequality in n-dimensional space, J. Reine Angew. Math. 428 (1992), 161-176. [HM] C. Huh, S.G. Mason, Effects of Surface Roughness on Wetting (Theoretical), J. Colloid Interface Sci. 60, 11-38 [JC] J.F. Joanny, P.G. de Gennes, A model for contact angle hysteresis, J. Chern. Phys. 81 (1984) [LJ] L. Leger, J.F. Joanny, Liquid Spreading, Rep. Prog. Phys. (1992), 431-486.
CAPILLARY DROPS ON AN INHOMOGENEOUS SURFACE
20 1
[T] G. Talenti, The Standart isoperimetric Theorem: Handbook of Convex Geometry (Vol. A) . Editor P.M. Gruber, J.M. Wills. North-Holland 1993. DEPARTMENT OF MATHEMATICS, UNIVERSITY E-mail address: caffarel@math . utexas . edu
OF
TEXAS AT AUSTIN, AUSTIN, TEXAS
78712
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF TEXAS AT AUSTIN, AUSTIN, TEXAS 78712 Current address: Department of Mathematics, University of British Columbia, Vancouver, BC
E-mail
address: mellet@math . ubc . ca
Contemporary Mathematics Volume 446, 2007
Diffusive Lagrangian Transformations, Navier-Stokes Equations and Applications Peter Constantin To
Haim Brezis, with respect and admiration.
ABSTRACT. We present
a description of the Navier-Stokes equations in terms of diffusive Lagrangian t ansfo atio s and relate it to the behavior of the inviscid invariants.
r
rm
n
1. Introduction Stretching and folding of fluid parcels and dissipation of small scale eddies are physical processes in incompressible flows that efficiently mix tracers. The mathematical expression of stretching and folding is present in the Euler equations of ideal, frictionless, incompressible fluids. Smooth solutions of the Euler equations possess an infinite number of constraints embodied in conservation law::;. The incompressibility, the conservation of circulation, the conservation of helicity are such constraints. Folding of flJlid elements is the phenomenological descrip tion of the mathematical fact that the evolution of fluid invariants is obtained by composition with volume-preserving transformations. Stretching of fluid elements is quantitatively expressed by the change in the magnitude of the gradients of the direct or inverse Lagrangian maps. The diffusive dissipation of small eddies is not present in smooth solutions of the Euler equations. The coexistence of the two qualitative features, stretching and folding on the one hand and diffusive dissipation of small eddies on the other, is expressed in the Navier-Stokes equations. Ideal Euler flows preserve the distinct identity of fluid parcels that start out distinct. Whcn the flow is turbulent then distinct regions of fluid are quickly drawn near or apart. Diffusion mixes the identity of parcels, and this can happen no matter how small is the molecular viscosity. What happens 1991 Mathematics Subject Classification. Primary 76D05, 60K40. Key words and phrases. N avier-Stokes equations, diffusive Lagrangian transformations. The a.uthor was supported in part by NSF Grant #DMS-0504213.
203
©2007 American Mathematico.l Society
204
PETER CONSTANTIN
then to the invariants? Can one make any sense of them? \Ve discuss below how one can make reasonably good sense of the invariants, in an appropriate diffusive Lagrangian frame.
2. Euler equations We start by recalling briefly facts about the Euler equations of incompress ible fluid mechanics ( [16] ) , in order to both establish notation and pose I,he main problems. The equations are
Dtu + \lp 0 with \l · u = O. The function u = u(x, t) is the velocity of an ideal fluid at the point in space at the moment t in time. The fluid is assumed to have unit density. The velocity is a three-component. vector, and x lies in three dimensional Euclidean space. The requirement that \l u = 0 reflects the incompressibility of the fluid. The material derivative ( or time derivative along particle trajectories ) associated to the velocity U is ( 2.1 )
=
x
.
(2 . 2)
x
Dtu.
at time t is The Euler The acceleration of a particle passing through in the form equations are an expression of Newton's second law, F = The only forces pmsent in the Euler equations are the pressure forces keeping the fluid incompressible. These forces are not local: the pressure obeys
= ma,
-
\lp
Dtu.
-£lp = \l . (u · \lu) = Tr {(\lU)2 } = OiOj (UiUj) . In the whole space, with vanishing velocity boundary conditions
p = RiRj(UiUj)
where £l ) are lliesz transforms. ( We always sum repeated indices, unless we specify otherwise. The pressure is defined up to a time dependent constant; in the expression above we made the choice of zero average pressure. ) Differentiat ing the Euler equations one obtains:
R; = ai(
-
-
G = (\lu)
1
2
DtG + G2 + (R ® R) {TrG2} = 0
p
is the matrix of derivatives. We used the specific: expression for where written above for the whole space with decaying boundary conditions. Thls equa tion is quadratic and it suggests the possibility of ( infinite gradient ) singularities in finite time, by analogy with the ODE 0 . In fact, if one replaces Dt + by at , then in the resulting equation
%t G G2 = atG + G2 + (R ® R) {TrG2 } = 0
one does indeed get blow up ([2]) , but this blow-up is due to a subtle compressible effect. Blow up in the true Euler equations is elusive, perhaps non-existent, but noticeable gradient amplification is well-documented (numerically and physically) . A physical manifestation of gradient amplification is stretching of vortex lines. Vortex lines are by definition smooth curves whose tangent at every point is parallel to the vorticity vector x 7L at that point. The vortex lines are material lines ( i.e. are carried by the fluid) . By "carried by the fluid" we mean that the vortex lines are obtained from initial vortex lines by composition with the Lagrangian map
w = \l a
H
X(a, f) ,
X(a, O)
= a.
DIFFUS l VE LAGRANGlAN TRANSFORMATIONS
205
For fixed a , the Lagrangian trajectory X (a , t) is a characteristic of u : dX (a , t) = u(X(a, i), t). dt If a = I'o (so) is a vortex line at time t = 0 parameterized by arclength So, then x = I't ( so) with I't ( so ) = X (')'0 (so) , t) is a vortex line at time l. The time independent parameter So,ceases to represent the arclength of the curve at time t, in most cases. The fact that the new curve is a vortex line is true because of the classical Cauchy formula
(2.3) w(X(a, t ) , t) = ('\7 aX(a, t) ) wo (a) . The Cauchy formula is obtained directly from the definition of the trajectories X and the chain rule, at ( '\7 aX) = ('\7 xU 0 X) ('\7aX) the fact that the vorticity obeys the vorticity equation, ,
(2.4)
Dtw
=
('\7xu) w,
and the uniqueness of solutions to ODEs. Ertel's theorem
[Dr, w , '\7]
=
0
also follows from the vorticity equation and implies the fact that vortex lines are material. The unit direction vector field w �= Iw l is defined in the region {x; w( x , t) =1= o}. This region is material. The Cauchy formula (2.3) can be read as
Iwo ( a) 1 � (X (a, t ) , t) = "-----,-:'---'c-'----'-c'('\7x u 0 X) �° ( a) . , -'---' ) ) 1w (X ( a , t t I The chain rule in I't (so) = X(')'o (so ) , t) together with the form of the Cauchy formula above implies that the length element of the vortex line at time t, dSt is related to the length clement at time 0, dso , by IW(')'t( so) , t ) 1 dso, Iwo ho ( so )) 1 which can be expressed by stating that ds Iw l is a material object. The arclength of a vortex line is proportional to the magnitude of the vorticity and therefore stretching of vortex line elements and growth of I w l have identical meaning. The stretching of vortex lines is expressed in the equation obeyed by the mag nitude of the vorticity (2.5) This equation follows from the vorticity equation with a (x , t) = ('\7u) � . �.
The stretching factor a can be expressed ([3] , [4] ) as a principal-value singular integral of Iw I . The kernel of the singular integral has a factor that is depleted when
PETER CONSTANTIN
206
� is smooth, turning the principal-type integral into a less singular operator. This property was e,xploited ( [11]) in conjunction with the well-known Beale-Kato-Majda ( [1]) criterion to give sufficient conditions for the absence of finite time singularities based on the geometry of vortex lines. The eonditions require that the velocity of the flow remain locally bounded and that the vortex direction field � remain locally Lipschitz continuous. Given such conditions, the stretching rate a is finite and consequently the total vortex line length growth is controlled, which implies by the Beale-Kato-Majda criterion that no singularity occurs. See ([9]) for application to the Navier-Stokes equation and ( [14) ) for additional recent developments in this subject. Smooth solutions of the Euler equations conserve total kinetic energy, helicity and circulation. The total kinetic energy is proportional to the L2 norm of velocity. This is conserved for smooth flows. The Onsager conjecture ( [19) , [13)) states that this conservation occurs if and only if the solutions are smoother than, roughly speaking, Holder continuous of exponent 1/3. The "if" part was proved ([8] ) . The helicity integrals are
L hex, t)dx = c
constants of motion, for any vortex tube T. Vortex tubes are open domains in R3 whose boundary is foliated by vortex lines. A helicity density h is
h=u·w
The helicity densities are obviously defined only up to the addition of terms V' . (cpw) for any scalar cp. The circulation around a closed curve I (not necessarily a vortex line) is ,
C-r = If we write ,et) = X(-y, t) then
d d C-r(t)
t
u . dx.
= O.
The Euler equations can be expressed as a first order system using the inverse Lagrangian map: (2.6)
DtA = 0, Dtv = 0, u = W[A, v]
The inverse Lagrangian map A(x, t) = X-l ex, t) starts from the identity map: (2.7)
A(x, O) = x.
A(x, t) is the map that assigns to a spatial position x the label a = A(x, t). This
is label is determined by the requirement that the the Lagrangian particle path X (a, t) starting at t = 0 from a reaches x at time t, x = X ( a, t). The equation DtA = 0 means that, as one follows a particle path, one does not change the place from where the particle originated. The function v is the virtual velocity. As it does not change along particle trajectories, it is obtained by composition
v(x , t) = (uo o A)(x, t) = uo ( A(x , t ) )
DIFFUSIVE LAGRANGIAN TRANSFORMATIONS
207
where uo(a) is time independent: it is the fluid's velocity at t = O. The expression W is the Weber formula ( [20] ) (2.8)
W[A, v] = p { (V'xA) T11) }
The matrix (V'xA)T is the transposed of (V'xA) and P is the orthogonal projection on divergence-free functions. The Weber formula is the mathematical reason why the Euler equations conserve circulation. If one considers the map X (a, t) and pulls back the one-form W[A, Uo 0 A] . dx then one obtains
X(·, t)* {W[A, uo 0 A] . dx} - uo · da = dn This is conservation of circulation:
1 W[A, Uo 0 A] . dx = �oA
Uo da •
for loops homologous to zero. The Cauchy invariant in ideal flow is
( = wo o A. This is the Lagrangian curl of the virtual velocity v
( = V'A
X
v
where V'A is the pull-back of Vx under A . The classical Cauchy formula (2.3) expresses the current vorticity in terms of the Cauchy invariant and the Lagrangian map. One can rewrite it as (2.9)
w
= CIVA, (] C[V'A, (] = (Det(V'A))(V'A) -I(
noting that Det(V'A) = 1. Because of the fact that one can recover velocity from vorticity, it follows that one can write the Euler equations as
DtA = 0, Dt( = 0, u = V' x (-A)-lC[V'A, (]
(2. 10)
The system (2.10) is obtained from (2.6) by taking the label curl V'A x because time and label derivatives commute
[V'A , Dt] = O. The Cauchy formula is obtained from the Weber formula by applying V'A x to it. The systems (2.6) and (2.10) are active scalar systems ( [3], [4)): The functions A, v and ( are transported frozen in the flow of a velocity they create. Knowledge of the active scalars at any instant of time is sufficient to determine the velocity at that instant of time by a time-independent nonlocal equation of state. The formulations can be used to provide local existence ([5)). Using the Cauchy formula, the Beale Kato-Majda criterion becomes: a solution of the Euler system (2.6) or (2. 10) with smooth initial data is smooth on the time interval [0, T] if and only if
T
o
208
PETER
CONSTANT1N
3. Diffusive Lagrangian Formalism The N avier-Stokes equations are D u + Vp = 0, together with the incompressibility condition V . u (3 . 1 )
(3.2)
"
=
Dv = Dv (u, V) = at + U · V
O. The operator Du
-
v�
describes advection with velocity u and diffusion with kinematic viscosity v > O . When v = 0 we recover formally the Euler cquationH (2.1), and Dv1v=0 = Dt. The vorticity w = V x u obeys an equation similar to (2.4): (3 . 3) Dvw = w . Vu. The Eulerian-Lagrangian equations (2.6) and (2. 10) have viscous counterparts ([6]). The equation corresponding to (2.6) is Dv A
=
0, D"v = 2vCVv, u = W[A, v]
(3.4)
The u = W[A, v] is the Weber formula (2.8), the same as in the ca.<>e of v = O. The equation D" (u, V)A = 0 describes advection and diffusion of labels. The right hand side of (3. 4) is given terms of the connection coefficients C;:;�i =
( VA ) - I )
ji
(Oj Ok Am ) .
The detailed form of the virtual velocity equation in (3 . 4) is Duv;
=
2vCriokVm' ,
The connection coefficients are related to the Christoffel coefficients of the flat Riemannian connection in R3 computed using the change of variables a = A (x, t}:
;
( A t) a x , CJ:\(x, t) = -rj7(A( x , t)) X k
The Eulerian-Lagrangian label gradient VA is the pull back of the Eulerian gradient under A.
vt
=
« VA )-l){aj.
The connection coefficients are commutator coefficients,
[V t , OkJ
=
Cri V! ,
and ( note case v = 0)
[Du(u, V), VfJ
=
2VCr;ak V!.
As in the inviscid ease, we associate to the virtual velocity v the Eulerian-Lagrangian curl of v (3.5)
( = VA
X v.
The viscous analogue of the Eulerian-Lagrangian Cauchy invariant active scalar system (2. 10) is
(3.6)
DvA = 0,
Dv(q = 2vG�k {)k (P + vT:(P, u = V X (_L\)-l (e [VA, (])
209
DIFFUSIVE LAGRANGIAN TRANSFORMATIONS The Cauchy formula
l C[V'A, (] = (det(V'A » (V' A) ( . is the same as the one used in the Euler equations, in the form (2.9). The specific form of the two terms on the right hand side of the Cauchy invariant's evolution are ·gckim Gpqk - up '" CkqiP ' (3 .7 ) and -
Tq - EqJ"t'rmp ern p kji er ./kjj " The system (3.4) is equivalent to the Navier-Stokes system. When v = 0 the system reduces to (2.6). The system (3.6) is equivalent to the Navier-Stokes system, and reduces to (2.10) when v = O.
(3.8)
-
. .
THEOREM 1 . The system
Dv (u, V')A = 0,
A(x , O) = x,
Dv (u, V')v = 2vCV'v, v(x, O) = uo (x) , u = P ( V'A?'v) has local smooth solutions. The function u given by the Weber formula above obeys the incompressible Navier-Stokes equations OtU + u . V'u - vflu + V'p = 0, V' . u = O. 4. Diffusive Lagrangian Invariants
One cannot expect conservation of all invariants in the presence of viscosity. In order to describe the behavior of the invariants, let us recall that, under the change of variables A = x + P., the Euclidean Riemannian metric becomes g mn (a ) = (ok A m ) (Ok An)
A(x, t). We say that a function F is diffusively Lagrangian under the Navier-S tokes flow if F = rjJ A and rjJ obeys a a second order linear parabolic PDE with coefficients determined locally by the Euclidean Riemannian metric induced by the change of variables A, and which vanish when v = 0: orjJ(a, t) = v£[g, oal rjJ(a, t) ot where erg , o"jrjJ = g ij 0i2j
a =
0
£[I, oal
=
vfl
so, in variables a the function F obeys a parabolic equation that starts out as the heat equation with diffusivity v. The metric itself is not diffusively Lagrangian. The inviscid invariants become diffusively Lagrangian in the presence of viscosity: THEOREM 2. The virtual velocity v, the Cauchy invariant ( the Jacobian determinant det(V'A) associated to solutions of the Navier-Stokes equations are diffusively Lagrangian. The helicity has a diffusively Lagrangian density. ,
210
PETER CONSTANTIN
The virtual velocity is expressed as vex, t) = v(A(x, t), t)
and the Cauchy invariant as ({x, t) = z(A(x, t), t) .
Using DvA = 0, the operator Dv becomes Dv(j A) = ({at - vgij aiaj ) J) A 0
0
Applying this observation to the equation for the virtual velocity (3.4) one can write with ease the equation for v:
atVi = vgmn 8m 8n Vi - 2vV;mJ 8m vj
with TTmj
Vi
j m kr -9 ik·
_
The equation for z is obtained from (3.6):
8tz q = vgijai8j zq + 2vw�k ak Zn + vT:zP
with
wQk pn l n = _8Q gkrrprp + g kprQ r rm gap .. r Tpq - CqJ,tr mp <>j (ji n
In order to verify the statement about. the helicity we first need to consider the notion of diffusively Lagrangian density. Under the change of variables X f->
a = A{x, t); da = (Det(\7A))dx,
it is natural to define a density pdx to be diffusively Lagrangian if p(x, t) = f(x, t) (Dct(\7 A))
with f a diffusively Lagrangian function. In order to verify that the helicity has a diffusively Lagrangian density it is convenient to use the helicity density w·w
where w = {\7A)Tv. The fact that this is a helicity density follows from the Weber formula (2.8) . From the Cauchy formula (2.9) we have that w
. w = (Det(\7A)) (v . ().
Now, from the definition it is easy to verify that products of diffusively Lagrangian functions are diffusively Lagrangian, and therefore v . ( is diffusively Lagrangian. The definition is reasonable, because in view of vex) = veal, ((x ) = zeal
it follows that T
(v.! . w)dx =
A(T)
V · zda z
expresses the helicity in terms of the solution v . of a parabolic equation which starts as a perturbation of the heat equation with diffusivity 1/.
211
DIFFUSIVE LAGRA NGIAN TRANSFORMATIONS
5. Resettings and vortex reconnect ion The determinant of
A
obeys
D.,(u, \7 ) (log
(5.1 )
When v =
\7
Det(\7A»
=v
{ CL CZ;i } .
0 this equation states the conservation of incompressibility,
as the initial
data is zero, and the evolution does not change this. In the presence of viscosity
numerical calculations
( [1 7)), ([18])
show that viscosity has a dramatic effect, and
the non-dimensional integrals V
tt
{ Ck;s CZ;J
to
dt
can be bOWlded away from zero far vaflishingly small viscosity. This is of course a statement about the development of large second order derivatives of the diffusive Lagraflgian flow map, but it is also related to the physical phenomenon of vortex reconnect ion . One can consider resetting times
o<
tl <
n.
t ..
.. <
_
and a group expansion defined as follows: On each interval
[ti, ti+tl
we solve
D,, (u, \7)v = 2vC\7v, u = P ( \7A)Tv)
with resetting values
A(x, til = x,
ti
ti
v ex, + 0) = « \7A)T v) (x, - 0). The resetting values are such that similar const.ruction based on be the criterion for resetting? small in an analytic norm:
If T =
Ei{tHl - t.)
can prove that
u
is continuous in time. One can implement a
(3.6) instead of (3 . 4). The question is: what should A possibility ( [7]) is to require \7e = (\7 - I to be
A)
sup tE[ti ,tH' !
J e>-Ikl £(k) dk <
t
< l.
then we say that the group expansion converges on
[0, T] .
One
3. ([7]) Let Uo E HI be divergence-free. Let T > 0 be given. Then, there exists T > 0 such that - ti > T and the group expansions con verges on [0, T] if and only if the solution of the Navier-Stokes equations obeys SUP09
and
tHl
This condition is sufficient for regularity. In the numerical calculations of
( [17] )
( [18])
When
the resetting criterion was based on the magnitude of
this magnitude reached a pre-assigned small value Because of the maximum principle in
(5.1)
det(\7A)(x, t) > exp
-v
one has t
ti x
sup
€
Det{\7A).
one marked a resetting time.
{ CL Cz;J
dO'
so that a lower bound for the length of time between two resettings is given interms of the local resetting frequency
l/
{ ct. Ck;i }. It turns out in numerical experiments
PETER CONSTANTIN
21 2
that the resetting frequency is large in periods of topological change, when distinct vortex tubes merge and exchange pieces.
6. Stochastic Representation The deterministic Navier-Stokes equations can be given a stochastic representa tion, based on the fact that the diffusive Lagrangian transformations arise naturally as Fokker-Planck equations associated to diffusion processef:i with drift u and dif fusivity v. The difficulty is conceptual, not technical: the drift depends on the the diffusion process and so the representation is obtained only implicitly, as a fixed point:
THEOREM 4. ( [15]) Smooth NSE is equivalent to the system: dX
=
u(X, t )dt + �dBt, X (O) = a.
with Bt �tandard three dimensional Brownian motion,
A (x, t) with
=
E(X- 1 (x, t))
X- 1 (x, t) the spatial inverse of the random map X
=
X (a, t),
U = W[\7A, v] and with
v ex, 0) = uo, smooth.
VI/ (U, \7)v
=
2vC\7v
This result represents the Navier-Stokes equation exactly but is highly implicit . Clearly, any explicit stochastic representation needs to include interacting particles. Consider also the active system
dX
(6.1 )
=
udt + ../2VdB, X(a, O ) = a, A = X-1 u = EP ( C\7A)T(Uo 0 A) } ,
This system has a direct equation of state producing the deterministic drift from the stochastic map. Although it is a more explicit representation, it f:itill requires a fixed point to be defined. Remarkably, it does repref:ient exactly the solutions of the Navier-Stokes equations:
THEOREM 5. ([15] , [12]) Let U be a solution of the periodic Navier-Stokes equation with smooth initial data cr+1 with r > 1 , r noninteger, and period L. Let U be the solution of the system (6.1) above, with the same initial data. The stochastic system (u. l) has a strong solution for at least a short time, the solution 'U
is smooth and it coincides with u.
References J.T. Beale, T. K ato , A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Commun. Math. Phys. 94 (1984), 61-66. [2] P. Constantin, Note on loss of regu larity for solutions of the 3D incompressible and related equations, Commun. Math. Phys. 106 (1986), 311-325. [3] P. Constantin, Geometric and analytic studies in turbulence, in Trends and Perspectives in Appl. M ath. , L Sirovich ed., Appl. Math. Sciences 100, Springer-Verlag (1994). [4J P. Constantin, Geometric statistics in turbulence, SIAM Review 36 (1994), 73-98.
[1]
DIFFUSIVE LAGRANGIAN TRANSFORMATIONS
213
[5J P. Constantin, An Eulerian-Lagrangian approach for incompressible fluids: local theory, Jour nal of the AMS, 14 (2001), 263- 278. (6) P. Constantin, An I!:ulerian-Lagrangian approach to the Navier-Stokes equations, Commun. Math. Phys. 216 (2001), 663-686. (7) P. Constantin, Near identity transformations for the N avier-Stokes equations, in Handbook of Mathematical Fluid Dynamics, Volume 2, S. Friedlander and O. Serre Edtrs, Elsevier (2003) . [8J P. Constantin,W. E, E. S. Titi, Onsager's conjecture on the energy conservation for solutions of Euler's equations, Commun. Math. Phys., 165(1994), 207-209. [9) P. Constantin and C. Fefferman, Direction of vorticity and the problem of global regularity for the Navier-Stokes equations, Indiana Univ. Math. Journal, 42 (1993), 775. [10] P. Constantin, C. Foias, Navier-Stokes Equations, The U niversity of Chicago Pr"",s, Chicago, 1988. 111] P. Constantin, C. Fefferman, A. Majda, Geometric constraints on potentially singular solu tions for the 3D Euler equations, Comm. POE 2 1 (1996), 559-571. 112] P. Constantin, G. Iyer, A stochastic Lagrangian representation of the 3-dimensional incom pressible Navier-Stokes equations Commun. Pure AppL Math., to appear (2006). 1 13] G. Eyink, Energy dissipation without viscosity in the ideal hydrodynamics, L Fourier analysis and local energy transfer , Phys. D 3-4 ( 1 994) 22 2- 240 . [14] J. Deng, T. Hou, X. Yu, Improved geometric conditions for non-blowup of the 3d incompress ible Euler equations, Commun. POE, to appear (2005) . [15) G. Iyer Phd thesis, The UniverSity of Chicago, 2006. (16) A. Majda and A. Bertozzi, Vorticity and incompressible flow, Cambridge Texts in Applied Mathematics, Cambridge University Press 2002. [17) K Ohkitani, P. Constantin, Numeric..\ st udy of the Eulerian-Lagrangian formulation of the Navier-Stokes equations, Phys. Fluids 15-10 (2003), 3251-3254. [18J K Ohkitani, P. Constantin, Numerical study of the Eulerian-Lagrangian analysis of the Navier-Slokes turb ulence, work in preparation. [19) L. Onsager, Statistical Hydrodynamics, Nuovo Cimento 6(2) (1949), 279-287. [20] W. \Veber, Uber eine Transformation der hydrodynamischen Gleichungen, J. Reine Angew. Math. 68 (1868) , 286-292. ,
,
,
DEPAIITMENT OF MATHEMATICS, THE UNIVERSITY OF CmCAGO. E-mail address: constlDmath . uchicago . edu
Contemporary Mathema.tics Volume 446, 2007
.
'
Some open problems on the control of nonlinear partial differential equations Jean-Michel Caron
It is my pleasure and honor to write a paper in this volume, to celebrate Haim Brezis's 60th birthday. I was very fortunate to do my Ph.D. under his guidance and to collaborate with him later on. Hai'm, with his constant support, has strongly helped me. One of Hai'm's many qualities, which make him a prominent adviser, is his ability to propose numerous, seminal and fascinating open problems. In order to celebrate Hai'm's birthday, I will try to follow in my master and friend's steps, in proposing a few open problems on the control of nonlinear partial differential equations. I will also give some new results on the controllability of the viscous Burgers equation. 1. Semilinear I-D heat equation Let L > ° fixed and f : � boundary control system
---
R be a function of class C2. Let us consider the
Yt Yxx = fey), y(t , 0) = 0, yet, L) = u(t),
(1)
-
where, at time t , the state is y ( t , .) : [0, L) --- R and the control is u(t) E R Let us first give a precise definition of a solution to the Cauchy problem (2) (3 ) (4 )
(t , x) E (0, T) x (0, L), y eO, x) = Yo (x), x E (0, L ), y(t , O) = 0, y et, L) = u(t) , t E (0 , T) .
Yt
-
Yxx =
fey ),
where Yo is given in Loo (O, L) and u is given in LOO (O, T) . The function y is required to be in LOO « O, T) x (0, £)) . Equation (2) has to be understood in the sense of distribution in (0, T) x (0, L). From the requirement that y E Loo « O, T) x (0, L» and (2), we get (5)
(6)
Yt E L2« 0, T) , H-2(0, L» ,
Yxx E L2« 0, L), H - 1 (0, T» .
2000 Mathematics Subject Classification. Primary 93B05, 93C20, 93ClO; Secondary 35K50, 35K55, 35L50, 35Q55. Key words and phrases. Control theory, nonlinear partial differential equations.
215
©2007
American Ma.thematica.l Society
2 16
JEAN-MICHEL CORON
J:.rom (5), we obtain
Y E C([0, T] , H-2(0, L»
(7) From (6), we get
.
y E C([O, LJ, H - 1 (0, T» .
(8)
Hence, for the definition of (3) , we require y (O, ·) = Yo in H-2 (0, L), and, for the definition of (4) , we require
y(-, O) = 0, y (-, L) =
'U
in H-1 (0, T) .
With this definition of a solution to the Cauchy problem (2)-(3)-(4), it is a classical result, that given Yo E LOO(O, L) and U E LOO (O, T), this problem has at most one solution. For initial data whlch are not in Loo (O, L) and weaker definitions of solution, we refer to [7] by Brezis and Cazenave. When f is superlinear there may be no solution. Indeed it is well known that, if, for some Ii 0, yf(y ) � ol y l l+d for I y l large enough, then blow-up phenomena may occur in finite time for the Cauchy problem Yt - Yxx = f ey), (t, x) E (0, T) x (0, L), (9 ) y(t, O) = 0, yet, L) = 0 , t (0, T) , y ( O, x ) = yo(x), x E (O, L) .
>
E
For example, if I(y) = y3, for every T > 0, there are initial data yo in LOO(O, L) and T E (0, T) such that the unique solution to the previous Cauchy problem exists on (0, 'T') X (0, L) for every T' E (0, T ) and satisfies
)��_ I! yl !L=«O.T')X (O.L))
=
+ 00,
for instance [6, 8, 33, 53, 57, 61, 73] and references therein. Concerning the global controllability problem, one of the main results of Fabre, Puel and Zuazua in [27] asserts that if f is globally Lipschitzian then this control system is approximately globally controllable, see also [29, 74] when there is a convective term. Moreover, Imanuvilov has proved in [49, 50] that one has in fact global exact controllability, still if I is globally Lipschitzian. When I is superlinear, the problem of global controllability is still widely open. One may ask if, acting on the boundary of x = L with a suitable control u, one could avoid the blow-up phenomenon. Actually the answer to thls question is negative in general, as it has been proved by Fernandez-Cara and Zuazua in [32]: for some nonlinear functions I satisfying see
I f ( y ) 1 l y l logP ( l + Iyl ) as I yl � +00, with p > 2, and for any time T > 0, there exist initial data which lead to blow-up before time T, whatever the control function U is. Notice however that if '"
I f (y) 1
=
0
( ly l log3/2 ( 1 + Iyl»)
as Iyl
--+
+00,
then the blow-up (which could occur in the absence of control) can be avoided by means of boundary control, see [32J. Nevertheless in the first case where the blow-up phenomenon cannot be com pensated by means of boundary control the situation is not completely desperate: we can always move from any given steady-state to any other one belonging to the
OPEN PRORJ,F;MS IN CONTROL THEORY
217
Harne connected component of the set of steady-states. More precisely, let us define . the notion of steady-state. DEFINITION
(1) if
1 . A function y E dl([O, L] )
is
a steady-state of the control system
d2y f e y) 0 , y e O) = O. + dx2 We denote by S the set of steady-states, endowed with the C2 topology. =
One has the following theorem, which is proved in [20] .
THEOREM 2 . Let Yo and Y1 be two steady-states belonging to a same connected component of S. There exist a time T > 0 and a control function u E H l (0, T) such that the solution y in Loo « O, T) x (O, L)) of
Yt Yxx = f e y) , y(t , O) = 0, y et , L) = u(t) , y eO , x) = Yo (x), -
(10) satisfies y e T, ) = Y1 ' REMARK
.
3. This is a (partial) global exact controllability result. The time needed in the proof of Theorem 2 given in [20) is large, but on the other hand there are indeed cases where the time T of controllability cannot be taken arbitrarily small. For instance in the case where f ey) = _ y3 , any solution of (1 0) in £"" « O, T) x (0, L) ) starting from 0 satisfies the inequality
10 (L - x)4y(T, x)2dx � 8LT, L
and hence, if YO = 0, a minimal time is needed to Teach a. gi'IJen Yl i= O . This result is due to Bamberger [47] , see also [36, Lemma 2.1] . Note that Diaz has found in [23] an obstruction to global controllability even in large time. In [20) it is proved that PROPOSITION 4. If Yo and Yl belong to di.�tl:nct connected components of S, then it is actually impossible to move either from Yo to Yl or from Yl to Yo, whatever the tim.e and the control are. The result of Theorem 2 may be achieved directly by using repeatedly a lo cal exact controllability theorem, see Fursikov-Imanuvilov [36, Theorem 4.4] or Imanuvilov [50, Theorem 3.3] . But in [20) we pre�ent a new controllability strat egy, based on a feedback stabilization procedure, which is therefore more robust to perturbations. It is clear also that this approach may be applied to other problems, without requiring controllability of the linearized system around an equilibrium; see [14] for an example. Let us now turn to our open problems related to the previous results. We take
( 11) As mentioned above, it is one of the cases where, for every T > 0, there are initial data Yo E LOO(O, L) such that Cauchy problem (9) ha.s no solution y E L oo ( (O, T) x (O, L)) . One may ask, if for this special nonlinearity, the blow-up phenomenon can be avoided with a control at x = L, that is
21 8
JEAN-MICHEL CORON
OPEN PROBLEM 1 . Let us assume that (11) holds. Let Yo E LOO (O, L) and let T > O . Does there exists U E LOO(O, T) .mch that the Cauchy problem (2)- (3)-(4) has a solution
Note that it seems to be a case where one cannot use the proof of [32, Theorem 1 . 1] to get a negative answer to Open Problem 1 . One could be even more "am bitious" and ask if our control system is exactly controllable in an arbitrary time, i.e. ask the following question OPEN PROBLEM 2. Let us assume that (11) holds. Let T > O. Let Yo E Loo(O, L) and let y E LOO « O, T) x (0, L)) be a solution of Cauchy problem (2)-(3) (4) for some initial data Yo E L oo (O, L) and some control u E Loo(O, T) . Does there exists u E Loo(O, T) such that the solution y E L= « O, T) x (0, L)) of the solution of Cauchy problem (2)-(3)-(4) exists and satisfies such that y(T, x) = y(T, x), x E (O, L)?
Let us recall ( see Remark 3) that, if one replaces (11) by f ey) :- _y3, the answer to this question is negative. One could also ask questions about the connectedness of S in higher dimen sion. Indeed the proof of Theorem 2 given in [20] can be easily adapted to higher dimension (see also [21]). Let us recall ( see the proof of Proposition 3.1 in [20]) that, for every smooth bounded open subset n JR, c
is a connected subset of 02 (0).
In particular:
One may ask if the same holds in larger dimension.
OPEN PROBLEM 3. Let 0 be a smooth bounded open set of JR2 . Let
Is 6(0) a connected subset of C� (O) ? 2. Viscous Burgers equation
In this section we consider the following control Burgers ( 12)
Yt - Yxx + YYx = 0, (t, x)
E (O, T) x (0, 1). :
For this control system, the state at time t is y (0, 1) JR. We consider two case::; (i) The ca::;e where one requires also that yet, 0) = 0 (Subsection 2.1), ( ii) The case where no more condition is required ( Subsection 2.2). We could also consider the case when one requires (besides (12)) y(t, 1) O. But this case can be reduced to the case where one requires y(t, 0) = 0 just by making the following change of variables x := 1 x and fj := -yo ->
=
-
2 19
OPEN PROBLEMS IN CONTROL THEORY
2.1. Control at x = 1 . In this subsection the control systeIJl considered is Yt - Yxx + YYx y(t, O)
( 13)
= =
0, 0,
(t , x ) E (0, T) x (0, 1), t E (0, T),
where, at time t E (0, t), the t;tate is y(t , ·) : (0, 1) --+ R For the control at time t , we can take, for example yet, 1 ) . But other choices are possible, as, for example, Yx (t , I ) and we prefer to not specify the choice: the control system (40) is just considered as an underdetermined equation . The problem of controllability is the following one. Given T > 0, g E L2« 0, T) x (0, 1 » satisfying ( 13) and yO E L2(0, 1), does there exists y E L2 « (0, T) x (0, 1 » satisfying ( 13) together with (14)
y (O , x )
=
yo ( x) , y(T, x )
=
y (T, x ) , x E (0 , 1 ) ,
REMARK 5. Throughout Section 2, the equation ( 15)
Yt - Yxx
+ YYx
=
0 , (t, x) E (0, T) x (0, 1),
stands, as usual, for ( 16)
Yt - Yxx +
=
° in V' «(O , T)
x
(0 , 1 » ,
where D' « (O, T) x (0, 1» denotes the set of distributions in (0, T) L2 « 0, T) x (0, 1 » satisfies (16), then ( 17)
x
(0, 1 ) . If y E
Yt E L2« 0, T), H-2 (0, 1 » ,
Yxx E L2 ( (0, 1), H-1(0, T» .
(18) From (1 7), one gets
Y E C([O , T] , H-2 (0, 1 » .
(19) From (18), one gets
Y E C([O, 1] , H- 1 (0, T» .
(20)
Hence, by (20), the boundary condition y(t , O)
has to be understood, as usual,
as
Y (-, O)
=
=
0, t E (0, T)
° in H-1 (0, T) .
Similarly, using now (19), (14) has to be understood, as usual, as yeO, . )
=
Yo in H-2(0, 1), y (T, ·)
=
Y(T, ·) in H-2(0 , 1 ) .
We use this convention, and similar usual conventions, throughout the whole paper. (We have already used such conventions in Section 1.) Let, for T > 0, W�,2 « 0, T) that
x
(0, 1 ) be the set of y E L2 ( 0, T) x (0, 1 » such
Yt and Yxx al·e in L2 ((0, T) x (0, 1)).
One has the following result of local controllability, due to Fursikov and Imanuvilov [37, Theorem 6.1].
JEAN-MICHEL CORON
220
THEOREM 6. Let T > 0, let i)
E
Wg,2 « 0 , T)
(0, 1» be such that (1 3) holds (for y : = y). Then there exists c > ° such that, for every Yo E HI (O, 1) such that Yo(O) = ° and lIyo yeO, ') I IHl(o,l) < c, there exists y E Wi ,2 « 0. T) (0, 1» x
x
-
satisfying (13) together with
yeO , x) = yo (x ), yeT x ) = yeT, x), ,
x E (0, 1),
(In fact [31, Theorem 6.1] deals with interior control but the proof can be easily adapted to treat the case uf boundary control. See also [36, Section 5 . 1].) Fernandez-Cara and Guerrero have proved in [30] the following property: what ever is £ > 0, there exists T > ° and Yo E L2(0, 1) with I IYollL2(O,I) � c Sllch that there is no y E L2« 0, T) x (0, 1» satisfying ( 13) together with
y(o, i ) = yo (x) , y (T, x )
(2 1 )
=
0, x E (0, 1 ).
In fact, if '1'(c) is defined as the infimum of the time T such that, for every yo E L2(0 , 1) such that II Yoll p(o,l) � c, there exists y E L2« 0, '1') x (0 , 1» satisfying (13) together with (21), they have got the behavior of T(c) as c 0+: --)
THEOREM 7. (130] .) There exists 8 > ° such that, for every c < {), 8
T (c) � � In(I/c)
1 8 1n(l/c) '
Let us make some comments on this theorem. Nole that Theorem 6 is a local controllability result for a fixed time T > 0. Theorem 7 shows that, as '1' 0+ , the local controllability becomes quite bad: One has, if we denote by c(T) the (best) c; in Theorem 6 for fj := 0, --)
lim £('1') = O. T-+OT We expect that this type of behavior holds for many nonlinear partial differential equation.". In fact, it already holds for many control systems in finite dimension. For example, let us consider the following very simple control system
(22)
x=
(23)
sin(u),
where the control is u E JR and the state is x E JR. The linearized control system at the equilibrium (x , u ) : = (0, 0) is the linear control system
x = u,
(24)
•
where the control is u E R and the state is x E JR. The linear control system (24) is controllable. Hence, by a classical theorem (see, e.g., [16, Theorem 3.8]) the nonlinear control system (23) is locally null-controllable in arbitrary t ime T 0 , 0, there exists c;(T) > 0 such that, for every a E R such that Le., for every T lal � c(T), there exists u E LOO (O, T) such that the solution x E C([O, '1'] , IR) of the Cauchy problem
>
>
(25) (26 ) satisfie!:l
(27)
x
=
sin(u{t» ,
x(O) = a, x ( T) = O.
OPEN PROBLEMS IN CONTROL THEORY
221
(In fact, for this very simple nonlinear control system, this null�controllability result is obvious.) From (25) and (27) , one has T I sin (u( t» ldt � T. Ix(O) 1 � o
Hence, taking a = c:(T), one gets
c:(T) � T. In particular (22) holds. The reason for this phenomenon is not mysterious and is the following one. Let us start with a :;tate which is not O. In order to steer the linearized control system from this point to 0 in small time, we need to usc large controls. However, if one uses large controls, the linearized control has nothing to do with the nonlinear control system and the nonlincarity is going to play a crucial role. In this example, the nonlinearity is in the control. Nevertheless, the same phenomenon holds for the control affinc system Xl = sin(x2 ) ' X2 = u,
where the control is u E lR and the :;tate is ( Xl > X 2 ) E IR? (For this control system, the nonlinearity is in the state.) In conclusion, the nonlinear terms play a crucial role in general for t.he behavior of e(T) as T 0+. Concerning the global null controllability, it has been proved by Fursikov and Imanuvilov in [36J that, for every Yo E L2(0, 1), there exists T > 0 depending on Yo such that there exists y E L2« 0, T) x (0, 1» satisfying (13) together with (21) . The next theorem, which, as far as we know, seems to be new, tells us that one can choose T independent of Yo (and so there exists C > ° such that T(r) � C for every r > 0). ---+
THEOREM 8. Ther'e e.rists T > 0 such that, for every Yo E L2(0, 1), there exists y E L2« 0, T) x (0, 1 » satisfying (13) together with (21). The key lemma to prove Theorem 8 is the following one. LEMMA 9. Let T' > 0 and let y E L2« 0, T') (28)
+ YYx = 0, (t, x)
x
(0, 1» be such that
(0, T') X (0, 1), y (t, 0) = y et, 1) = 0, t E (0 , T').
Yt - Yxx
(29)
Then
-I
(30)
t
-x
�
y(t, x)
E
x
�
t ' (t , x) E (O, T' )
x
(0, 1 ) .
Let us prove this lemma. Let T' > 0 and let. Y E L2 « 0, T') x (0, 1» be such that (28) and (29) hold. By density of Loo (O, 1) in L2 (0, 1) we may assume that y(O, ·) E LOO(O, L). Let B > 0 he such that
(31) Let
c:
> O. Let y
E
yeO, x) COO ([O, T'J
y(t, x) _
:=
x
� B, x E (0, 1).
[0, 1]) be defined by
Be + x
c+l
,
Vet, x) E [0, T ] I
x
[0, 1].
222
JEAN-MICHEL CaRON
Then (see (29) for (33) and (3 1 ) for (34)) Yt - Yxx
(32) (33) (34)
+ YYx = 0, in [0, T']
x
[0 1 ] ,
0 = yet, 0) � yet , 0), 0 = yet, 1 ) � y et , 1), t E (0, T') , y ( O, x ) � y (O, x), x E (0, 1).
From (28) , (32), (33) , (34) and the maximum principle for parabolic equations, one gets y(t, x) � y(t, x) �
(35)
Bf: + X
, (t, x)
E (O, T ) I
x
(0, 1).
t Letting f: 0+ in (35) one gets the second inequality of (30) . Finally the first inequality of (30) follows from the second one applied to the function (t , x) E (0, T') X (0, 1) t--> - yet , 1 - x). This concludes the proof of Lemma 9. .....,
REMARK 10. Inequalities (30) are independent of the viscosity: They hold with (28) replaced by Yt - VYxx + yYx = 0, whatever is v > 0. When v = 0, they are the Oleinik inequality (see, e.g. [71, (2.32)] or [55, (3.31)]), which hold for enf1'opic solutions of Yt + YYx = 0.
These inequalities hold for much more general nonlinearity, even with nonlinear viscosity adapted to the flux (see, e.g. , [26]). Let us go back to the proof of Theorem 8. Note that (30) implies that
(36)
1 lIy (T , X) IIL2(O, 1 ) � " T I
By Theorem 7, there exists v > ° such that, for every Yo E L2 (0, 1) such that II Yo llu(O , l) � v, there exists y E L2 « 0, 1) x (0, 1)) satisfying (13),
(37) (38)
yeO , x ) = yo(x ) , y ( l , x) = 0, x E (0, 1), y (t, O) = 0, t E (0, 1).
Then, taking T' = l/v and using also (36), on gets that Theorem 8 holds with T : = T' + 1 = ( l/v) + 1 . In contrast with Theorem 8 one has also the following results, due to Diaz [24, Theorem 1], and to F'ursikov and Imanuvilov [37, Lemma 6.2 page 59], which give obstructions to global (approximate) controllability in a uniform time for the (viscous) Burgers control system (13).
THEOREM 11. [24 , Theorem 1] Let Yo E L OO ( O, 1 ) . Then there exists C > ° such that, for every T > ° and for every solution y E L OO « O, T) x (0, T)) of (1S) satisfying y (O x) = Yo (x), x E (0, 1), one has
,
ye t, x) �
C I-x
, (t, x) E (0, T)
x
(0, 1).
OPEN PROBLEMS IN CONTROL THEORY
,
223
THEOREM 1 2 . [37, 6. 2 59J There exists I< > ° such that, for every T > ° and for every solution of the control system (13),
Lemma page
d 1 (39) ( 1 - x)6yt (t: x)dx < I<: dt 0 where y+ (t, x) := max(y(t, x), 0) . 2.2. Control at x = ° and x = 1.
considered is
In this subsection the control system
Yt - Yxx + yyx = 0 , (t, x) E (O, T)
(40)
(0, 1 ). t y(t, ' ) : (0 1 ) -> R. (y(t, 0), yet, 1)). x
For this control system, the state at time is For the control , at time t , we can take, for example But other choices are possible, as, for example, (y(t, 0), y,,(t, 1)) and we again prefer to not specify the choice: the control system (40) is just considered as an underdetermined equation, as in Subsection 2 . 1 . Let us first emphasize, that it follows from Theorem 12 that for a fixed time T > 0, one cannot steer the control system (40) from ° to a state z with arbitrary large value for 101 ( 1 - x)64 (x)dx. This in contrast with what happens for the control system (40). Indeed, one has the following theorem, which seems to be new, THEOREM 13. For every T > 0, there exists M > ° such that, for every C E lR with ICf � M, there exists y E L2 « 0, T) x (0, 1)) satisfying (4 0) and such that
y(O, ·) = 0, y(T, ') = C.
(4 1 )
Let us give a proof of this theorem based on the Hopf-Cole transformation. This transformation tells us that, if is a solution of the heat equation w
Wt - Wxx = °
which does not vanishes, then y : = -2wx/w satisfies the (viscous) Burgers equation Yt - Yxx + yYx = 0.
Hence this transformation allows to get controllability results for the (viscous) Burgers equation from controllability results for the heat equation. This has been previously used by EI Badia and Ainseba in [25] . From classical controllability results on the heat equation (see Section 3 for references) one has the following proposition
PROPOSITION 14. Let 7 > 0. Then there exists I< := I< (7) such that, for every '{J LOO (O, 1), there exists z = Z'T ('{J) E LOO« 0, 7) x (0, 1)) such that Zx E
L2« 0, 7)
E
x
(0, 1 )) ,
(42) (43) (44)
Zt - Zxx = 0, (t, x) E (0, T)
(0, 1), z (O , x) = '{J (x), Z (7, x) = 0, x E (0 1 ) X
,
,
Let T > 0. Let us choose 71 > ° and T2 > 0 in such a way that T = 71 + 72 . Let us2 also point out that, if y E L2((0, T) x (0 , 1 )) satisfies ( 40) and (41), then y E L « 0, T) x (0 1 ) ) defined by yet, x) := -yet, 1 - x ) satisfies also (40) as well as ,
y(O, ')
=
0, y(T, · ) = -C.
JEAN-MICHEL CaRON
224
Hence, we may assume that C < O. Let h E Loo « O, T) requIrIng •
x
•
(0, 1 ) ) be defined by
ht - hxx = 0, (t, x) E (0, Td x (0, 1 ) , h(t, 0) = 0 , h(t, 1) = 0 , t E (0, TI ) , h (O , x )
= e 1c 1 / 2 - e I C l x /2
,
x E (0, 1).
By the maximum principle for paraholic equations, (45)
Note also that hx
(46) Let w
E
0 ::;; h(t , x ) ::;; e1c1 /2 , (t, x) E ( O, TI ) x (0, 1 ) . L 2 «0, T) x (0, 1 ) ) . Let us extend h to h , T] x [0, 1] by
h(t, x) : = ZT2 (hh , · ) )(t - TI , X) , (t , x) E ( TI T) x (0 , 1) . L OO « O T) x (0, 1)) be defined by 4 / w et , x ) : = e IC l2 t/ e lc lx 2 + h (t, x) , (t, x ) E (0, T) x (0, 1). ,
E
(47) One has
,
Wx E L 2« 0, T) x (0, 1)), Wt - Wxx = 0, (t, x) E (0, T) x (0, 1 ) ) .
(48) (49)
Let us cheek that, if ICI is large enough,
wet, x ) �
(50) By (4 5) and (47) ,
wet , x)
1,
(t , x) E
(a, T)
� e ICI ' t / 4elcl x/2 � 1 ,
(t, x)
x
(0, 1 ) .
E (0, TI )
By ( 44) , ( 15) , (46) and (47) , we have, on h , T) x (0, 1 ) ,
x
(0 1 ) . ,
2 C 1 1 117 (t , x) � e t/ 4 - K (T2) llh(Tl , ·) IILOO(O.1) � e l C l Td4 K ( T2 ) e 1 c 1/2 2
_
.
Henee (50) indeed holds if 101 is large enough. Therefore, if 101 is large enough,
y := -
2117x
w
E
L2 ( ( O, T)
x
(0, 1 ) ) .
Moreover, by the property of the Hopf-Cole transformation recalled above, (49) and (50), (40) holds. Finally it follows easily from our construction that, for every x in [0, 1 J , x/ 2 ) C I C l x/ 2 _ i 2 elcl 1 / x + e (e yeO, x) = -2 = 0, x E ( 0 1 )
y eT, x)
=
e1c1/2 (e IC I2T/4 e 1 c 1 x/ 2 ) " -2 2eICI2T/4 eIClx/2
,
,
= C, x E (0, 1).
Hence (41) also holds. This concludes the proof of Theorem 13. 15. One can provide a different proof of Theor·em 1 3 by proceeding in the following way: Take, for t E [0, T/ 2] y(t, 0) = y ( t, 1 ) = C. Then straightfor ward estimates show that Ily(T/2, ·) - CIILOO (O,I ) < ", -1 exp( -",C2 T) , for a suitable 1) independent of C > 1 and 0 < T < 1 with C2 T > 1 . Then one concludes by using an analogous of Theorem 6 for the control system (40) with fi = C, but with a careful estimate of the dependance of f. on C and T similar to the ones obtained in [19] . This new proof is more complicated, but is also much more flexible. In REMARK
,
225
OPEN PROBLEMS IN CONTROL THEORY
particular it works for many other equations where the Hopf- Cole transformation cannot be used. It also shows that, with the notations of Theorem 1 3, there p..'Eists A > ° such that one can take M = AIT (this can also been obtained with our Hopf- Cole 's approach).
Looking at Theorem 6 and at Theorem 13 leads naturally to the following open problem. 4. Let T > ° and C E IR. Does there exists y E L2 « 0, T) (0, 1 » satisfying (40) such that y(O, ') = 0, y(T, · ) = C? OPEN P ROBLEM
x
Recently Guerrero and Imanuvilov have got in [46] the two following theorems, which provide uncontrollability results for the control system (40).
16. There exist T > 0, yO E HI (0, 1) and c > ° such that, for every y E L2« 0, T) x (0, 1» satisfying (40) and THEOREM
y (O, ') one has
= yO,
lI y (t , ' ) II H1 (U,1) �
C.
17 . For evenJ T > 0, there exist yO E H 1 (0, 1), yl E H1 (0, 1 ) and c > ° such that, for every y E L2« 0, T) x (0, 1» satisfying (4 0) and THEOREM
y(O, .)
one has
= yO ,
I l y(t, ' ) - y I IlHl( ,l) � C. O
18. For the nonviscous Burgers equation (i. e. without the term -Yxx), results have been obtained by Ancona and Marson in [2] and by Horsin in [48] . Note REMARK
that it is surely important to have a good understanding of the contmllability of the nonviscous Burgers equation if one wants to study the global controllability of the viscous Burgers equation in small time. Indeed for large states and fixed time or fixed states and small time the nonlinear term y Yx is surely a key term compaTl�d to Yxx ' Similarly, it is by using results on the controllability of the Euler equations [11, 13, 42] that one gets global results for the contmllability of Navier-Stokes equations (see [12, 18] and section 6).
3. Singular optimal control: A linear I-D parabolic-hyperbolic example
Let ( T, L, M) control system f: ,
(51)
E (0, + 00)3
X
R.
We consider the following parabolic linear
(t, x) E (0 , T) Yt - f: Yxx + M Yx = ° y(t , O) = u(t) , yet, L) = ° t E (0, T), y (O , x) = yO (x) x E (O, L),
x
(0, L) ,
where the state is y(t, ·) E L2(0, L) and the control is u(t) E R We are interested in the dependence of the cost of the null controllability of system (51) with respect to the four parameter::; T, L, M. It is already known that, for every T > 0, the control system (51) is controllable to the null final state at time t = T. This means that, for every yO E L2 (0, L) and E,
226
JEAN-MICHEL CORON
for every (e, T, M) E (0, +00)2 X JR, there exists u E L2(0, T) such that the (weak) solution of (51 ) satisfies yeT, . ) == O. This result is due to Fattorini and Russell [28, Theorem 3.3] . See also Imanuvilov [49, 50], Fursikov-Imanuvilov [37] , and Lebeau Robbiano [56] for parabolic control systems in dimension larger than 1. ( The last reference does not explicitly deal with transport terms; but the proof given in [56] can perhaps be adapted to treat transport terms. ) For yO E L2(0, L), we denote by U (c, T, L, M, yO) the set of controls u E L2 (0, T) such that the corresponding solution of (51 ) satisfies y(T, · ) = O. Next, we define the quantity which measures the cost of the null controllability for system (51) : K(e, T, L, M)
(52)
e
-->
:=
lIyO I I L' (O , L ) � l
sup
{ min{ll u ll £2 (O,T) : u E Ute, T, L, M, yO) } }.
In this section we are looking for estimates on K(e, T, L, M), in particular as 0+ . Let us point out that simple scaling arguments lead to the relations K(e, T, L, M)
(53 )
=
1
M / 1 2 c, a T, a L, 1 / K 1 /4 a 2 a
and (54)
for every (a, e, T, L, M) E (0, +00)4
X
JR.
In order to understand the behavior of K(c, T, L, M) as e -> 0+ , it is natural 0+. This is to look at the limits of trajectories of the control system (51) as c done in the following proposition, proved in [19] . --t
PROPOSITION 19. Let (T, L, M) be given in (0, +00)2 xJR' and let yO E L2 (0, L) . Let (Cn)nEN be a sequence of positive real numbers which tends to 0 as n -> +00. Let (U,,)nEN be a sequence of junctions in L2(0, T) such that, for some u E L2(0, T), (55)
Un
converges weakly to U in L 2 (0, T)
as
n
--t
+00.
For n E N, let us denote by Yn E CO([O, T) ; H- 1 (0, L» the weak solution of (56) (57) (58)
Ynt - Cn Ynxx + M Ynx = 0, (t, x ) E (0, T) x ( O, L) , Yn (t, O) = un (t) , Yn (t, L) = 0, t E (0, T),
O Yn (O, X) = y (x ), x E ( O, L ) .
For M > 0, let Y E CU([O, T] ; L2(0, L)) Yt + M yx = 0 y (t, O) = u(t) (59) yeO, x ) = yO (x ) and, for M <
be the weak solution of (t , x) E (O, T) t E (0, T), x E (0, L),
0, let y E CO ([0, T] ; L2(0, L)) Yt + M yx
=
0
Then Yn
-'
(O, L),
be the weak solution of
(t , x) E (O, T) x (O, L ) , t E (0, T),
y(t, L) = 0 O y(O, x ) = y ( x) x
(60)
x
E
(0, L) .
Y weakly in L2 ((O, T)
x
(0, L)) as n -> 00.
.'
OPEN PROBLEMS IN CONTROL
It follows directly from Proposition LIIMI, one has
19
227
THEORY
that, for every (T, L, M) with T <
lim K(e, T, L, M) = +00. t-+O+ For M > 0, the divergence of K stated in (61) is of course due to the fact that the time T under consideration is less than the characteristic control time needed for the limiting hyperbolic control equation (59). Let us emphasize that for M < 0, there is not control for the limiting equation (60) The following theorem, proved in [19] , gives a lower rate of divergence in (61).
(61)
.
JR*
,
THEOREM 20. There exists G\ > 0 such that, for every (e , T, L, M) E (0, +00)3 X
(62) K(e, T, L, M) � C1
l -3/2 -1/ 2 2 e T L M /2
(63)
1
L3 M3 + -".e3
exp
M (L 2c:
_
TM)
_
7r2eT L2
if M > 0,
C3 /2 T- l/2 L2 1 M1 1/2 T MI ( I 2L T I M I) 7r2 c: K(e, T, L, M) � C1 exp if M < o. 2c: L2 1MI'-3 L 3 '-0;-1 + -' 3 e Concerning upper bounds of K(c:, T, L, M), let us point out that • If M > 0 and T > LIM, the control u = 0 steers any state to 0 in time T for the control system (59), where the state is y(t, ·) E L 2 (0, L) and the control is u(t) E R ( This means that, if M > 0, T > LIM and u = 0, then, for the function y defined in Proposition 19, y(T, ·) == 0 whatever is yO E L2 (0, L).) • If M < 0 and T > LIIMI then, for the function y defined in Proposition 19, y(T, ·) = 0 whatever is yO E L2 (0, L ) . This could have given the hope that, for every CT, L, M ) E (0, +00) 2 X JR" with T > L/IMI, _
_
K(c:, T, L, M)
(64)
-->
0
as
e
-->
0+ .
As shown by Theorem 20, this turns out to be false for M < 0 and T E (LI I M I , 2L/IMI). Our next theorem, proved in [19] , shows that (64) holds if TIMI/L is large enough.
THEOREM 2 1 . There exists C2 > 0 such that, for every (e, T, L, M) E (0, +00) 3 x R* with
(65) •
If M > 0 and
(66) (67)
then
L T � (4 . 3) M'
3
-
4
2 2TM 2.61 1 3L -
•
228
JEAN-MICHEL CORON •
If M
<
0 and L
T ;' (57.2)
(68)
IMI '
then
3 -
4
•
The proof of this theorem relies on a decay estimate of the solution of (51) when u 0 and a new Carleman estimate for the solutions of the adjoint system of (51). Of course, one can already see in [19] , the numerical constants appearing in Theorem 21, namely 4.3, 2.61, 57. 2 and 18.1, are not optimal. OUf open problem in this section is on improvements of the constants 4.3 and 57.2. In particular, it is natural to ask =
as
OPEN PROBLEM 5. ASS'(Lme that
M > 0, TM > L. Is it true that
K(c:, T, L, M)
Assume that
M
Is it true that
<
--+
0 as c:
-+
O?
0, TIMI > 2L.
K (c:, T, L, M)
--+
0 as c:
--+
O?
4. Schrodinger equation
Let I (70) (7 1 ) (72)
=
( - 1 , 1).
\Ve consider the following Schrodinger control system. 'If;t = i'lf;xx + iu(t)x'lf;, (t, x) E [0, T] x I , 'IjJ (t , - 1 ) = 'If;{t, 1) = 0 , t E (0, T) , Set) = u(t), D(t) = Set), t E (0, T) . •
•
This is a control system, where, at time t the state is ('IjJ(t, ), Set), D(t» E L2 (1, q x JR JR with II 1'IjJ (t , x)1 2dx 1 for every t E [0, T], the control is u(t) E JR. This system has been introduced by Rouchon in [69] . It models a non relativistic charged particle in a I-D moving potential well. The control is the acceleration of the well. We want to control at the same time the wave function 'If; of the particle, the speed S and the position D of the box. Let us first recall some important trajectories of the above control system when the control is O. Let •
x
.
•
(73)
'If;n (t, x) :=
In (73) , An := (mr) 2 /4 are the eigenvalues of the operator A defined on D(A) := H 2 (I, q n HJ (I, q by A
-
"
=
229
OPEN PROBLEMS IN CONTROL THEORY
and the functions r.pn are the associated eigenvectors: r.pn (x) := sin(mrx/2), n E 1'1*, if n is even, (74) (75) r.pn (X) := cos(mrx/2), n E 1'1* , if n is odd. Let us also introduce the following notations. § := { r.p E L2 (I , C) ; 11r.p llu (J,q = I}, HloP, C) := { r.p E H7 (I, C); r.p(2 k) (-I) = r.p(2k) ( 1 ) = °
for k = 0, 1 , 2, 3} . With these notations, one ha.s the following result, proved in [5] .
THEOREM 22. For every n E 1'1" , there exists TJn > ° such that, for every nO, nf E N", for every ('l/Jo, So, Do), (1Pf , Sf, Dj ) E [§ n H(O) (I, C)) x lR x R with (76) (77) there exists
( 78 ) (79)
(80)
a
I I'l/Jo - r.pno ll H7 + I So l + I Do l < TJno ' II 'l/Jj r.p", llw + I Sf l + I Dj l < TJnf ' time T > ° and ('l/J , S, D, u ) such that 'l/J E CO([O, T] , H2 n HJ (1, C) n C1 ( [0, TJ , L2 (1, C) , u E HJ (O, T), S E Cl ([O, TD , D E C2 ( [O T]) , -
(70) , (71) and
(81)
(82)
,
(72) hold,
('l/J (O) , S(O ) , D(O» = ('l/Jo, So, Do), ('l/J (T) , S(T) , D(T) ) ( 'l/Jj, Sf , Dj). =
(83)
Thus, we also have the following corollary.
CO ROLLARY 23. For every no, nf E W" , there exists a time T > ° and ('l/J, S, D, u ) satisJ'l/ing (78) to (81) such that ('l/J (O), S(O), D(O» = ( r.pn , O , O) ('l/J(T), S(T), D(T» = a
(r.pn" O, O) .
,
Note that, if one does not care of S and D and if (no , nf ) = (1, 1), Theorem 22 is due to Beauchard [4] . There are lea.st three points that one might want to improve in Theorem 22: (i) Remove the a.ssumptions (76) and (77) on the initial data and the final data, (il) Weaken the regularity a.ssumptions on the the initial data and the final data, (iii) Estimate the time T of controllability. Concerning (i), on can propose, for example, the following open problem. OPEN PROBLEM 6 . Let ('l/Jo, So, Do ) , ('l/Jf, Sj, Df) E [§ n H(o) (I, C l] Does there e.'lits T > 0 and ( 'l/J , S, D, u) such that (78) to (83) hold?
x
lR x
R
Concerning (il), one can propose, for example, the following open problem. OPEN PROBLF,M 7. Let
Hro) ( I, C) ; = {r.p E H7 (1, C); r.p(2k) (0)
=
r.p(2kl (l)
=
0 for k
=
0, 1}.
Let ('l/Jo, So, Do), ('l/Jf , Sf, Df) E [§ n HroP' C)] x lR x R Does there exits and ('l/J , S, D, u) with u E L2(0, T) such that (78) to (83) hold?
T>°
230
JEAN-MICHEL CaRON
REMARK The regularity conjectured in Open Problem 7 comes from the regularity for the controllability of linearized control systems (see, in particular, [4, 336]). 5, 862] and [5,
24. Theorem page Proposition 2, page Concerning (iii) , since the speed of propagation for the Schrodinger equation is infinite, one could expect that the following small-time local controllability property. For every c > 0, there exists T} > ° such that, for every (,00, So , Do ) , (,0f, Sf, Df ) E [§ n H[O) (I, C)] x R x R with 11'1/10 - 11'1 (0, ' ) IIH7 + I So l + I Do l < T},
(84) (85)
II'I/If - 'I/Il ( c, ' ) llw + I Sf l + I Df l < T} ,
there exists ('1/1, S, D, u) such that (78) to (83) hold with T = c and Il u IIHl (U,e)
:(
e.
However this is false. Indeed, one has the following theorem ([16, Theorem 9.8]; see also [15, Theorem 1 .1] for a weaker result) . THEOREM 25. Let T > ° be such that
T<
(86)
2
2 ' 7r
Then there exists 10 > ° such that for every D =F 0, there is no u E L2 ((0, T) ; ( - 10 , e» such that the solution ('I/I, S, D) E CO ([O, T) ; H6 (I; C) x CO([O, T]; R) x C1 ( [0, Tj ; R) of the Cauchy problem
(87) (88)
'!/Jt
(89) (90) (91) satisfies
(9 2 ) (93)
=
i'!/Jxx + iu(t)x'!/J, (t, x) E (0, T) x I,
'I/I(t , - 1 ) = '!/J (t, 1 ) = 0, t E (0, T) , Set) = u(t), D(t) = Set), t E (0, T), '!/J(O, x) = '!/J] (0, x), x E I, S(O) = 0, D(O) = 0, t/J(T, x) = t/Jl (T, x), x E I, SeT) = 0, D(T) = D,
where '!/Jl is defined in (73) and (75) .
Looking at Theorem 25, two natural open problems can be raised: OPEN PROBLEM 8. Is it possible to remove assumption (86) in Theorem 25'1
Looking at Theorem 22 and Theorem 25, one may also ask if one has �mall-time local controllability with large controls, i.e.,
OPEN PROBLEM 9. Let T > 0. Does there exist "I > ° such that, for every (t/Jo, So, Do ) , (t/Jf' Sf, Df) E [§ n H[oP, C)] x R x R with
II'!/Jo - t/Jl (O) lIw + I So l + I Do l < 17, II'I/If - '11'1 (T) IIH7 + I Sf l + I Df l < "I,
-
OPEN PROBLEMS IN CONTROL THEORY
2
31
there exists ('IjJ, S, D, u) such that
'IjJ E CO( [O, T] , H2 n HJ (J, IC)) n 01( [0, TJ , L2 (I, IC) ) , u
E HJ ( O, T) ,
S E C1 ( [0, T] , JR) , D E C2 ( [0, T] , JR) , (70), (71) and (72) hold,
('IjJ(0) S(O), D(O)) = ('ljJo, So, Do), ,
('IjJ(T) , S(T), D(T))
= ('ljJf, Sf, Df)?
Concerning the proof of Theorem 22, let us just mention that it relies on the following methods 1 . The return method, a method we have introduced in [ 1 0] for a stabilization problem and in [1 1] for a controllability problem. This method has been used also for the controllability of the following partial differential equations: Euler equations of incompreHHible fluids; see [1 1, 13] , and [42] by Glass, Navier-Stokes equations of incompressible fluids; see [12, 18], and [40] by Fursikov and Imanuvilov, Nonviscous Burgers equation by Horsin in [48] , Saint-Venant equation (or shallow water equations) in [14], Vlasov-Poisson equation by Glass in [43] , SchrOdinger equations by Beauchard in [4] , I-D isentropic Euler equation by Glass in [44] . 2. Quasi-static deformations. This method has been introduced in [14] . It has also been used for A heat equation in [20], A SchrOdinger equation by Beauehard in [4] , A wave equation in [2 1 J , Couette flows by Schmidt and Thelat in [70]. 3. Power series expansions. This method has been introduced in [17] for a Korteweg de Vries equation. See also Section 5. 4. Nash-Moser process, a method introduced in control theory by Beauchard in [4]. • •
•
•
•
• •
•
•
•
•
5. Korteweg-de Vries equation
Let us consider the following Korteweg-de Vries control system (KdV)
Yt + Yx + Yxxx + YYx = 0, t E (0, T), x E (0, L) y (t , O) = y e t , L) = 0, t E (0, T).
For this control system, L ° is given, the state at time t is y(t, ' ) : [0 , L] JR, and for the control at time t one can take, for example, u(t) = yx (t, L) E JR. Sec [67] by Rosier for other boundary conditions than the one appearing in (KdV). The Korteweg-de Vries equation serves as a model for propagation of small amplitude long water waves in a uniform channel. We are interested in the local controllability of (KdV) around 0. Rosier has proved in [66] that the control system (KdV) is locally controllable around ° provided that the length of the Hpatial domain is not critical: >
-+
232
JEAN-MICHEL CORON THEOREM 26. (166,
Theorem l . 3] . )
Let T > 0, and let us assume that
' 2 + 12 '1 . '" L rf. N := 211" J + J ; J, / E N' 3
(94)
Then ther-e exists r > 0 such that for any Yo, YT E L2 (0, L), with II Yo I i L'(U, L ) < r and I i YT I iL2(0,L) < r, there exists Y E G([O, TJ, L2 (0, L» n L2 ( (0, T) , Hi (0, L» satisfying (KdV) such that y(O, ·) = Yu and y (T, ·) = YT . Let
k E N* .
Note that
an infinite number of
k,
L = 2k11"
is in
N
(take
j=I=k
in (94» . However, for
one still has local controllability for these critical lengths.
Indeed, the following theorem is proved in
[17] .
THEOREM 27. Let T > 0 and k E N \ to} . Let us assume that (95) (i + 12 + jl = 3k2 and (j, l) E (N \ { 0}) 2 ) =} (j = I k) . =
2k1l". There exists r > 0 such that, for every Yo , YT E L2 (0, L) with Let L II Yo l i ,,2 (0,L) < r and Ii YT IIL2(0 ,L) < r, there exists y E G([0 , TJ , L2 (0, 2k11"» n L2 ( (0, T), Hi (O, L» satisfying (KdV) such that y(O, ·) Yo and y(T, ' ) = YT . =
REMARK 28. Assumption (95) holds for k � 6. There are an infinite number of positive integers k such that (95) does not hold: see [16, Remark 8.2) . How ever, there are also infinitely many positive integers k such that (95) holds: see [16, Proposition 8.3)}. Unfortunately, we have forgotten assumption (95) in [1 7, Theorem 2] . Thi,s is a mistake: the proof of [17, Theorem 2) requires (95). =
When
L = 2k1l"
0
the linearhled control system of (KdV) around
Yt + Yx + Yxxx = 0 , y(t, O) = y(t, 2k11") = o.
(KdVL) It has been shown by Rosier in
is
[66] that this linear control system is not controllable.
To prove that the nonlinear term
yYx
gives the local controllability, a first idea
could be to use the exact controllability of the nonlinear equation around nontrivial stationary solutions proved in
[22)
and to apply the method introduced in
[14]
(that
is, use the return method together with quasi-static deformations) . But, with this method, we could only obtain the local exact controllability in proving Theorem 2 7 one uses in
[17]
large
time.
For
a different strategy that we briefly describe
now. One first points out that in this theorem we may assume that
Yo = 0:
this
follows easily from the invariance of the control system (KdV) by the change of variables
r
= T - t , � = 2br - x.
Then one uses the following result for the linear
control system.
THEOREM 29 . (166,
Remark
Let T > 0 and let H :=
3.6
ii)l ')
y E L2 (0, 2k11"),
Let k E N \ to} be such that (95) holds.
2k1r
o
y ( l - cos(x»
dx
=0 .
For' every (Yo, YT ) E H x H, there exists y E G([O, T] , L2(0, 2k11"» nL2((0, T), HI (O, 2k1!"» satisfying (KdVL) such that y(O, ·) = Yo and y (T, . ) = YT . Next one can see that the nonlinear term
yYx
allows us to "go" in the two
directions ± ( l - cos(x» which are missed by the linearized control system Finally one derives Theorem
27 by means
of a fixed point theorem.
(KdV L) .
OPEN PROBLEMS IN CONTROL THEORY
233
For the other critical lengths, the situation is more compLicated: there are now four noncontrollable (oriented) directions of the linearized control system around ° and there are Less explicit than ±(1 cos(x)). But our guess is that Theorem 27 also holds for all the other critical Lengths, that is the answer to the following open problem should be positive. -
10. Let L > ° and T > 0. Does there exist r > ° such that, for every YO, YT E L2(0, L) with IIYo l l u (O,L) < r and IlyT il L2 (0,L) < 1', ther'e exists Y E G(IO, TJ, L2(0, L)) n L2« 0, T), HI (0, L)) satisfying (Kd V) such that y(O, · ) = 110 and y (T, · ) = YT . OPEN PROBLEM
All the previous results are local controllability results. Concerning global controllability results one has the following result due to R.osier [65J .
T
THEOREM 30. For every Yo E L2(0, L), for every YI E £2(0, L), there exist > ° and 11 E G([O, T], L2(0, L)) n L2« 0, T), HI(O, L)) satisfying
such that y(0,
.
Yt + Yx + YXXI + YYx ) = Yo and y eT, . ) = YI ·
=
° in D'«
O, T) x (0, L))
Note that in this theorem (1) One does not require y(t, 0) = yet, L) = 0, (2) A priori the time T depends on Yo and YI . lt is natural to see if one can remDve these restrictions. For example, one may ask
T > 0 , Yo E L2(0, L) and YI E L2(0, L) . Docs there exists Y E G(IO, T], L2 (0, L)) n L2« 0, T), HI (O, L)) satisfying (KdV) such that OPEN PROBLEM 1 1 . Let L > 0,
y(O, · )
= Yo and y (T, ') = Yl ?
If one does not care of T (Le., if we allow T to be as large as wc want, depending on Yo and Yl), a classical way to attack this open problem is the following one (this is the way which is already used by R.osier to prove Theorem 30). Step 1 Use the reversibility with respect to time of the equation to show that one may assume that YI = 0. (In fact this part holds even if one deals with the case where one wants T > ° to be small.) Step 2 Use a suitable stabilizing feedback to go from yO into a givcn ncighborhood of O. Step 3 Conclude using a suitable local controllability around y : = O. Step 1 indeed holds (perform the change of variables (i, x) : = (T - t , L - x)) . Let us assume that L
0, t E (0, +00), E (0, L) , (97) y ( t, O) = yet, L) = Yx(t, L) = 0 , t E (0, +00), y(O, x) = yo (x), x E (O, L). (98) where Yo is given in L2(0, L) and one requires (96)
(99)
Yt + Yx + YXII + YYx
=
x
234
JEAN-MICHEL CORON
y is smooth enough, ly(t,xWdx -l yx (t, OW·
Straightforward computations shows that, at least if
d L dt 0 Hence one can expect that the Cauchy problem (96)-(97)-(98) has a solution. This turns out to be true: It has been proved by Perla Menzala, Vasconcellos and Zuazua i n [64] that the the Cauchy problem (96)-(97)-(98) has one and only one solution. (See also [17, Appendix A] .) However, it is still not known (even for L r;. N) if the following open problem has a positive answer. =
OPEN
PROBLEM
( 1 00)
12. Does one have, faT every Yo E L2(0, L), lim r L 1 y (t, xWdt = 0? Jo t->+oo
There are available results showing that (100) holds if there is some interior damping, more precisely with (96) replaced by
Yt + Yx + Yxxx + YYx + a(x)y
where a E LOO (O, L) is such that
=
0,
t E (0, +(0), x E (0, L),
a(x) � 0, x E (O L )
the support of a has a nonempty interior. ,
,
Tbese results are in [64] by Perla Menzala, Vasconcellos and Zuazua, in [63] by Pazoto, and in [68] by Rosier and Zhang. 6. Navier-Stokes equations In this section we present some results and open problems on the Navier-Stokes equations of incompressible viscous fluids. Let us introduce some notations. Let I E {2, 3} and let f! be a bounded nonempty connected open subset of ]Rl of class ceo. Let fo be an open subset of f af! and let f!o be an open subset of 12. We assume that
:=
(101)
The set f 0 is the part of the boundary f and 120 is the part of the domain 12 on which the control acts. The fluid that we consider is incompressible so that the velocity field satisfies div On the part of the boundary f\fo where there is no control, the fluid does not cross the boundary: it satisfies
y
(102)
y = O.
n
where denotes the outward unit normal vector field on f. Besides (102), the fluid satisfies on f\fo, , some extra conditions which will bc specified later on. For the moment, let us just call by Be all the boundary conditions satisfied by the fluid on r\fo. For simplicity, often we omit to specify the regularities of the functions considered (one can find these regularities iII the papers mentioned) . Let us introduce the following definition.
(102))
(including
OPEN PROBLEMS IN CONTROL THEORY
235
DEFINITION 31 . A trajectory of the Navier-Stokes control system on the time interval [0, T] is a map y : [0, T] x f/ Rl such that, for some function p : [0, T] x f/ lit, ---+
---+
(103)
Yt
-
t1y + (y 'V) y + 'Vp = 0 in [0, T] .
div Y = 0 in [0, T]
(104)
x
TI,
x
(f/\f/o),
(105 ) y(t, . ) satisfies the boundary conditions Be on 1'\1'0, "It E [0, TJ . J.-L. Lions's problem of controllability is the following one: let T > 0, let Yo : f/ lItl and Yl : f/ lItl be :mch that div Yo = 0 in 0, (106) ---+
--t
(107) ( 108)
div Yl
=
° in 0,
Yo sat.isfies the boundary conditions BC on n1'o ,
(109)
Yl satisfies the boundary conditions Be on n1'o , does there exist a trajectory y of the Navier-Stokes or the Euler control system such that
y(O, · )
(1 10)
=
Yo in f/,
and, for an appropriate topology -see [59] , [60]-,
(111)
y(T, ) is "close" to Yl ? .
That is to say, starting with the initial data Yo for the velocity field, we ask whether there are trajectories of the Navier-Stokes control system considered which, at a fixed time T, are arbitrarily close to the given velocity field Yl . If this problem has always a solution one says that the control system considered is approximately controllable. Note that (103), ( 104) , (105) and (1 10) have many solutions. In order to have uniqueness one needs to add extra conditions. These extra conditions are the controls. We will see below a way to replace (111) in order to recover a natural definition of (exact) controllability of the Navier-Stokes equations. Let us now specify the boundary conditions BC. Three types of conditions are considered: • Stokes' boundary condition, • Navier's boundary condition, • Curl condition. The Stokes boundary condition is the well-known no-slip boundary condition
(112) which of course implies (102). The Navier boundary condition [62] imposes, condition (102) , which is always assumed, and
( 11 3)
l7y.T + (1
-
(7)ni
TJ
=
0 on 1'\1'0,
236
JEAN-MICHEL CORON
where (1 is a constant in [0, 1), n = (n1 , , nl ) and T = (7 1 , . , Tl ) is any tangent vector field on the boundary r. In (113) we also have used the usual summation convention. Note that the Stokes boundary condition (112) corresponds to the case (1 = 1 , which we will not include in the Navier boundary condition considered here. The boundary condition (113) with a ° corresponds to the case where the fluid slips on the wall without friction. It is the appropriate physical model for some flow problems; see [41] for example. The case (J E (0, 1) corresponds to a case where there the fluid slips on the wall with friction; it is also used in models of turbulence with rough walls; see, e.g. , [54]. Note that in [9] F. Coron has derived rigorously the Navier boundary condition (113) from the boundary condition at the kinetic level (Boltzmann equation) for compressible fluids. Let us also recall that Bardos, Golse, and Levermore have derived in [3] the incompressible Navier-Stokes equations from a Boltzmann equation. Let us point out that , using (102), one sees that, if I = 2 and if T is the unit tangent vector field on 80. such that ( T, n) is a direct basis of JR2, (113) is equivalent to (1y.T + curl Y = 0 on r\ro, •
.
.
.
.
=
with 0'
E COO(r; JR) defined by
2(1 - a) lI: (x) - a \..I r ) . _ _ vX E , 0' ( x .-
( 1 14)
1 - 0'
,
where II: is the curvature of r defined through the relation 8n/8T = 11:7. Finally the curl condition is considered in dimension 2 (l = 2). This condition is condition (102), which is always assumed, and curl y = ° on
( 115)
r\ro.
It corresponds to the case (J = ° in (114). Due to smoothing properties of the Navier-Stokes equations, one cannot expect to get y(T, ') = Yl , at least for general Yl . For these equations, as for the heat equation and the vi::;cous Burgers equation considered above, the good notion for exact controllability is not passing from a given state Yo to another given state Yt : the good definition for exact controllability is passing from a given state Yo to a given trajecto'T7J y. This leads to the following, still open, problem of exact controllability of the Navier-Stokes equation with the Stokes, or Navier, or curl condition.
O PEN
13. Let T > 0. Let y be
a
tmjecto'T7J of the Navier-Stokes control .5y8tem on [0, T] . Let Yo : 0. -> R!) satisfying (106) and (108). Does there exist a trajecto'T7J y of the Navier-Stokes control system on [0, T] such. that
( 1 16) (117)
PROBLEM
E 0., = y(T, x) , \Ix E m
y(O, x) = Yo (x) , 'Ix y(T, x)
Let us point out that the (global) approximate controllability of the Navier Stokes control system is also an open problem. Related to the Open Problem 13 one knows two types of results • local results, • global results, which we briefly describe in the next ::;ubsections
OPEN PROBLEMS IN CONTROL THEORY
237
6.1. Local results. Let us introduce the following (informal) definition. The Navicr-Stokes control system is locally exactly controllable if, for every (smooth enough) trajectory y on [0, TJ of the Navier-Stokes control system, there exists (' > ° such that, for every ( Hmooth enough) Yo : D Il�J satisfying (106) , (108) and -t
I Yo - yeO , ) 1 < € , for a suitable norm I . I , there existH a trajectory y of the Navier-Stokes control sYRt.em un [0, TJ satisfying (1 16) and ( 1 17) . Then one has the following results (with precise choices of I . ! and of smoothness requirements) . .
THEOREM
32. The Navier-Stokes control .�ystem is locally exactly controllable.
The case of the curl condition has been obtained by Fursikov and Imanuvilov in [39J for l = 2. The case of the Navier boundary condition in every dimension has been obtained by Guerrero in [45J . The case of the Stokes condition has been obtained by Imanuvilov in [51 J and [52J (see also the prior work by F'tU'sikov in [34] as well as the paper [31] by Fernandez-Cara, Guerrero, Imanuvilov and Puel for less regular spaces) . 6.2. Global results. Let d E CO(D; JR) be defined by
d(x) = dist (x, r)
=
Min { I x - xi i ; x' E r}.
In [12] tbe following theorem is proved. T HEOREM 33. Let (10"/) hold and
T > 0, ld Yo and YI in HI (D, JR2) be such that (1 06) and
curl Yo E LOO(D) and cud YI E L= (D), Yo ' n = 0, YI n = 0 on l' \ ro o .
Then, thc7'c e.xists a sequence « yk , Pk ) ; k E N) of solutions of the Navier-Stokes control system on [0, TJ with the Navier boundary condition (1 13) such that (1 18) ( 1 19)
yk and Pk arc of class COO in (0, TJ yk E L2« 0, T) , H2 (D?) ,
(120)
y�
(121)
Pk E L2« 0, T) , HI(D» ,
and such that, as k (122)
-t
+00,
E
x
D,
L2 « 0, T), L2(D?),
l dl1 l yk (T, ·) - Yl ! -t 0, 'riM > 0,
0, lI yk (T, . ) - YI l l w-l . � (rl) and, for all compact K included in D u ro , (124) Il yk (T, · ) - YI IIL� (K) + II curl yk (T, ·) - curl YI II L� (K)
(123)
-t
�
O.
In this theorem, W -I,OO (D) denotes the usual Sobolev space of first derivatives of functions in LOO (D) and II Ilw- l'�(rl) one of its usual norms, for example the norm given in [1, Section 3 . 12, page 64] . As in the proof of the controllability of the 2-D Euler equations of incompress ible inviscid fluids [11, 13J , one uses the return method in order to prove Theorem
JEAN- MICHEL CORON
238
33. Let us recall that it consists in looking for a trajectory of the Navier-Stokes control system y such that y (O, ·)
(125 )
=
y (T, ·)
=
0 in 0,
and such that the linearized control system around the trajectory y has a control lability in a "good" sense. With such a y one may hope that there exist.s y close to y satisfying the required conditions, at. least if Yo and Yl are "small" . Note that the linearized control system around y is -
-
(126)
at
oz -
6.z + (y . V) z + (z . V) y + V7r = 0 in [0, T] x (0\0 0 ) ,
(12 7)
divz = 0 in [0, T]
z.n = 0 on [0, T]
(128) ( 129)
�Z.T
+ curl z
=
x
x
0,
(r\ro ) ,
0 on [0, T]
x
(T'\ro).
In [38, 35] Fursikov and Imanuvilov have proved that this linear control system is controllable (see also [58] for the approximate controllability). Of course it is t.empting to consider the case y = O. Unfortunately, it is not clear how to deduce from the controllability of the linear system ( 126) with y = 0, the existence of a trajectory y of t.he Navier-Stokes control system (with the Navier boundary condition) satisfying ( 1 10) and ( 1 1 1) if Yo and Yl are not small. For this reason, one does not use y = 0 , but y similar to the one constructed in [13] to prove the controllability of the 2-D Euler equations of incompressible inviscid fluids; these y are chosen to be "large" so that, in some sense, "6." is small compared to "(y . V) + ( . 'V')y" . REMARK 34. In fact, with the y we use, one does not have {125}: we have only the weaker property
(130)
y(O, ')
=
0, y (T, ·) is "close " to 0 in 0 .
However the controllability of the linearized control system around y is strong enough to take care of the fact that y(-, T) is not equal to 0 but only close to O. A related observation can be found, in a different setting, in the paper [72] by Sussmann. With the notation of [72], y plays the role of �. in the proof of [72 Theorem 12] and the Euler control system plays for' the Navier-Stokes control system the role played by 9 for F. ,
Note that (122) , ( 123) , and (124) are not strong enough to imply
( 131)
ii yk (T . ) - Yl iiv(!!) ,
-t
0,
i.e. to get the approximate controllability in L2 of the Navier-Stokes control system. But, in the special case where ro = r, (122), (123), and (124) are strong enough to imply (131). Moreover, gluing together the proofs of Theorem 32 and of Theorem 33, one gets
THEOREM 35. ([IB, 40] ') The Open Problem 13 has a positive answer when ro = r .
239
OPEN PROBLEMS IN CONTROL THEORY
6.3. Control on the tangential component of the velocity on the boundary. It would be interesting to know if the Navier-Stokes equations are controllable even if the control is only on the tangent component of the velocity on roo For example, in the case of the Stokes boundary condition, this leads to the following open problem, where ro is assumed to be a non empty open subset of r. OPEN PROBLEM 14. Let T > 0. Let fi
and P E L2 « 0, T), Hl (n» be such that
E £",° « 0, T) , Hl (n» n L2 « 0, T), H2(0»
iit E L2«
Let Yo
E
0, T) , L2(0» , div y = ° in (0, T) x 0, fi(t, x) = 0, (t , x ) E (0, T) x (r \ r a), fi(t, x) · n (x) = 0 , (t, x) E (O , T) x ro, Yt - �Y + ( Y 'V)ii + 'Vp = 0 in (0, T) x n. .
Hl (Oi Rl ) be such that
div Yo = ° in 0 , Yo(x) = 0, Vx E r \ ro, yo(x) · n(x) = 0, Vet, x) E [0, TJ
x
roo
Does there exist y E Loo « O, T), Hl (0» nL2« 0, T) , H2(0» andp E L2 « 0, T), Hl (n» such that Yt E L2« 0, T) , L2 (n» , div y = ° in (0, T) x 0, y(t, x) = 0, (t, x) E (O, T) x (r \ ro ) , yet, x) . n(x) = 0, (t , x) E (0, T) x ro, Yt - b.y + (y . 'Il)y + 'Ilp = 0 in (O, T) x 0, yeO, x) = Yo (x) , x E 0, y(T, x) = fi(T, x), x E m
Acknowledgement. We thank Eduardo Cerpa and an anonymous referee for useful comments.
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91 405
ORSAY, FRANCE , WWW . MATH . U-PSUD . FRrCORDN/
E-mail address:
Jean-Miche l . CoronCimath. u-psud . fr
Contemporary Mathematics Volume 448, 2007
The I-Laplacian, the oo-Laplacian and Differential Games Lawrence C. Evans
This paper considers the p-Laplacian PDE for p = 1, 00 and some interesting new game theoretic interpretations, due to Kohn-Serfaty [K-S] and to Peres-Schramm-Sheffield-Wilson [P-S-S-WI. ABSTRACT.
1. Introduction Thill informal, expository paper discusses the degenerate/Ringular diffusion PDE div ( I DuIP -2 Du) = 0 (Ll) in the extreme limiting cases p = 1, p ao , and explains also some interesting game theoretic interpretations due to Kohn-Serfaty [K-S] and to Pems-Schramm Sheffield-Wilson [P-S-S-W] . I additionally lace the exposition with various general comments, taking this occasion to preach a bit about some guiding principles of nonlinear analysis and nonlinear PDE. --->
This is a recapitulation and follow-up to my lecture in Paris during the sum mer of 2004, at the conference in honor of Haim Brezis. It is a real pleasure to contribute a paper to this volume: Haim has long been one of my mathematical heros, starting with my reading as a graduate student of his monograph Operateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert and continuing to this day. Of his countless contributions to our field, I would par ticularly like to single out Haim's service as editor of the major Birkhauser series Progress in Nonlinear Differential Equations and Their Applications. 2. Degenerate diffusions, extreme cases
2.1 Overview. It seems to me that within nonlinear analysis we should always
be looking for unifying, guiding principles that cut across the boundaries of specific problems. And it is by studying in detail appropriately simple models that we often stumble upon general principles. 1991 Mathematics Subject ClassiJicu.tion. 35J15, 91A20. Key words and phrases. Minimax, nonlinear diffusion PDE, differential games. Supported in part by NSF grant DMS-05004 52.
245
@2007
American Mathema.tical Society
246
LAWRENCE Co EVANS
The p-Laplacian for I
is the nonlinear PDE
< p < 00 .
A greatly studied such model problem
(2. 1) holding in some open set U C Rn , where we write .6.pu := div( IDuIP- 2 Du) for the p-Laplacian. This is of course the Euler-Lagrange equation for minimizing the functional (2.2) subject to given, but here unspecified, boundary conditions. Our quasilinear PDE is elliptic, but is degenerate wherever Du = 0 for 2 < p < 00 and is singular wherever Du = 0 for 1 < p < 2. It is easy to show there exists a minimizer u up belonging to the Sobolev space W1,p, but examples show u ¢ C2 in general. When 2 < p < 00 , K. Uhlenbeck and N. Ural'ceva independently showed that u E C1,e> for some ex = ex(p); and years later I gave another proof in [EI] . The idea of these proofs is roughly as follows: On the set { I Du l > O} the PDE is nondegenerate and the usual strong elliptic estimates apply. On {IDul O} the PDE degenerates, but there we know that in fact the gradient Du = O. (This insight is far less useful for a PDE that degenerates also for other values of Du.) =
=
2.2 Extreme cases. One important principle of mathematics is that extreme cases reveal interesting structure. We illustrate this bit of mathematical faith by putting p = 1, p -> 00 in (2.1).
The I-Laplacian. Setting p
1 in (2.1) is easy and gives us the PDE Du .6.1u : = div -0 I Du l
(2.3)
=
-
,
assuming of course u is smooth and IDul =1= O. The nonlinear operator .6.1 is called the 1-Laplacian. For later reference, we rewrite into nondivergence form:
(2.4)
1 .6.1 u := IDu l
UX,Xj
Since the vector .. v .-
=
O.
Du
IDu l is perpendicular to each level set of u, we see that our I-Laplacian PDE (2.3) describes "isotropic diffusion within each level surface, with no diffusion across different level surfaces." Of course in general a solution of a PDE like (2.4) must be interpreted in some suitable weak sense, in this case as a viscosity solution, as for instance in [E-Sp] .
The oo-Laplacian. Sending
into nondivergence form:
p
-t
00
is trickier. For this, first rewrite (2. 1 )
p I Du l - 2.6.u + (p 2) I Du lp-4ux, ur; UXiXj = O. Divide by IDuIP- 2 (p 2) and send p -t -
-
(2 .5)
00 :
THE I-LAPLACIAN , THE oo-LAPLACIAN AND DTFFEllENTlAL GAMES
247
The nonlinear operator .6.00 is called the 00- Laplacian. Observe that the 00Laplacian PDE (2.5) formally describes "diffusion across different level surfaces, with no diffusion within each level surface." This interpretation is somehow "dual" to that for the I-Laplacian. Again, a solution of (2.5) must rigorously be interpreted as a viscosity solution: see Aronsson-Crandall- Juutinen [A-C-J] for details. The next sections review some of the literature on the PDE
(2.3), (2.5) .
3. More interpretations of the 1- and oo-Laplacians 3.1 The I-Laplacian. Look now at the boundary-value problem u 0 in U .6. 1 (3.1) in au, u=g =
for a given function 9 : aU
-+
R.
Variational interpretation. This nonlinear boundary value problem for
mally corresponds the variational problem of minimizing h [v]
=
subject to the boundary condition that
L IDvl dx
v = 9 on aU.
Geometric interpretation. Now let
fc := {x E e l u(x) = c} denote a level set of u, which we suppose is a smooth hypersurface, with unit normal is v = g:I ' The mean curvature of r
I
c
H
= div(v) = div
•
So our PDE .6.1 u = 0 asserts that "each level surface of u has mean curvature zero" .
Estimates and regularity. The foregoing geometry involves each level set r
c
of 'U, and is consequently unchanged if we relabel the level sets. This means that our PDE is invariant under an arbitrary change of dependent variable: (3 . 2 )
If u solves .6.1 u = 0 and
w
= t/>(u), then .6.1 w = 0 also.
But this invariance is bad for analysis, since it implies that there can be no interior a priori estimate on a modulus of continuity of u or its gradient Du. We discuss next similar insights for the oo-Laplacian.
3.2 The oo-Laplacian. As a kind of dual problem to (3. 1) consider now the boundary value problem
(3.3)
.6.oo u = 0 u=g
in U on au.
Variational interpretation. Assume that 9 : au
R is Lipschitz continu ous. G. Aronsson [AI] has proposed the "L oo variational problem" of finding an extension u such that for each open set V C U, (3.4) ess-supv I Dul < ess-supv IDvl for all v = u on aV. -->
248
LAWRENCE C. EVANS
It
1.1
turns that requiring to satisfy (3.4) is equivalent to asking u to be a viscosity solution of (3.3). The rcally basic problem concerns the uniqueness of functions satisfying (3.4) , which Bob Jensen proved in his important paper [J] . See AronsHoll-Crandall-Juutinen [A-C-J] for more. Geometric interpretation. I am unaware of any geometric interpretation of (3.3) and pose this as an open question. Consider the ODE OUt.
x
= vex)
=
�� (x) , I I 1.1 .
with flow lines normal to the level surface of Is there any purely geometric way to explain the corresponding trajectories? Whatever answer the reader comes up with should in particular explain the flow for Aronsson's solution (3.5), mentioned later. Estimates and regularity. Regularity issues for the oo-Laplacian are very hard. O. Savin [S] has recently made a major breakthrough, proving that if n = 2 ann is the viscosity solution of (3 . 3), then 1L
It
remains an outstanding problem to prove CI-regularity in dimensions n > 3. The Harnack ineqnality states that for each V CC U there exists C C(V) such that if u > 0, then max u < Cmin1L. I proved in [E3] that if u is smooth, then I Dul also satisfies a Harnack inequality: m�x I Dul < CmJn IDul; and this has the remarkable consequence that a smooth, nonconstant solution of (3.3) can have no c,..itical point within U. Aronsson had earlier proved this for n 2 dimensions. However, this conclusion is dead false for general weak ( that is, viscosity) solutions, a fundamental example of which for n = 2 is =
v
v
-
=
1.1
(3.5)
-
XS1 4
x23 , 4
-
which Aronsson found by separating variables. Blow-up limits. Crandall, Gariepy and I have shown in [C-E-G] and [C-E] that for each point E U that lim u(xo+r;x)-u(xo) if the limit v(x) := rj-O exists, rj (3.6) then vex) (a, x) is linear. This is an interesting assertion in view of the basic open question of regularity for weak solutions. Note carefully that (3.6) does not assert to be differentiable at the point since different rescaled, blow-up sequences could possibly converge to different linear functions. Xo
=
Xo
u
THE
THE
I-LAPLACIAN,
oo-LAPLACIAN
249
AND DIFFERENTIAL GAMES
4. Game theory interpretations 4.1 Minimax formulas_ I again indulge myself in this section with a bit of
philosophy concerning nonlinear problems. me
review convexity. A key point is that convexity is a kind of "one-sided" linearity. More precisely, if : �n -+ � is convex, we can write
Convexity. First, let
(x )
(4.1 )
=
max {(a"', x) + ba }
for appropriate a'" E �n , b" E R This representation formula suggests why con vexity is a fundamental hypothesis for many nonlinear problems: linear and affine functions are simple, and "max" is a simple operation. Conseqllent.iy, convex func tions are (relatively) simple to study. General nonlinearities. Suppose now IJI : ]Rn � is an arbitrary, say Lipschitz continuous, function. Then we can write C<
-+
lJI(x) = mjn c.13 (x) ,
where each function CI3(-) is convex. For example, we can touch each point of the graph of u from above by a round cone. But as above we have CI3(x)
=
,
for appropriate a"',13 E ]Rn and
max '"
{ (aQ,,I3, x) + bCt.f:I }
b°"tJ E �; and hence lJI(x) min max {(a"',I3, x) + b"',I3 } .
(4.2)
=
13
,
"
This formula shows that an essentially arbitrary nonlinearity can be written maximin of affine mappings. I found a more explicit formula in [E2] :
(4. 3)
lJI(x)
=
max min
oElRn fJERn
f (DIJI« Ol J
and it turns out that "minimax
n
=
l
-
A)a + )..{3 ), x
-
a) dA + W (a)
as
a
•
,
maximin" for this particular representation.
'With these formulas in mind, we can extend the heuristics above. Linear and affine functions are simple, "max" is a simple operation and "min" is a simple operation. Consequently, since essentially any nice function admits the maximin representationR (4.2) or (4.3), we in principle have a general tool for nonlinear analysis.
Control theory and game theory. There is another way to view these
comments. For convex problems, with max-type structure, we expect to have opti mization or control theory representation formulas for solutions. And likewise for wide elasses of problems with minimax- and/or maximin-type structure, we can expect to find game theoretic iuterpretations. (See for instance [E-Sol . ) In the next sections, we illustrate this principle for the 1- and oo-Lapladans.
4.2 A random "tug-of-war" game_ Following Peres-Schramm-Sheffield Wilson [P-S-S-W] let us introduce for each c 0 a random tw�person, zer�sum game, in which Player I wants to minimize the expected payoff and player wants
>
to maximize the expected payoff. The rules of play are:
Step 1: At each time tk
=
k€, players
I and II flip a fair coin.
II
250
LAWRENCE C. EVANS
Step 2: The winner selects a unit vector v E Rn and moves from the current
position Xk to the new position
Step 3: The game ends when our process first exits U, that is, when x* Xk+l 'I. U. The payoff is then g(x* ) , for a given function 9 : au R Dynamic programming and PDE. The method of dynamic programming =
--+
lets us cast this problem into analytic terms. Let given we start at x = Xo . Then
(4.4)
u' (x)
=
u'(x) denote expected payoff,
1 max u' + min u' . 8B(x.,) 2 8B(x.
The idea is that if player I wins the coin flip, he moves the point on aB(x, E ) where u' is smallest; whereas if player II wins the coin flip, she moves the point on 8B(x, E) where UC is largest. These events occur with equal probability 1/2. (This functional identity (4 . 4) had been previously noted by Oberman [OJ and Le Gruyer [LGJ.) It is not so hard to guess from (4.4) that if u ' --+ u as E 0, then u should solve the boundary-value problem --+
.6.oou = O u=g
(4.5 )
in U on au.
To see this, note that if Du'(x) =I 0, then 8B( x ,,)
max
8B(x,E)
min
u' � u' (x + EDu'(x)/IDu' (x) l)
uE � u'(x - EDu' (x)/IDuE (x) l).
(4.4) implies (4.6) u' (x + EDu' (x)/ I Du' (x) l) + u' (x - EDu' (x)/ID'u'(x) I) - 2u' (x) � O. Divide by E2 and send E 0, to obtain (4.5). Hence the functional equation --+
This very rough derivation is suspicious, since we in fact do not know the functions u' to be smooth, nor that the error term in (4.6) is of order o( (2 ) . Never theless the conclusion is correct, provided we understand u as a viscosity solution: see Peres-Schramm-Sheffield-Wilson [P-S-S-WJ (or [B-E-J]) for more. And very recently Peres and Sheffield in [P-SJ have introduced a stochastic tug-of-war inter pretation of the p-Laplacian operator for 1 < p < 00.
4.3 A deterministic "pusher-chooser" game. Kohn and Serfaty [K- SJ
have introduced a different two-person, zero-sum game, but one with clear similar ities. Again, player I wants to minimize the expected payoff and player II wants to maximize the expected payoff. The rules of play are now: Step 1: At each time tk = kc2 , player I selects a unit vector v E R2 .
Step 2: Player II then selects b E { I , - I } and moves from the current position Xk to the new position
Xk+l ;= Xk + v2Ebv. Step 3: The game terminates when x* = Xk+l 'I. U, and the payoff is t* = time the process exits U.
THE I-LAPLACIAN , THE co-LAPLACIAN AND DIFFERENTIAL GAMES
251
In other words, player I wants to exit from U as quickly as possible, and player II wants to prevent this as long as possible.
Dynamic programming and PDE. As before, let u'(x) denote the payoff,
given we start at
x = Xo . By analogy with (4.4), we have the functional equation u'(x) = 1"1=1 min max {E2 + 1 b=±
(4.7)
u
'
{
x
+
V2Ebv) } .
Again, the interesting question is understanding what happens as € -> O. Note firstly that if Du'(x) f= 0, then Player I �hould certainly select v = ±Du« x).l /IDu« x) l , a unit vector perpendicular to Du'(x). Otherwise, Player II could by appropriately choosing b = ± I force a move to a new position where u' is strictly larger than its value at x. For definiteness, assume player I takes v = Du'(x) l/IDu' (x) l . Then the func tional equation (4.7) gives
and then
our
expanding
u' in a Taylor series about x shows
(4 .8 ) Now assume that the PDE
u'
->
u.
Our dividing
1
IDul2 Since we are in n (4. 9)
(4.8) by E2 and letting
.1 2 .1 Du , D uDu ) (
=
t ->
0 yields
1.
= 2 dimensions, we can rewrite, t,o obtain - IDul�lU = 1 u=O
in U on au.
Our formal derivation is again very suspicious; but the conchlsion is valid, as proved by Kohn and Serfaty, provided we interpret u as a viscosity solution. J.
Spruck and I in [E-SpJ have previously in vestigated the boundary-value problem (4. 9). The geometric interpretation is that the level curves of u evolve by curvature motion, starting at au = {xlu(x) = O}. Consult Kohn and Serfaty's paper [K-SJ for some pictures and more explanation ali to why curvature motion is in fact relevant for their pusher-choose game.
Geometric interpretation.
Questions. In summary, boundary-value problem (4. 9) for the pusher-chooser
(4.5)
for the tug game entails the I-Laplacian, whereas the corresponding problem ot�war game involves the oo-Laplacian. Is there a philosophical explanation for this? Are these problems somehow "dual" ? In view of my previous comments about minimax representations, H is perhaps not so surprising thaL our PDE have game theoretic interpretations, but in what sense are these particular games "natural" ?
252
LAWRENCE C. '"VANS
5. Generalizations
Generalization: other dynamics, running costs. Barron, Jensen and I in [B-E-J] have generalized the Lug-of-war problem by assuming now the dynamics
to be
d:�) d��)
(5. 1 )
= f(X(T) , 1J (T» =
for player I for player II,
-f(X(T) , « T»
for times 0 < T < t" . We also introduce both a terminal payoff and a running cost:
P(1J(') ' « '» := g(x(T:,,» +
(5.2)
o
t*
±h(X(T), O (T» dT.
Here 0 ( ' ) denotes 1J (-) when I iii playing and « . ) when II is playing. The "±" term means that the running payoff is -h(X(T) , 'T}(T» when Player I controls the dynamics, and is h(X(T), ( T» when Player II controls the dynamics. The dynamic programming formula analogous to (4 . 4) for the value function now reads
(,').:�)
uf (x)
1 =
max [u (x + d(x, y» - f.h(x, y)]
2 Ivl< l
'
+ min [uE(x - Ef(x, z» + Eh (x , z)] .
Izl
We make the standing hypothesis that {(f(x, w), -h(x, w» : Iwl S; I} iii a bounded and uniformly convex subset of IRn+ l . So for each p E IRn, there exists a unique pair (f(x w) , -h (x, w» that maximi:;>;eR the expression (f(x, w) , p) - h(x, w). u uniformly, as E -+ O. It turns out that then the Now suppose that u' function u is a viscosity solution of the PDE . . (5.4) - !' (x, w ) P (x, w) UX i Xj = O in U, ,
�
for each W E argmax { (f(:c, w), Du) - h(x, w) : Iwl < I } .
We can interpret (5.4) as a generalization of the oo-Laplacian.
Aronsson's equation, forward and hackwards Hamilton·Jacobi flows. Another generalization of the oo-Laplacian is Aronsson's operator'
(5 .5)
defined for H : IRn
�
R, H = H(p) .
We sketch a situation where the Aronsson operator arises. Given H consider the pair of Hamilton-Jacobi equations
(5.6)
Vi
+ H (Dv) = 0 v=g
(t > 0) (t = 0)
0
( t > 0) (t = 0)
and
(5 .7 )
Wt - H(Dw)
=
w=g
as
above,
THE I-LAPLACIAN, THE oo-LAPLACIAN AND DIFFERENTIAL GAMES
253
A degenerate nonlinear wave equation. We formally differentiate (5.6) with respect to t and Xk , to find
for k =
1 , . . . , n.
Vll + Hpk VXk t = 0, V"'kt + Hpj VXkXj = 0
Substitute the second equation into the first, to discover that 1!tt -
AH [v]
=
0,
for the Aronsson operator (5.5) . Similarly, we deduce thaL So Lhe solution� v, tv of the "forward" and "backwards" Hamilton-Jacobi equa tions (5 .6 ) and (5.7) formally solve the same nonlinear wave equation; a deduction that is certainly in general false, since v and tv are not usually smooth. But we do have the rigorous conclusions that v = 9 - tH(Dg) + w as
"2 AH[g] + o(e) t2 = 9 + tH(Dg) + "2AH[g] + 0(t2 )
0, valid for smooth initial data equations to knock out the OCt) terms: t
--7
t2
g.
And we can average these expansions
A nonlinear parabolic PDE. To be more precise, we now introduce nonlinear semigroup notation, writing v : = R(t)g to denote the unique viscosity solution of (5.6), and tv : = 8(t)g to denote the unique viscosity solution Qf (5.7) . Next, set
F(t)
R e .;t) + S e .;t) 2
:=
(t > 0) ,
to record an average of the two dynamics, over the long time scale .;t.
\Vhat happens if we repeatedly apply t.he nonlinear operator F over shorter and shorter time intervals?? In [B-E-J] we demonstrate that the limit 71. : =
lim F(t/n)ng
n � DC
(t > 0)
exists uniformly for eaeh time t > 0; and u is the unique viscosity solution of the parabolic equation Ut -
�AH[U]
=
0
u=g
(t > 0 ) (t = 0) .
The proof uses one of my favorite tools of nonlinear analysis, the "Chernoff formula" for nonlinear semigroups, due to Brezis and Pa:q [B-P .
]
254
LAWRENCE C. EVANS
References G. A ronsson , Extension of functions satisfying Lipschitz conditions. Ark. Mat. 6 (1967) 551-56l. (A2] O. C. Aronsson, O n the partial differential equation ux�xx + 2uxuy uxy + Uy�yy Ark. Mat. 7 ( 1968) , 395-425. G. Aronsson, M. Crandall , and P. Juutinen A tour of the theory of absol utely mini [A- C-J ] mizing functions , Bnll. Amer. Math. Soc. 41 (2004), 439-505. [B-E-J] E. N. Barron , L. C. Evans and R. Jensen, The infinity Laplacian, Aronsson s equation and their generalizations, paper to appear. H. Brezis and A. Pazy, Convergence and approximation of semigronps of nonlinear [B-P] operators in Banach spaces, J. Funct ional Analysis 9 (1972), 63-74. M .Crandall, L. C. Evans and R. Gariepy, Opt imal Lipschitz extensions and the infinity [C-E-G] Laplacian, Calculus of Variations and Partial Differential Equat ions 13 (2001), 123139. M .Crandall and L. C. Evans, A remark on infinity harmonic functions, Electronic [C-E] Journal of Differential Equations, Conf. 06, (2001), 123-129. L. C. Evans , A new proof of local c1,a regularity of solutions of certain degenerate [E 1] partial differential equations, Journal of Differential Equations 45 (1982), 356--373. [E2] L. C. Evans, Some min-max methods for the Hamilton-Jacobi equation, Indiana U Math J 33 (1984), 31-SO. L. C. Evans, Est imates for smooth absolut ely m inimiz ing Lips ch itz extensions, Elec [E3] tronic Journal of D ifferent ial Equations 1 (1993) . L. C. Evans and P. E. Sougarridis, Differential games and representation formulas for [E-Sol solutions of Hamilton-Jacohi equations Indiana University Mathematics Journal 33 (1984), 773-797. [E-Sp] L. C. Evans and J . Spruck , Motion of level sets by mean curvature I, J ournal of Differential Geometry 33 (1991), 635-681 . R. Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradi [J] ent , Arch. Rational Mech. Analysis 123 (1 993) , 51 74 . R. Kohn and S. Serfaty, A deterministic control-based approach to motion by mean [K-S] curvature, to appear in Commun ications Pure and Applied Math [LG] E. Le Gruyer, On absolutely minimizing Lipschitz extensions and the PDE L:.oo (u) = 0, preprint, 2004 A. Oberman, Convergent difference schemes for the infinity Laplacian: construction of [0] absolutely minimizing Lipschitz extensions, preprint Y. Peres and S. Sheffield, Tug of war with noise: a game theoretic view of the � [P-S] Laplacian preprint, 2006. [P-S-S-W] Y. Peres, O. Schramm, S. Sheffield and D. Wilson, Tug-of-lVar and the infinity Lapl",. cian, preprint , 2005. O. Sav in , C1 regularity for infinity harmonic functions in two dimensions, Arch. Ra [S] tional Mech . Analysis 176 (2005), 3,; 1 - 361.
(AI]
=
,
'
,
-
,
OF MATHEMATICS, UNIVERSITY OF CALIFORNIA, B ERKELEY , CA 94720 E-mail address: evanslDmath . berkeley . edu
DEPARTMENT
Contemporary Mathematics
Volume 446,
2007
Probabilistic Approach to a Class of Semilinear Partial Differential Equations Jean-Franc;ois Le Gall Dedicated to Profe.... or
Hairn Bn!zis.
We discuss the recent progress about positive solutions uf the semi linear equation .6.'U = 'UP in a domain, which has involved a combination of probabilistic and analytic methods. We emphasize the main ideas that have been used in the probabilistic approach. Special attention is given to the boundary trace problem, which consists in obtaining a one-to-one correspon dence between the set of all solutions and a suitable set of admissible traces on the boundary. A few important open questions are also listed.
ABSTRACT.
1.
Introduction
It has been known for a long time that properties of random systems of branch ing particles are related to solutions of certain semilinear partial differential equa tions. In the last 15 years, these connections have given rise to fruitful developments in the setting of the theory of measure-valued branching processes, also called su perprocesses. A major step was accomplished by Dynkin [7J, who provided a simple probabilistic representation of the solution of the Dirichlet problem for the equation �u = uP, 1 < p < 2 in a domain of jR d , in terms of the so-called exit measure of the associated superprocess (see Theorem 2.1 below) . A very interesting feat.ure of this representation, in contrast to other probabilistic approaches, is its robustness: A formula that is a priori only valid for solutions with a given continuous boundary value can be generalized, by means of various limiting procedures, to yield similar representations for many other solutions whose behavior at the boundary can be very singular. In fact, as will be explained in Section 6 below, a generalized ver sion of the probabilistic formula applies to any nonnegative solution of �u = uP, 1 < p :::; 2. Our goal in this work is to give an account of these developments, including the important recent contributions of Dynkin and Kuznetsov from the probabilistic side, and Marcus and Veron from the analytic side. We made no attempt at exhaustivity, 1991 Mathematics Su bject Classification. Primary
35J60, 35J65; Secondary 60J45, 60J80. Key words and phrases. SemiJinear partial differential equation, removable singularity, boundary blow-up, boundary trace, superProcess , Brownian snake, exit measure, polar set. ©2007
250
A meric an Mathemat ical Soci�ty
256
JEAN-FRANQOIS LE GALL
ami for instance we do not discuss parabolic equations which can also be handled by the same probabilistic tools. Rather, we try to explain as simply as possible the basic probabilistic ideas and the way these ideas can lead to analytic results. For this reason, we often concentrate on the particular ca.�e p = 2, where the random process called the Brownian snake can be used in the probabilistic representation of solutions. The Brownian snake was introduced in [22] , and its connections with equation �u = u2 were first discussed in [24] (see also the monograph [27]) . The Brownian snake is a simpler object than superprocesses and is sometimes more tractable for analytic applications, even though most of the analytic results that have been obtained for p = 2 via the Brownian tinake could then be extended to the case 1 < p < 2 using superprocesses. We do not discuss analytic methods herc. The reader who is interested in the analytic approach to the problems discussed below should look at Laurent Veron's recent paper [40]. Above all, we tried to emphasize the nice interplay between analytic and prob abilistic concepts. Already in [7] , Dynkin used the characteri�ation of removable singularities from Baras and Pierre [2] to solve the important problem of the de scription of polar sets fur super-Brownian motion (see Theorem 4 . 1 below) . In the reverse direction, we give examples of theorems that were finit proved for p = 2 via the probabilistic approach, and then ext.ended to arbitrary p > 1 by analytic methods. See Thcorem 3.3 and its generalization Theorem 3.4 by Labutin, or The orem 6.1 and its generalization Theorem 6.2 by Marcus and Veron. Obviously, the probabilistic approach, which docs not apply to the case p > 2, does not replace analytic methods. Still we believe that in some particular cases the probabilistic intuition can help guessing or even proving new analytic results, which can then be generalized. Section 2 below gives a brief presentation of the Brownian snake and i:itates the key Theorem 2.1, from which the different probabilistic representation formulas can be deduced. This section should provide sufficient background to understand the probabilistic ideas that are explained in the remainder of the paper. Analytic ques tions, namely solutions with boundary blow-up, removable singularities, solutions with measure boundary data, and the trace problem are discussed in Sections 3 to 6. We have emphasized the boundary trace problem, which has given rise to recent major advances by Dynkin, Kuznetsov, Mselati, Marcus and Veron.
2. A probabilistic tool: The Brownian snake
In this section, wc give a brief presentation of the Brownian snake, which will be our main tuol in the probabilistic analysis of semilinear partial differential equations. We refer to the monograph [27] for a more detailed presentation. At an informal level, our aim is to construct a "tree of Browrnan paths" originating from a given point x E JRd . More precisely, we will construct a collection (W8) of random paths, indexed by a real parameter s varying iII some interval. For each fixed value of the parameter s, Ws = (W.(t) ° < t < (s ) is thus a finite path in ]Rd starting from 1.: : , with lifetime denoted by (s . If s i= s', the paths W8 and Ws' coincide over an interval of the form [O, m(s, s')], where m(s, s') < (8 /\ (." In this sense, the collection (�V., ) forms a "tree" of paths. Assuming that x belongs to a domain D, a key role in our applications is played by the set E D of all exit points from D of the paths Ws (more precisely of thoi:ie paths W8 that do exit D), and by the exit measure from D, whkh is a finite measure supported on ED ,
PROBABILISTIC APPROACH TO SEMILINEAH
257
PDE
Let US turn to more rigorous definit.ions. The Brownian snake is a Markov process taking values in the set of finite paths in ]Rd . By definition, a finite path in lR.d is a continuous mapping w : [0, (] -7 lR.d . The number ( ((w ) > 0 is called the lifetime of the path. We denote by W the set of all finite paths in Rd. ThiH set is equipped with the distance d (w, w' ) = I ((w) - ((w') I + sup Iw(t II ((w» - w' (t II ((w,» I · -c
t> o
Let us fix x E ]Rd and denote by Wx the set of all finite paths with initial point w(O) = x. The Brownian snake with initial point x is the continuous strong Markov process W = (Ws , s > 0) in Wx whose law is characterized as follows. 1 . If ( = ((w.) denotes the lifetime of W., the process (( s > 0) is a reflecting Brownian motion in ]R + . 2. Conditionally on ((. , s > 0), the process W is a (time-inhomogeneous) Markov process. Its conditional transition kernels are described by the following properties: For s < s' , • W., (t) = W.(t) for every t < m(s, s' ) : = infIs , s'l (,. ; • (Ws. (m(s, s') + t) - Ws, (m(s, s'» , O < t « s, - m(s, s'» is a standard Brownian motion in ]Rd independent of W Informally, one should think of W.. as a Brownian path in lR.d with a random lifetime (. evolving like (reflecting) linear Brownian motion. When (s decreases, the path Ws is "erased" from its tip. When (. increases, the path W. is extended (independently of the past) by adding "small pieces" of Brownian motion at its tip. From this informal explanation, it should be clear that the evolution of the Brown ian snake generates a "tree of Brownian paths" in the sense that was explained at the beginning of this section. Denote by x the trivial path in Wx with lifetime O. It. is immediate that x is a regular recurrent point for the Markov process W. We denote by Nx the associat.ed excursion measure. Under Nx the law of W is deHcribed by properties analogous to 1. and 2., wit.h the only difference that the law of reflecting Brownian motion in 1. is replaced by the (infinite) Ito measure of positive excursions of linear Brownian motion (see [27] ) . In other words, the "lifetime process" ((s, s ). 0) is under Nx a positive Brownian excursion: It starts from 0, comes back to 0 at a finite time TJ > 0 called the duration of the excursion (and then stays at 0 over the time interval [TJ, +00» , whereas between times 0 and TJ it takes positive values and behaves like linear Brownian motion. Knowing the lifetime process ((., s > 0) , the behavior of the Brownian snake under 1"1", is given by property 2., as was informally described above. From our definitions, it is clear that W. = x for all s > TJ' Nx a.e. Thus under 1"1"" we will only be interested in the paths Ws for 0 < s S 7]. We can normalize Nx so that, for every 10 > 0, Nx (sups>o (. > c) = (210) - 1 . Although Nx is an infinite measure, we have for every 6 > 0 .
"
•.
( 2. 1 )
Nx (
s>O, O
sup
I Ws (t)
-
xl > 6) = Cd 6- 2 < 00,
where Cd is a positive constant (see [27] , Proposition V.g) . For every fixed s > 0, conditionally on (.. Ws is distributed under Nx as a d dimensional Brownian path started at x and stopped at time ( If 0 < s < s ' < Ti, the paths W. and W.' coincide up to time m(s, s ' ) > 0, by Property 2., and this •.
JEAN-FRAN<;:OIS LB GALL
258
again corresponds to our concept of a tree of paths. In what follows, we always consider the Brownian snake under its excursion mflasure Nx , for some x E ]Rd . Next, consider a domain D such that x E D. For any finite path w E Wx, set T(W) inf {t E [0, ( w) l : wet ) ¢. D} with the usual convention inf 0 = 00. An important role in this work is played by the set of exit points [ D {WsCT(Ws» s > O, T(W.) < oo}. Notice that [ D is a random closed subset of aD. The exit measure ZD from D is a random finite measure supported on the set [D. This measure can be defined by the following approximation ([27], Chapter V) : =
:
=
(ZD ,
�f�! 10'" ds
for any continuous function
:
set
u(x) = Nx (1 - exp -(Z D , g) .
Then the function 11. is the unique nonnegative solution of the boundary value prob lem in .:lu = 4 u2 U 1 81) = 9 .
D,
( 2.2)
Th
e boundary value u l 8D = 9 should be understood in the pointwise seIlSe lim u(x) = g(y) x-..y ,xE D for every y E aD. The uniqueness of the nonnegative solution of (2.2) is an easy application of the maximum principle for elliptic equations. The factor 4 in (2.2) is an unimportant scaling constant due to our specific normalization. Theorem 2.1 is a reformulation in terms of the Brownian snake of a more general result due to Dynkin ([7] , Theorem 1.1). Dynkin's result, which is formulated in the setting of the theory of superprocesses, applies to the equation .:lu uP when 1 < p < 2. Here we chose to concentrate on the case p = 2, where the description of the relevant probabilistic objects is easier to understand. See [27] , Theorem V.6, for a proof of Theorem 2.1 in the present setting. For 1 < p :::; 2, and equation .:lu = 4 11.2 replaced by �u uP, the statement of Theorem 2.1 remains valid if we use the so-called p-stable Levy snake instead of the Brownian snake (see Proposition 4.5.1 in [6)). On the other hand, it seems that no such probabilistic tool can be used to handle equation .:lu uP when p > 2. Remark.
=
=
=
PROBABILISTIC APPROACH TO SEMILINEAR POE
259
In order to prove Theorem 2 .1, one establishes the equivalent integral equation
dy GD(X, Y) U2 (y) = J( KD ( x, dz) g (z) , x E D aD D where GD is the usual Green funct.ion (for � �) in D and Kv(x, dz) is the harmonic measure on aD relative to the point x. In probabilistic terms, this equation can be (2 .3)
u(x) + 2
rewritten in the form (2.4)
where (Bt, t � 0) is a d-dimensional Brownian motion that starts from x under the probability measure Px , and T = inf{ t � 0 : Bt ¢: D}. A computational way of proving ( 2 .4) is to expand the exponential in the formula u(x) = Nx(l exp - (ZD , g) , and then to use recursion formulas for the moments of (ZD , g), which follow from the tree structure of the Brownian snake paths. Let us summarize the contents of this section. Under the measure Nx , the paths (W., 0 � s < 'I) form a tree of Brownian paths started from x, each individual path Ws having a finite lifetime (s ' The set E D consists of all exit points from D of the paths Ws (for those that do e.xit D), and the exit measure ZD is in a sense uniformly spread over ED. We will also use the range R, which is defined by -
R
:= {y = Wa (t); 0 < s :::; '1), 0 < t < (a}.
This is simply the union of the Brownian snake paths.
3. Solutions with boundary blow-up According to Keller [18] and Osserman [38] , if D is a bounded smooth domain and 1jJ is a function that satisfies an appropriate integral condition, there exists a nonnegative solution of equation �u = 1jJ (u) in D that blows up everywhere at the boundary. This holds in particular if 1jJ(u) = uP for some p > 1 . This raises the following two questions: (a) For which non-smooth domains does there exist a solution that blows up everywhere at the boundary ? (b) Assuming that there exists a solution \vith boundary blow-up, is it unique ? The probabilistic approach turns out to be rather efficient in providing answers to t.hese questions. Let us start by reformulating in terms of the Brownian snake two key theorems again due to Dynkin [1]. THEOREM 3 . 1 . Let D be a bounded domain. Assume that D is Dirichlet regular.
Then Ul (X) = Nx(ZD i= 0), x E D is the minimal nonnegative solution of the problem in D �u = 4u2 (3. 1 ) UraD = +00 . The proof of this theorem is ea.')y from Theorem 2.1. Simply consider for every n > 1 the function vn(x) = Nx( 1 - exp -n(ZD , 1) that solves (2.2) with 9 = n. Clearly, Vn r Ul as n r 00 , and it follows that Ul also solves .6.u = 4u2 in D. Since Ul > Vn for every n, we have ullaD = + 00. Finally, if U is any (nonnegative) solution of (3. 1 ) , the maximum principle implies that u > Vn for every n and so
u > Ul.
260
JEAN-FRANCOIS LE GALL
To state the second theorem, recall our notation R for the range of the Brownian snake.
3.2. Let D be any open set in ]Rd and U2 (X) = Nx(R n DC t= 0) for x E D . Then Uz is the maximal nonnegative solution of !.::m = 1u2 in D (in the sense that u < 1L2 for any other nonnegative funci't on u of class C2 in D such that l!.1/. = 4'u2 in D). THEOREM
Remark. By combining Theorem 3.2 and (2.1), we recover the classical a priori bound u(x) < U2 (X) < cd dist(x, oD)-2 , x E D which holds for any nonnegative solution of l!.U = 4 u2 in D. Again the proof of Theorem 3 . 2 is relatively eai:iY from the preceding theorem. One can argue separately on each connected component of D, and thus a.."lm me that D is connected (notice that by construction the range R is also connected) . It is then easy to construct an increasing sequence (Dn )n> l of bounded Dirichlet regular subdomains of D such that Dn C Dn+ 1 for every n, and D = UDn. By Theorem 3.1, ui' (x) = Nx(ZDn t= 0) is a solution in Dn with infinite boundary conditions, On the other hand, from the probabilistic formulas of Theorems 3.1 and 3.2, one can check that. ul(x) 1 U2 (X) as n i 00, for every x E D. It follows that Uz is also a solution of l!.U = 4u2 in D. Moreover any other nonnegative solution 'U is bounded above by uf in Dn (by the maximum principle in Dn) anu therefore is bounded above by U2 . The previous theorems already shed some light. on questions (a) and (b). From Theorem 3. 1 , a solution with boundary blow-up exists as soon as D is Dirichlet regular. Unuer this assumption, question (b) reduces to giving conditions ensuring that Ul = U2 ' In the case when D is not Dirichlet regular, it. is easy to construct examples where Ul t= uz . Let B(x, r) denote the open ball with radius r eentered at x. Then if d = 2 or 3 and if D = B(O, 1 ) \{0} is the punctured unit ball (which is not Dirichlet regular) , the fUIIcLion U2 blows up near the origin, as a consequence of Theorem 3.3 below, whereas the function Ul stays bounded near the origin, because the exit measure "does not see" the origin, To state conditions ensuring that Ul = U2 , assume that d > 2 (the case d = 1 is trivial) and denote by Cd-Z (K) the Newtonian capacity (or the logarithmic capacity if d = 2) of a compact subset K of Rd. According to Theorem IV.9 of [27], the answer to question (b) is positive under the following assumption: For every y E aD, there exists a positive constant c(y) such that the inequality (3 . 2)
holds for all n belonging to a sequence of positive density in N (here B (x, 1') is the closed ball of radius r centered at x ) . See also Marcu� and Veron [29] for related results obtained by analytic methods for the more general equation l!.U = vY . It is interesting to compare (3.2) with the classical Wiener test, which gives a necessary and sufficient condition for the Dirichlet regularity: The bounded domain D is Dirichlet regular if and only if for every y E aD, (3.3)
261
PROBABILISTIC APPROACH TO SEMILINEAR PDE
Clearly, assumption C3.2) is stronger than C3.3). However, it is very plausible that (3.2) is not the best possible assumption, and this leads to the following question.
Open problem. Is the solution with boundary blow-up unique in the case of a
general Dirichlet regular domain ? ( ) JRd, the that t the
Let us discuss question a . Here Theorem 3.2 immediately tells us that for a general open set D in existence of a nonnegative solution of t;.u = u2 in D blows up everywhere a boundary of D is equivalent to I.he property DC =J. 0) = +00 (3.4) n lim Nx(n x-Y,xE D for every y E aD. It is not hard to see that this condition holds if and only if for every y E aD, Ny(Ws (t) r;. D for some s > 0 and t E ( 0, (8]) = +00. In this form, (3.2) is quite similar to the probabilistic version of the characterization of the Dirichlet regularity, with the difference that a single Brownian path started from y is replaced by a tree of Brownian paths started from the same point.. In order to sLaLe the next result, we need to introduce some notation. If > 0 and K is a compact subset of JRd, we define the capacity Ca (K) by setting l inf 1/ v (dY) !I(dz)fa C l y - zl) ) Cu(K) = ( vEM,(K) wehere M l (K) is the set of all probability measures on K, and if 0 , = = faCr) if > 0 . THEOREM 3.3. [4] Let D be a domain in IRd . Then the following two properties a
a
a
are equivalent. (i) The problem
(ii
in t;.u = u2 u l8D = + 00 has a nonnegative solution. ) Either d < 3, or d > 4 and for every y E aD,
D
L 2n(d-2) Cd_ 4 (DC n B (y, Tn » 00
= +00.
n=l
This theorem, which was proved in [4] by probabilistic methods involving the Brownian snake, thus gives a complete answer to question (a) ahove. Theorem 8.:3 wa.'> generalized a few years later by Labutin [21] using purely analytic methods. To state Labutin's result, and in view of further statements, we introduce Bessel capacities in JRd. For every I > 0, consider the classical Bessel kernel dt 2 -t 1[" 1 1 G�d) (x) = ) (3 .5) t 2 exp ( 2 o 41[" t where a-y = (41[") --y /2r ('y/ 2) -1 . For any compact subset K of IRd, and every p > 1 , we then set, P/P' (3.6) C-y,p(K) = sup ( Jr dx ( J /1(dy) G�d) ( y _ x ) r') }.tEM ,(K) llf.d a
-y
00
7- d
x
262
JEAN-FRANQOIS LE GALL
where ! + ; = 1 as usual. The capacity Cr,p can also be viewed associated with the Sobolov space W')" P: See Theorem 2.2.7 in [1) . ,
as
the capacity
1.
THEOREM 3.4. [21J Let D be a bounded domain in ]Rd d > 3 and p > Let p' be defined by ! + ; = 1 . Then the following two pmperties are equivalent. ,
,
(i) The problem
�u = uP in D U l 8D = +00 has a nonnegative solution. (ii) Either p < �2 ' or p > d�2 and for every y E aD, d
.L 2n(d-2 ) C2 ,p' (D C n B( y, 2- n) ) = +00 . DC
'0.=1
In the case p = 2, we recover the preceding result. Indeed, a few lines of calculations show that, if d > 4, there exist two positive constants al and a2 such that, for every compact subset K of the unit ball,
(3 7) .
4. Removable singularities
Let K be a compact subset of ]Rd. We say that K is an interior removable singularity for �u = u2 if the only nonnegative solution of �u = u2 in JRd \K is the function identically equal to O. This turns out to be equivalent to saying that for any open set 0 containing K , any nonnegative solution on O\K can be extended to a solution on O. From the probabilistic point of view, interior removable singularities correspond to (interior) polar sets. The compact set K is said to be polar if for every x E ]Rd\K,
NxCR. n K # 0) = o.
In other words, the cornpaet set K will never be hit by the tree of Brownian paths which is the range of the Brownian snake.
THEOREM 4.1. Let d > 4 and let K be a compact subset o/ Rd. The following
are equivalent. (i) K is an interior removable singularity for �u = u2 . (ii) K is polar. (iii) Cd - 4 (K ) O. =
In dimension d < 3, the equivalence (i){c}(ii) also holds trivially, since (i) or (ii) can only be true if K is empty. The equivalence (i){c}(ii) is an immediate consequence of Theorem 3.2 above applied with D = KC• The equivalence (i){c}(iii) was obtained by Baras and Pierre [2): More generally, Baras and Pierre have shown that K is a removable :;ingularity for �u = uP if and only if C2 ,p' (K) = 0 (see also [3J for an earlier discussion of removable singularities for sernilinear equations). From the probabilistic viewpoint, it is worthwile to look for a direct proof of the equivalence (ii){c}(iii). A simple argument gives the implication (ii)=> (iii) (this implication was first obtained, independently of [2] , by Perkins [39) , and later the
263
PROBABILISTIC APPROACH TO SEMILINEAR PDE connection with [2] was made by Dynkin [7] ) . Indeed, suppose that Cd- 4 (K ) and so that there is a probability measure v supported on K such that
> 0,
JJ v(dy)v(dz)fd_4(ly - zl) < 00
(4 . 1 )
where fd- 4 (r) is as above in the definition of Ca (K) . Let h be a radial nonnegative continuous function on ]ftd with compact support contained in the unit ball, and for every c E (0, 1]' set he (x) = c- d h(x/c). Finally let I be the "total occupation" measure of the Brownian snake defined by
" t (I, g ) = 10 ds g(Ws ) -
-
where Ws = Ws «(s) is the terminal point of the finite path W. , and we recall that TJ is the duration of the excursion under Nx • Notice that by construction I is supported on R. If x E ]ftd\K is fixed, explicit moment calculations using (4. 1) give the bounds
Nx «I, he * v» > C1 > ° Nx ( (I, he * v) 2 ) < C2 < CXJ , where the constants Cl and C2 do not depend on c E (0, 1] (see Chapter VI in [27] for details). Let Ke denote the closed tubular neighborhood of radius c of the set K. From all application of the Cauchy-Schwarz inequality, it follows that ,
* c� . h I « v) N ? e , x ( 0) * > N (R n K -'v) > > T 0) > N « I h Nx( (I, hE * v) 2 ) C2 By letting c go tu 0, it follows that Nx (R n K i= 0) > 0, and thus K is not polar. In view of the simplicity of the preceding argument, une would expect that similar probabilistic proof should also give the converse implication (iii)=>(ii). Sur prisingly t his is not the case, and the only known way to obtain this implication is ' via Baras and Pierre's result (i),*(iii). Open problem. Give a direct probabilistic proof of the implication (iii)=>(ii) in Theorem 4-1. Finding such a proof would be of interest for other related problems where the analogues of the results of [2] are not always available. An example of such problems is provided by the notion of boundary removable singularity. From now on, consider a bounded domain D in ]Rd , with a smooth (COO) boundary aV. A compaet subset K of av is called boundary removable for Llu = uP (in V) if the only nonnegative function u of class C2 in V such that. Llu = uP and u tends to ° pointwise at every point of aD\K is the function identically equal to 0. Boundary singularities were studied first by Gmira and Veron [17] , who proved in particular that singletons are removable if p 2 �±t. To introduce the corresponding probabilistic notion, recall that £ D is the set of all exit points from V of the Brownian snake paths. The compact set K c av is said to be boundary polar if x
e
-
x
,
E
-
-
N", (£D n K i= 0) = 0
for every x E V. The following analogue of Theorem 4 . 1 was obtained in [25] , confirming a eonjecture of Dynkin [8] . THEOREM 4.2. Suppose that d > 3 and let K be a compact subset of ]Rd. Then the following are equivalent.
JEAN-FRANQOIS LE GALL
204
(i) K is a boundary removable singularity for .6.u = 1.12 in D . (ii) K is boundary polar. (iii) Cd-3 (K) = o. If d < 3, (i) and (ii) only hold if K = 0. The equivalence (i){o}(ii) is an immediate consequence of the following lemma (Proposition VII.I in [27] ) , which is analogous to Theorem 3.2. LEMMA 4.3. If K is a compact subset of D, the function
UK(X) = N:x: ([D n K # 0),
xED
is the maximal nonnegative solul'ion of the problem .6.1.1 = 41.12
(4.2)
in D
UI8 D\ K = O.
Lemma 4.3 is essentially a consequence of Theorem 2.1 above. Roughly speak ing, one can find a sequence a sequence (gn) of continuous functions on aD, such that (ZD, gn) converges to +00 on the event {C D n K # 0}, and to 0 on the com plementary event. It follows that UK solves .6.1.1 = 41.12 and it is also not hard to l>p.e that UK vanishes on aD\K. A suitable application of the maximum principle gives the maximality property stated in the lemma. Coming back to Theorcm 4.2, the implication (ii)=>(iii) can be p.stablished in a way very similar to the probabilistic proof of (ii)=>(iii) in Theorem 4. 1 that was described above (compute the first and second moments of (ZD , g) for suitable func tions 9 that vanish outside a small neighborhood of K). The implication (iii)=>(i) was obtained in [25] by Fourier analytic methods, using some ideas from [2J . The analytic part of Theorem 4.2, that is the equivalence (i){o}(iii) , can in fact be extended to equation .6.1.1 = up. This extension again involves the Bessel capac ities that were introduced above, but now considered for subsets of the boundary aD. If K is a compact subset of aD, we set ' / P P C�� (K) = sup ( 4. 3) a(dx) f.L(dY) G�d - l) (y _ X) (JD /LEM , ( K ) where a(dx) stands for Lebesgue measure on aD, and the Bessel kernels G(d) were defined in (3.5). This is of course analogous to (3.6), but lRd is replaced by the (d - I )-dimensional manifold aD, and consequently G(d) is replaced by G(d- l) .
(
(J
r' )
THEOREM 4.4. Let K be a compact subset of aD . Then the following are equivalent. (i) K is a bounda.ry removable singularity for .6.u = uP in D. (ii) cggp, (K) = O. In the case p = 2, we recover the preceding theorem, since an easy calculation shows that Cff (K) = 0 if and only if Cd- 3 (K) = O. Theorem 4.4 was proved in the case 1 < p < 2 by Dynkin and Kuznetsov [13J using a combination of probabilistic and analytic techniques (in the ca..<;e 1 < P < 2, boundary removable singularities still have a probabilistic interpretation in terms of superprocesses with a p-stable branching mechanism). The case p > 2 of Theore m 4.4 was obtained by Marcus and Veron [31) . Rather surprisingly, the analytic techniques of [31] did not apply to the case p < 2 treated in [13J . In a subsequent paper [32J , Marcus and Veron •
265
PROBAI3ILISTIC APPROACH TO SEMILINEAR PDE
developed a different approach that allowed them to give a unified treatment of all cases of Theorem 4.4.
5. Solutions with measure boundary data In this section, as well as in the next one, we keep assuming that D is a bounded domain in lRd with a smooth boundary aD. Many of the subsequent results hold under weaker regularity assumptions on D, but for the sake of ::;implicity we will omit the precise minimal assumptions. We are now interested in the problem in D , D.u = uP (5.1) Ul 8D = V , where v is a finite (positive ) measure on aD. Similarly as for (2.2) , the boundary condition Ul 8D = v may be interpreted via t,he int.egral equation (5.2)
u(x) +
1
dy GD (x , lI) UP (y) =
v(dz) PD (x , z) ,
xED,
2 D 8D where CD is as in (2.3) the Green function of D, and PD is the Poisson kernel of D (in the notation of (2.3) , KD(X, dz) = PD (x, z)a(dz)). (5.2 ) makes it obvious that u is bounded above by the harmonic function PDv. Conversely, any nonnegative solution of D.u = uP in D which is bounded above by a harmonic function solves a problem of the type (5.2), for some finite measure v on the boundary: See e.g. Proposition 4.1 in [25], for an argument in the case p = 2 which is easily extended. A nonnegative solution that is bounded above by a harmonic function will be called moderate. Gmira and Veron [17] considered the problem (5.1) ( in fact. for more general nonlinearities ) . They proved in partieular that (5.1) has a unique solution for any 'o finite measure f.1. on the boundary if p < ��i. Notice r,hat this conditi n corresponds to the case when singletons are not boundary polar. We fix p > 1 and to simplify terminology, we call boundary polar any compact subset K of aD that satisfies the equivalent conditions of Theorem 4.4. This is of course consistent with our preceding terminology for p 2.
5. 1 . Suppose that p > �+� and let f.1. be a finite measure on aD. The following two conditions are equivalent: (i) The problem (5. 1), or equivalently the integral equation (5.2), has a unique nonnegative solution. (ii) The measure f.1. does not charge boundary polar sets. Consequently, there is a one-to-one correspondence between the set of all moderate solutions of D.u = uP in D and the class of all finite measures on aD that do not charge boundary polar sets. =
THEOREM
In the case p = 2, this theorem was proved in [25] (again confirming a conjecture of Dynkin [8]) using both analytic and probabilistic arguments. In thaI. case, there is a probabilistic representation of the solution in terms of the Brownian snake: This is analogous to Theorem 2.1 with the difference thaL the quantity (ZD , g) should be replaced by a suitable additive functional of the Brownian snake. Similarly as for Theorem 4.4, the general form of Theorem 5.1 was obtained by Dynkin and Kuznetsov (see [13] and [14]) when 1 < p < 2 and by Marcus and Veron [31] when p > 2. A unified treatment was provided in [32] .
266
J EAN-FRAN!;:OIS LE GALL
6. The boundary trace probleID The classical Poisson representation states that nonnegative harmonic fWlctions h in D are in one-to-one correspondence with finite measures v on aD, and this correspondence is made explicit by the formula h = PD v, where PD is as above the Poisson kernel of v. We may say that the measure v is the trace of the harmonic fundion h on the boundary. Our goal in this section is to discuss a similar trace representation for nonneg ative solutions of �u = uP in D. ¥le will deal separately with the sub critical case p < ��i (where there are no nonempty boundary polar sets ) and the supercritical d±l . case p > - d-l
6.1. The subcritical case. We first consider p
2, so that the Brownian snake approach is available. Then the subcritical case holds if and only if d < 2 . Since the case d = 1 is trivial, we concentrate on d = 2 , where we have the folluwing theorem ([23], [26]). Recall that a(dz) denotes Lebellgue measure on aD. =
THEOREM 6.1 . Ass'llme that d = 2 , There is a one-to-one correspondence between nonnegative solutions of �u = 4 u2 in D and pairs (K, v), where K is a (possibly empty) compact subset of aD, and v is a Radon measure on aD\K. If a solution u i,� given, the associated pair (K, v) is determined as follows. For eveTY z E aD, denote by Nz the inward-pointing normal unit 1Jector to aD at z, then: (i) A point y E aD belongs to K if and only if, for every neighborhood U of y in aD,
lim a(dz) u (z + rNz) = +00. dO u (ii) For every contimtous function g with compact support on aD\K,
r
a(dz) u(z + rNz ) g (z) lim rJO JaD \ K
=
j V(dZ) 9(Z) '
Conversely, if the pair (K, v) is given, the solution u can be obtained by the fonnula (6.1 )
whel'e (ZD(z), z E aD) is the continuous density of the exit meaSU1'e ZlJ with respect to Lebesgue measure a(dz) on aD.
The pair (K, v) will be called the trace of u on the boundary. Informally, K is a set of singular points on the boundary (this is the set of points where u blows up as the squarc of the inverse of the distance from the boundary) and 11 is a mea.'mre corresponding to the boundary value of u on 8D\K. The formula (6. 1) contains as special cases the other probabilistic representations that have appeared previously. The formula of Theorem 2.1 corresponds to K = 0, v(dz) = g(z)a(dz). The function U2 of Theorem 3.2 (which here coincides with Ul of Theorem 3 . 1) is obtained by taking K = aD, More generally the functions UK in Lemma 4.3 correspond to the case v = O. Finally, the moderate solutions of Theorem 5.1 are obtained when K = 0. Let us outline the proof of the probabililltic representation formula (6, 1 ) , as suming for simplicity that D is the unit disk of the plane. Fix a sequence rn of
.
.
PROBABILISTIC APPR.OACH TO SEMILINEAR PDE
2 67
real numbers in (0, 1) �Ilch that rn i 1 as n i 00. For every n > 1 and x E D, set Un (X ) = r�u(rn x), so that we have .6.un = 4n� in D. Since Un obviously has a continuous boundary value on aD, we may use Theorem 2.1 to write, for every x E D, (6 . 2)
( )
Un X
=
Nx ( 1 - exp _ (ZD, un )
=
Nx (1 - exp
-
J
u
(dz) ZD (z)un (z»
using the fact that the exit measure ZD has a continuous density ZD with respect to u (this property only holds when d = 2). Note that we can identify aD with lR/Z. Using a compactness argument and replacing (rn) by a subsequence if necessary, we may assume that for every open subinterval ] of aD with rational ends, we have
nlim -co
I
u
(dz) un ( z)
=
a (J)
where a(I) E [0, +00] . We then set
+00 if y E I}. Replacing again (rn ) by a subsequence, we may also assume that the sequence of measures la \K (z)un(z)O"(dz) converges to a limiting measure v(dz), in the sense D of vague convergence of Radon measures on aD\K. From the definition of K and v, it can then be proved that, for every x E D, K
(6 .3 ) and (6.4)
j. n--.oo a lim
r
--+oo }a
lim n
D
D
=
{y E aD : a (I)
O"(dz) un (z ) ZD(z)
u(dz) un (z) ZD (z)
=
=
=
+00
(v, ZD)
,
Nx a.e. on
{En n K # 0}
, Nx a.e. on
{ED n K = 0}.
Indeed, (6.4) is easy if we observe that the support of ZD is contained in aD\K, on the event {E D n K = 0}. The proof of (6.3) reduces to checking that on the event {ED n K # 0} there is a (random) point z E aD such that ZD(z) > o. Using (6.3) and (6.4) , we can pass to the limit n ----> 00 in the right-hand side of (6.2), and we arrive at the representation formula (6. 1 ) . The other assertions of Theorem 6.1 then follow rather easily. Let us come back to the general case of equation .6.u = up. Marcus and Veron ([30] , Theorem 1) proved that, for any p > 1 and in any dimension d � 2, the trace (K, v) of a nonnegative solution U of .6.u = uP in D can be defined by properties (i) and (ii) of Theorem 6.1. Independently, Dynkin and Kuznetsov [15] gave a slightly different but equivalent definition of the trace. The one-ta-one correspondence between solutions and their traces can in fact be extended to the general subcritical case. The following theorem was proved by Marcus and Veron [30] . THEOREM 6.2. Assume that d < ��i . Then the mapping u • (K, v) associating with u its trace (K, v) (definp4 by (i) and (ii) of Theorem 6. 1) gives a one-to-one correspondence between the set of all nonnegative solut'ions of .6.u = uP in D and the set of all pairs (K, v), where K is a (possibly empty) compact subset of aD, and v is a Radon measure on aD\K.
When 1 < p < 2, the probabilistic representation formula (6.1) can be extended to this more general setting: See Theorem 1 .3 in [28] (which elaborates on preceding results of Dynkin and Kuznetsov [15], [16] ) .
JEAN-FRANC;OIS LE GALL
�68
6.2. The supercritical case. The supercritical case p > �+� is more compli cated and in a sense more interesting. As Weu> mentioned above, properties ( i) and (ii) of Theorem 6. 1 r:an still be used to define the trace of any nonnegative solution of �u = uP in D. However, the fact that there are nontrivial boundary polar sets now suggests that all pai rs (K, v) cannot occur as possible traces. More precisely, Theorem 4.4 indicates that the pair (K, O) cannot be a possible trace if K is boundary polar, and similarly, Theorem 5.1 suggests that, v should not charge boundary polar sets in order for (0, v) to be a possible trace. The characterization of possible traces was obtained independently by Marcus and Veron [31) and Dynkin and Kuznetsov [15) (the latter in the case p < 2) .
THEOREM 6.3. Let K be a compact subset of aD, and let v be a Radun measure on aD \ K. Then the pair' (K, I) ) is lhe trace of a nonnegative solution of �u = uP in D if and only if: ( i) The measure 1/ does not charge boundary polar sets. ( ii) The set K is the union of the two sets
K; = {y E K K n U is not boundary polar for every neighborhood U of y} and
:
avK = {y E K : v ( K n U)
=
00 fur evel'g neighborhood U of y}.
Another problem in the supercritical case is the lack of uniqueness of the so lution corresponding to a given (admissible) trace. To give an example of this phenomenon, consider the case p = 2, d 2: 3. Let (y,, ) be a dense sequence in aD and, for every n, let (r�, k = 1 , 2, . . . ) be a decreasing sequence of positive numbers. For every k > 1 , set
Hk
=
00
U { y E aD
n� l
:
iy
-
Yn i < r� } ,
and x E D.
Then it is easy to see that, for every k > 1 , 'Uk is a solution with trace (aD, O). On the ot.her hand, the fact that singletons are boundary polar implies that Uk 1 0 as k i 00 , provided that the sequences (r�, k = 1 , 2 , . . . ) decrease sufficiently feu>t. Therefore infinitely many of the fUIlctions 'Uk must be different. Tn view of this nonuniqueness problem, Dynkin and Kuznetsov [19J , [16] have proposed to use a finer definition of the trace, where the set K is no IOIlger closed with respect to the Euclidean topology. We will explain this definition in the general case of equation �u = up. 'vVe first need to introduce the analogue of the singular part for the fine trace of a solution u. Let b be a nonnegative continuously differentiable function on D. We can then eonsider the Poisson kernel (Pb(x , y) , x E D y E aD) associated with the operator �u bu in D (see Section 11.1.2 in Dynkin [9] for a detailed construction of pi»). A point y of the boundary aD is called singular for b if Pb (x, y) = 0 for some, or equivalently for every, x E D. Informally, this corresponds to points of rapid growth of b. A simple equivalent probabilistic definition can be given a::; follows. If ,
-
PROBABILISTIC APPROACH
TO
269
SEMILINEAR POE
( Bt , O < t < T ) is under Px-y a Brownian motion started from x and conditioned to exit D at y (in the sense of [5]), the point y is singular for b if and only if
dt b(Bt} = +00 )
Px-y a.s.
Consider now a nonnegative solution u of �u = up . The singular set of 'U, which is denoted by SC (1t ) is the set of all boundary points that are singular for up- I . Note that SG(u) is a Borel subset of aD, but needs not be closed in generaL We denote by N the set of all finite meaSUrf'B on the bowldary that do not charge boundary polar sets. For every v E N, we denote by u" the unique solu tion of the problem (5 . 1 ) , or equivalently the solution associated with v via I.he correspondence of Theorem 5.l. o
DEFINITION 6.4. Let u be a nonnegative solution of �n = 'uP in D . The fine trace of u is the pair (r, 'l) that is defined as follows: (i) f = SG(u). (ii) /1 is the O'-finite measure on DD\f sllch that, for every Borel subset A of 8D\f,
(6.5)
i1C4)
=
sup{v(A) : v E N, UV < u} .
Remark. It is not obvious that formula (6.5) defines a measure. Sec Theorem in [16]. It is clear from (ii) that v docs not charge bowldary polar sets. It can be
1 .3
checked that in the subcritical case this definition is equivalent to the one given by (i) and (ii) of Theorem 6.1. The interest of this definition comes from the following theorem (Theorem 1 .4 in [16] ) .
THEOREM 6.5. [16] Let us call O'-moderatc any nonnego,tive sulution of �u = uP that is the inc'{'f>.asing limit of a sequence of moderate solutions. Then O'-moderate solution are characterized by their fine traces.
This theorem shows that the lack of uniqueness mentioned above disappears if one considers the fine trace instead of the (rough) trace discussed in the previous subsection. Dynkin and Kuznetsov [16] also give a description of those pairs (r, v) that can occur as fine traces of solutions. Provided one considers only iT-moderate solutions, the fine trace thus yields a one-to-one correspondence between solutions and admissible pairs (f, v) . The obvious question, which was stated in the epilogue of [9] is thus:
Are all nonnegative solutions O'-moderate
?
This question was answered positively first in the case p = 2 in Msclati's thesi� [36], [37] . In addition, Mselati's work gives 1-1. probabilistic representation of solu tions, which is analogous to Theorem 6.1. To state this representation, we need to introduce some additional notation. Let v E JV, and let ltv be the harmonic function in J) associated with 1/ (h" = FD V in our previous notation) . Then, if (Dn ) is an increasing sequence of smooth subdomains of D such that Dn C Dn+l and D = UDn, we can define
Z" := lim (ZD" , h,,) ,
Nx a.e.
n T oo and the resulting variable Zv does not depend on the choice of the sequence (Dn ) . Note that the existence of the limit defining Zv is easy because (ZDn ' hv) is a nonnegative martingale. Then, if v is a O'-finite measure on aD that does not
270
JEAN-FRANyOJS LE GALL
charge boundary polar sets, we can find an increasing sequence (Vk ) in N such that v = lim 1 Vk, and we set ZI/ = lim i ZVk ( again this does not depend on the choice of the sequence (Vk)).
4
THEOREM 6.6. [37] All nonnegative solutions of .6.1.1 = 1.12 are a-modemte. Moreover, if 1.1 is a solution and (r, v) is its fine tmce, we have for every x E D, u(x ) = Nx ( 1 - 1{£Dn r=0} exp( - Zv ) .
)
(6.6)
A major step in the proof of Theorem 6.6 was to prove that the solution UK defined in Lemma 4.3 is O"-moderate, for any compact subset K of aD. The proof depends on delicate upper bounds on UK near the boundary, and analogous lower bounds for certain a-moderate solutions, which are obtained via probabilistic meth ods. Motivated by Mselati's work, Marcus and Veron [33] , [34] were able to obtain very precise capacitary estimates in the general case p > 1 . Part of their results is summarized in the following theorem.
THEOREM 6.7. [34J Consider the general case p > 1 . Let K be a compact
subset of aD and assume that K is not boundary polar. Let UK be the maximal nonnegative solutwn of .6.1.1 = uP in D that vanishes on aD\K . Then UK is 0" moderate. Moreover, for every x E D, UK (X) <
(6.7)
Cl
p(x) PK (X ) -1-2/(p-l) Cf/�,p, { K/PK (X ) )
where Cl is a positive constant, p(x) = dist (x, aD) and PK (X) = dist (x, K ) .
In addition to the upper bound (6.7), the main result of [34] also gives a sharp lower bonnd, which we omit here. Recently, Dynkin [10] ' [1 1 ] , [12] was able to extend to extend Mselati's result to all values p E (1, 2].
THEOREM 6.8 . [12] If 1 < p < 2 , all solutions of .6.u = uP are O"-moderate. Moreover, if U is a solution with fine trace (r, v) , we ha1Je U = ur EB UI/
where: •
• •
ur is the supremum of the functions UK for all compact subsets K of r. Uv is the supremum of the function8 1L/" for all measures f.I. E N such that
f.I. S.
v.
The notation urEBuv stands for the maximal solution dominated by 1Lr+ul/ .
See the monograph [12] for a detailed proof. In addition to some ideas taken from [37], an important role is played by an upper bound similar to (6.7), which was obtained by Kuznetsov [20] independently of [34] . A probabilistic representation formula analogous to (6.6) also holds in the setting of Theorem 6.8. Open problem. Extend Theorem 6. 8 to the case p > 2. The very recent paper [35] by Marcus and Veron collt.ains important progress towards the solution of this open problem.
References [lJ D.R. ADAMS, L . 1. HEDBERG (1996) Function Spaces and Potential Theory. Springer. [2J P . BARAS, M. PIERRE (1984) Singularites eliminables pour des Cquations semilineaires. Ann. Inst. Fourier 34, 185-206.
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(2004) SuperdifJusions and Positive Solutions of Nonlinear Partial DifJerential Equations. University Lecture Series Vol. 34. Amer. Math. Soc., Providence. K B . DVNKIN, S.E. KUZNETSOV (1996) Superdiffusions and removable singularities for quasi linear P. D.E. Comm. Pure Appl. Math. 49, 125-176. u" dominated by L-harmonic E . B . DYNKIN, S.E. KUZNETSOV (1996) Solut ions of Lu functions . J. Analyse Math. 68, 15-37. E . B . DVNKIN, S.E. KU ZNET SOV (1998) Trace on the boundary for solutions of nonlinear different ial equations. Trans. Amer. Math. Soc. 350, 4499-4519. E . B . DVNKIN, S . E. KUZNETSOV (1998) Fine topology and fine trace on the boundary associ ated with a cl ass of quasilineac differential equations. Comm. Pure Appl. Math. 5 1 , 897-936. A. GMIRA, L . VERON (1991) Boundary singularities of solutions of some nonlinear elliptic equations. Duke Math. J. 64, 271-324. J . Il . KELLER (1957) On solutions of �u = feu). Comm. PUrF; A ppl. Math. 10, 503-510. S. E . KUZNETSOV ( 1998) a-moderat e solutions of Lu = u" and fine trace on the boundary. C.R. Acad. Sci. Paris, Sme I 326, 1189-1194. S.E. KUZNETSOV (2004) An upper bound for positive solutions of the equation �u = u" . Electron. Res. Announc. Amer. Math. Soc. 10, 103-112. D . LABUTIN (2003) Wiener regularity for large solutions of nonlinear equations. Ark. Mat. 41, 307-339. J . F . LE GALL (1993) A class of path-val ued Markov processes and it s applications to super processes . Probab. Th. Rei. Fields 95, 25-46. J . F LE GALL ( 1 993) Solutions positives de �u = u2 dans Ie disque unit e . C. R. Acad. Sci. Paris 311, Serie I, 873-878.
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exit measure of super- B rownian motion. Ann. Probab. 33 , 194-222. [29] M . MARCUS, L. V ERON (1997) Uniqueness and asymptotic behavior of solutions with bound ary blow up for a class of nonlinear elliptic equations. Ann. Inst. H. Poincare Anal. Non Lineaire 14, 237-274. [30] M. MARCUS, L . VERON (1997) The boundary trace of positive ""lutions of semilinear equations I: The sub critical case. Arch. Rat. Mech. Anal. 1 44 201-231. 128]
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VERON (1998) The boundary trace of positive solutions of semilinear equations II: The supercritical case. J. Math. Pures Appl. 77, 181-524. M. MARCUS, L. VERON (2001) Removable singularities and boundary traces. J. Math. Pures Appl. 80, 879-900. M . MARCUS, L. VF.RON (2003) Capacitary estimates of solutions of a class of nonlinear elliptic "'I"ations . C. R. Math. Acad. Sci. Paris 336, 913-918. M. MARCUS, L. VERON (2004) Capacitary estimates of positive solutions of semilinear elliptic equations with absorbtion. J. Eur. Math. Soc.. (.lEMS) 6, 483-527. M. :\(ARCUS, L. VERON (2006) The precise b oundary trace of positive solutions of the equation .6.u = 11," in the supercritical "ase. In this volume. B . MSELATI (:l002) Classification et representation probabilistc des solutions positives de .6.11, = u2 dans un domaine. These de doctorat, U niversite Paris VI. B . MSELATI (2004) Classification and probabilistic representation of the positive solutions of a semilinear elliptic equation. Memoirs Amer. Math. Soc. 168 R. OSSERMAK ( 1957) On the inequality .6.11, > feu). Pacific J. Math. 7, 1641-1647. E . A . P8RKTNS (1990) Polar sets and multiple points for super-Brownian motion. Ann. Probab. 18, 453-49l. L. VERON (:lOO4) Elliptic equations involvin g measures. Stationary partial differential equa tions. Vol. I , 593-712, Handb. Differ. Equ. North-Holland, Amsterdam.
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DMA-ENS,
L.
45 RUE
D'ULM
75005 PARIS, FRANCE
E-mail address : legalHldma . ens . f r
Contemporary Mathem1l.tic.s Volume 446, 2007
Variational methods in image processing Ali Haddad and Yves Meyer This paper is dedicated to Haim Brezis, with our deepest admiration and respect.
Several algorithms have been proposed to unveil the geometrical structure of a given image. We will mostly focus on the ROF algorithm designed by Leonid Rudin, Stanley Osher, and Emad Fatemi. The ROF model can be used to detect the objects which are contained in an image and also to analyze its textured components. We then comment on some improved versions of the ROF algorithm. ABSTRACT.
1. Introduction Conventional wisdom says that natural images contain many objects with .simpIe geometrical forms. Detecting these objects is a main issue in image processing. When they addressed this issue in 1985 David Mumford and Jayant Shah ([29), [30]) proposed to model objects by functions of bounded variation. In the Mum ford & Shah model any imagc f is decomposed into a sum u + v between a sketch u and a second term v which takes care of the textured components and of some addit.ive noise. The working hypothesis of the Mumford & Shah model says that the objects we are looking for belong to the sketch u. These objects are assumed to be delimited by contours with finite lengths and u is a geometric-type image. It is then natural to assume that u is a function of bounded variation. This assumption will however be questioned in this paper. Jean-Michel Morel [27) told us that Ennio de Giorgi elucidated the role of the space B V (consisting of functions of bounded variation) which was implicit in the Mumford & Shah model. '
In 1992, Stanley Osher, Leonid Rudin, and Emad Fatemi simplified the Mum ford & Shah model. Their model is named the ROF model and is also aimed at splitting a given image f into a sum f = u + v . In the Mumford & Shah model as in the ORF model the decomposition f = u + v minimizes a certain energy. 1991 Mathematics Subject Classification. Primary 68U10, 94A 08 ; Secondary 42C4U, 65T60. Key words and phrases. Functions of bounded variation, Image processing, RO F algorithm. AH acknowledges support from the University of California. YM acknowledges support from CNRS, France. 273
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ALI HADDAD AND YVES MEYER
In these notes we will mostly focus on the ROF model. Here is our first result: Let us assume that we a priori know the objects which are contained in an image f and that the ROF algorithm is applied to f. It will be proved that some pieces belonging to the (physical) objects are always incorporated into the v component by the algorithm (see Theorem 3 below) . That is why Stanley Osher, Luminita Vese, and the second author have been looking for a third model. This model is analyzed in these notes and, in some cases, performs much better than the ROF model, as proved in Theorems 10 and 1 1 . These notes are organized as follows. Sections 2 contains introductory remarks on image processing, neurophysiology, and modeling. Sections 3, 4, and 5 are devoted to the standard ROF model. A first alternative to the ROF model is studied in Section 6. The BV norm is replaced by a Besov norm in this new model. This does not change the norm of indicator functions of rectifiable domains. This alternative model leads to an algorithm which is similar to Donoho's wavelet shrinkage. Therefore atomic decompositions and variational approaches have been reconciled. In Section 7 to 9 the ROF model is applied to a class of textured images and the qualities and drawbacks of the ROF model are carefully analyzed. Section 10 is devoted to the new Osher-Vese model. This material is a continuation of a program initiated in [26].
2. The primary visual cortex
A black and white image E is defined as a function f(x) = f (XI , X2 ) which is named the grey-level of x E E. We set f(x) = 0 if the point x is black in thc given image and f (x ) = 1 if x is bright white. Then f(x) E [0, 1] and the value of f(x) is the grey-level at the point x. The pair (x, f (x» is named a pixel. The bounded function f(x) is not continuous in general. Jump discontinuities of f(x) are playing a key role in image processing. Such discontinuities are generated by the edges of the objects which are present in the given image. At this level of the discussion the grey-level f is a bounded measurable function. This immediately raises a fundamental issue, since it is clear that most bounded measurable functions do not correspond to natural images. Similarly a random se quence of letters is not a poem or a novel. A natural image has a meaning and meaningful images should be adequately modeled. A first approach to modeling natural images consists in studying the laws which govern their production. The geometrical organization of the surrounding world and the laws of optics are playing a seminal role. But one should not forget. to take in account the processing achieved by the human vision system. This processing can be used as a clue for modeling natural images. David Rubel writes in [22] : In collaboration with Torsten Wiesel and Margaret Livingstone, I have attempted to build upon the work of Ramon y Cajal in an attempt to obtain a detailed understanding of the physiology of one small part of the cerebral cortex-the striate cortex, o r the primary visual cortex. . . Our main effort has been to determine how the visual information coming from the eye is handled and transformed by the brain. . . Cells in the primary visual cortex, to
VARIATIONAl, METHODS IN IMAGE PROCESSING
, • •
275
which the optic nerve projects (with one intermediate nucleus in terposed) are far more exacting in their stimulus requirements. The commonest type of cells fires most vigorously not to a cir cular spot, but to a short line segment-to a dark line, a bright line, or to an edge boundary between dark and light. Further more each cell is influenced in its firing by a restricted range of line orientations: a line more than about 15 to 30 degrees form the optimum generally evokes no response. Different cells prefer d-iffc7'ent orientations, and no one orientation, vertical, horizontal or oblique is represented more than any other. These observations, made in 1 958, had not been predicted and came as a complete surprise . . . A cell in the left hemisphere might respond to a bright red line oriented at 45 degrees to the horizontal, in a small region of the right visual field, but fail to respond to vertical lines, horizontal lines, or white and black lines. Are these discoveries bridging the gap between neurophysiology and image pro cessing ? D. Hubel is providing us with two mesHages. Hubel emphasizes the role of edge detection in perception. This will agree with the ROF model or in the Mumford & Shah model. We will return to this point in Section 3. The second message is even subtler. D . Hubel says that edges are not captured as a whole by the primary visual cortex. They are split into tiny components or "atoms" . The criterion is given by the orientation. D. Hubel tells us that some cells of the primary visual cortex are detecting the simplest geometric entities which still have a mean ing in an image. Can this processing be viewed as an atomic decomposition ? Is it ' possible to model the cells in the primary visual cortex by some wavelets 1/J). which would combine the frequency localization of the Gabor wavelets [19] together with the localization of the Grossmann-Morlel wavelets in the space domain ? In a striking paper [:31] B. A. Olshausen and D. J. Field proposed the following paradigm: the highly specialized tasks of the cells st.udied by David Hubel could be the result of an evolutionary process which selected this solution. The selection criterion is concision and robustness. That is why Olshausen and Fields aimed at finding the most concise representation of the class of all natural images. Instead of processing all nat.ural images, they used some pictures of landscapes available in a data basis. For discovering the most concise representation, they used an algorithm named lCA, or Independent Component Analysis. References to ICA are given in [1] , [13], and [23] . It happened that the building blocks found by Olshausen and Fields are close to some of the wavelets belonging to Coifman's libraries. These wavelets are described in [24] . In [5] H. B . Barlow also suggests that mammals use concise and efficient representations to process natural images. Before closing this section , let us mention a sriking discovery by Emmanuel Candes and David Donoho (1 1], [12] where atomic decompositions and geometry are truly reconciled. In full contrast with Coifman's best basis search (see [24] ) , Candes and Donoho where looking for a single hasis ej , .i E N, which would optimally compress all cartoon images. A cartoon image is defined as followH. We are given finitely many regions Om, 1 < m < N, delimited by smooth boundaries rm ' The cartoon image is required to be smooth inside each Om with jump discontinuities
276
ALI HADDAD AND YVES MEYER
across the boundary I'm of Om . If ej , j E N, is an orthonormal basis, we denote by Pj (f) the error norm Ilf - fj ll2 where fj denotes the best approximation to f by a linear combination fj = (Yl en1 + . . . + lYjCnj of j vectors picked in this basis. Using a standard wavelet expansion, we obtain the est.imat.e Pi (I) = j-l/� . Using a Fourier series expansion, t.he error would have been j-1/4 and wit.h Donoho-Candes curvelets, the error (still computed in L2) is reduced to j-1 (log j)3/2 . Up to the logarithmic term, this convergence rate is optimal. I cannot resist quoting Ca.ndes and Donoho. In [12] they write In fact neuroscientists have identified edge-processing neurons in the ea.rliest and most fundamental stages of the processing pipeline upon which mammalian visual proce.�8ing is built. . . This article is motivated by fundamental questions concerning the math ematical representation of objects containg edges: what is the sparsest representation offunctions f(x1, X2 ) that contain smooth regions but also edges ?
It is time to conclude. D . Hubel is telling us that edge detcetion is playing a key role in perception. Edge detection is also pivotal in the seminal paper [29] by Mumford & Shah, which is fully consistent with D. Hubel's discoveries. Moreover D. Marr [25] and S . Mallat [24] successfully bridged the gap between edge detection and atomic decompositions. This issue will addreslled again in Section 6. Cross fertilization between neurophysiulogy and image processing is not a dream. 3. The Rudin-Osher-FateIlli Illodel
The ROF modcl is a simplified version of the Mumford & Shah model. In both models, a given image is optimally split into two parts u and v . The first component u takes care of the objects which arc included in the given image f. These objects can be human beings, animals, some furniture or other items. The working hypoth esill says j,hat objects are delimited by boundaries whith finite lengths. Therefore they can be drawn by a painter. Neurophysiology is replaced by art which also relies on some remarkable properties of the human brain. Textures and noise are included in the second component. The space BV of funct.ions of bounded varia tion will play a key role in the the Mumford & Shah and in the ROF models. The co-area theorem by De Giorgi will explain this role. In the Mumford & Shah model, the given image f is defined over a domain n, the u component helongs to the subspace SBV of BV, which consists of functions in BV whose distributional gradient does not contain a singular diffuse measure. In other words, this distributional gradient V'u is the sum between an L l function and a measure carried by a one dimensional singular set K. Then t.he Mumford & Shah penalty on the u(x) component is a sum J(u) between two terms. The first term is the one-dimcnsional Hausdorff measure of K. The second one is the square of the L2 norm of the gradient of u (x) calculated on the complement of this singular set K. The third term of the J (u ) functional is the square of the L2 norm of vex). Then the functional to be minimized is (1 )
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where H l (K) denotes the I-dimensional Hausdorff measure of K. The two positive parameters a and f3 need to be tuned effectively. Indeed too many objects are detected the Mumford & Shah model if 0: is small. The same will happen with the Osher-Rudin-Fatemi model if the parameter A is small. The the Mumford & Shah model is raising beautiful mathematical problems which have much to do with the theory of minimal surfaces [16] . In the Mumford and Shah model, textures are treated as noise. The ROF model is similar. An image f(x) is again a sum between two compo nents 7t and v . In the seminal paper [32] the ROF model is motivated by ill-posed inverse problems and the BV norm wa.'> viewed as a regularization which preserves edges. Denoising is a particular example of such ill-posed problems. We now list some properties of the space BV(R2 ) . Our definition of the space R V slightly differs from the usual ones since do not demand that a function in BV should belong to L1 . This only concerns the behavior at infinity. One is tempted to say that the indicator function XE (X) of a domain E belongs to the space BV if and only if its boundary oE has a finite length and that the BV norm of XE(X) equals the length of the boundary. This statement is clearly valid in the (;1 case but is not true in general. In order to treat the general case, De Giorgi defined the reduced boundary 0* E of a measurable set E and proved that the BV norm of XE is the I-dimensional Hausdorff measure of its reduced boundary. References can be fonnd in [2] , or [10] . For defining this reduced boundary, let us denote by B(x, 1' ) the ball centered at x with radim; r. We then follow De Giorgi:
DEFINITION 1 . The reduced boundary 0* E of E is the set of points x belonging
the closed support of i-' = VXE snch that the following limit exists i-'{ B (x r ) } = vex) 1i-'I{B(x, r) } ,
(2)
An indicator function XE belongs to BV if and only if 8* E has a finite 1dimensional Hausdorff measure. More generally a function f(x) defined on R2 belongs to BV if (a) f (x) vanishes at infinity in a weak sense and (b) the dis tributional gradient V f of f (:1:) is a (vector valued) bounded Borel measure. An apparently weaker definition reads as follows: f belongs to BV if its distributional gradient is a (vector valued) bounded Borel measure. Then is is easily proved that f = g + c where c is a constant and g tends to 0 at infinity in the weak sense. The condition at infinity says that f * rp tends to 0 at infinity whenever rp is a function in the Schwartz class. When the Rudin-Osher-Faterni model is being used, a specific definition of the BV norm is crucially needed. Indeed the ROF model amounts to minimizing a functional which contains this BV norm. We will impose that this norm be isotropic. Let Wl begin by the simple case where Vf belongs to L 1 (R2 ) . Then the BV-norm of f will be defined as IIfll Bv = flV f(x) 1 dx. We then write f E W l,l and this function space will be useful in what follows. If Vf is a general Borel
ALl HADDAD AND YVES MEYER
278
measure, our simply minded approach does not work but paves the way to the following definition. We write p,j = oj f and we define the Borel measure (j by (j = 1p,1 1 + 1p,2 1 · By the Radon-Nikodym theorem we have p,j = 6j (x)rJ, j = 1 , 2 , where 6j (x) are Borel functions with values in [-1 , 1] . Finally the Borel measure IV(f)1 is defined by (3) We can conclude:
DEFINITION 2. The BV norm of f it! the total mass of the Borel measure IV(f) I · With an obvious abuse of language, we write Ilfll BV = J vl p,1 1 2 + 1 p,2 12 dx. We now return to the RO F algorithm. It splits an image f into a sum u + v between two components. As it was already said the first component u takes care of what might be drawn by a painter using a pencil. Therefore this component is adequately modeled by a function in BV. The texture and the noise belong to the component v . Then u is a sketch of the given image and our working hypothesis says that this sketch u captures the main geometric fcatures of the given image f. The v componcnt is more complex and is not described by a functional Banach space in the ROF model. In the ROF model, v E £2(R2) since both f and �l are square-integrable. The RO F algorithm depcnds on a tuning parameter >. > O. Objects with size less than -ix will be treated as some texture and wiped out from 'U (see Corollary 2 of Theorem 1 ) . We now arrive to the definition of the algorithm.
DEFINITION 3. Let f E £2 (R2 ) . Then the ROF decomposition f = 'tt + V of f minimizes the functional J(u) = lI'IlII Bv + >' l l v ll� among all decompositions of f as a sum between a function u E flV and v E £2(R2 ).
Later on the ROF model will be generalized and the space BV will be replaced by other functional spaces. This will pave the way between the ROF algorithm and the famous wavelet. shrinkage.
4. Properties of the ROF algorithm As will be proved in these notes, The Rudin- Osher-Fatemi algorithm is indeed performing a thresholding which is similar in spirit to the wavelet .shrinkage defined by David Donoho and lain Johnstone. More precisely the ROF algorithm has the following property : there exists a norm I I . II . and a threshold 2\ such that an image f with a norm IIfli. < A is put to 0 and an image with a norm larger than the threshold is reduced by a fixed amount (see [26] or Theorem 2 and its two corollaries). This is similar to Donoho's wavelet shrinkage. However the ROF al gorithm and Donoho's wavelet shrinkage present the same drawback. Pieces which belong to the true objects contained in an image are wiped ont by the algoritlun and viewed as belonging to the textured component (see Corollary 3 of Theorem 1 ) . The Osher-Vese algorithm performs better (Theorem 1 1) . We now define the dual norm II . II Let us denote by W I , I the closure in BV of the linear space of smooth functions with compact support. In other words f E W1,l means Vf E £1 . Here W I . I denotes the homogeneous version of the •.
VARIATIONAL METHODS IN IMAGE PROCESSING
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standard Sobolev space. The dual space of W 1,1 is the Banach space G consisting of all generalized functions 9 which can be written as 9 =
divH
(4)
where H = (hI, h 2 ) E Loo x LOO. The norm of g in G is denoted by I l g ll . and is defined as being the infimum of I I H lloo where this infimum is computed over all decompositions (4) of g. Here and in what follows IIHlloo = sUPXER2 I H(x) l , I H I = v'l h112 + I h2 1 2 . Then L2 C G and the space Go is defined as the closure of L2 in G. We have [26] LEMMA l . The Banach space BV is the dual space of Go and the norm in BV is the dual norm. This implies the following property: if a sequence Uj of functions belonging to BV converges to U in the distributional sense and if I l uj IIBv < 1, then u belongs to BV and lI u l I BV < 1 . This weak compactness property implies the existence of the optimal ROF decomposition and uniqueness is standard. The following lemma will be needed. LEMMA 2. If both u and v belong to L2, then we have u(x)v(x) dx l < Il u l l Bv l l v ll *
(5 )
For proving it, it suffices to approach u in L2 by a sequence of functions in virl ,l. The details can be found in [26] . An image f satisfies 0 < f(x) < 1 and the u component inherits this prop erty. The proof of this remark is easy. If f = u + v i� the ROF decompo sition, we have f = B(f) = B(u) + w. We then have II B(u) IIBv < I l u l l Bv and I w(x) 1 = I B(f) B(u) 1 < I I(x) u(x) l · Therefore II w l l 2 < Il v l l z and the uniqueness of the ROF decomposition implies u = B(u) as announced. �
�
Antonin Chambolle made the following crucial remark. Let FA be the closed convex subset of L2 (R2) defined by I I , 11. < A. Then we have THEOREM 1 . The ROF decomposition I = u + v of a function I E L2(R2 ) is given by (6) v = Arg inf { 1 1 1 v11 2 ; v E FA } �
The proof of this theorem will be given in the following section. For the time being, let us comment on Theorem l.
COROLLARY 1 . Let 1 E L2 (R2). If 11/11* < A , then the ROF decomposition of f is given by u = 0, v = I. If 11111. > A, it is given by f = u + v where Il v l l . = \ 2 and Il f - v l1 is minimal under that constraint. 2 Corollary 1 is now rephrased in a more intuitive way :
COROLLARY 2. If 11/11* < A, then f can be interpreted as a texture corrupted by an additive noise and does not contain any object.
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Returning to Corollary 1 and assuming I l f l l . > -ft' we have 1 = u+v , IIvll. = A. and lIull 2 is minimal under these requirements. Another characterization of the optimal pair (u, v) is given by I = u + v, lIvll. = A, and J uv = IluIIBv llvll This leads to the following definition: • .
DEFINITION 4 . A pair (u, v) of two junctions in L2 (R2 ) is named an extremal
pair if u E BV and J uv dx = lIu Il Bv l l vll. ·
Theorem 1 can be rephrased into the following assertion
COROLLARY 3. If 1 1 1 11 . >
-ft'
the ROF decomposition f = u + v 01 f is chamcterized by the following two properties: (a) IIvll . = -A and (b) (u, v) is an extremal pair.
The reader is referred to (26] where Corollary 3 is given a direct proof. Corol laries 1, 2 and 3 are telling us that the ROF algorithm is a shrinkage where the threshold is }" .
5. The abstract formulation of the ROF algorithm A proof of Theorem 1 is given now. This proof is valid in a more general con text which reads as follows. Let H be a real Hilbert space. The norm in H is denoted by I . I and the corre sponding inner product is x y. Let F be a non-empty closed convex subset of H. Let us define p : H I-t R U { +oo} by .
p(x) = sup {x . y ; Y E F }
( 7) This functional p is convex, lower semi-continuous and satisfies p(>.x) = >.p(x) for ). > 0, x E H. Let I E H be given. Among all decompositions I = x + y of I we want to find the one for which the energy J(x) = p(x) + ).lyl 2 is minimal. We denote by I = x + y this optimal decomposition. Without loosing generality we can assume ). = 1 /2. It suffices to replace F by A. F to obtain the general case. Following Antonin Chambolle we have
THEOREM 2. With the preceding notations the optimal decomposition I = x + Y is given by (8) y = Arg inf{ 1 f - YI; y E F}
The proof of Theorem 2 is not difficult . It consists in applying von Neumann's minimax theorem to the functional 1 V(x, y) = x ' Y + I I - x I 2 , x E H, y E F ) (9 '2 The minimax theorem says the following. Let E be a compact and convex set. Let F be a convex set which does not need to be given a topological structure. We consider a functional V : E x F I-t R. We define P : E I-t R U { oo } and Q : F I--t { - oo} U R by (10) P(x) = sup{V(x, V) ; Y E F} and similarly
Q(y) = inf{V(x, V) ; x E E} (11) We obviously have Q(y) ::; V(x, y) < P(x) which implies sup{Q(y) , y E F} = (3 ::; Q: = inf{P(x), x E E}. The minimax theorem (Theorem 3 below) implies Q: = {3.
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THEOREM :3. Let u.s assume that x 1-+ VeX, y) is convex and lower semi continuous on E for every y E F. Let us also assume thai, y t-t V(:c, y) is concave on F for every x E E. Then there exists an element x E E such that P(x) = {3.
Let us fix a large R > 1 and let E c H be the closed ball 1 :1: 1 < R equipped with the weak topology. The functional V is defined by (9) . Theorem 3 says that P(x) = p(x) + � If - xl2 reaches its minimum a at x . Then Q( y) reaches its maximum f3 at y and y is defined by (8). We then have f = x + y and Theorem 2 is proved. The details of t.his proof arc left to the reader. For proving Theorem 1 it suffices to observe that p(u) = II u II Bv by Lemma 2 .
6. Replacing the BV norm by a Besov norm in the ROF algorithm. We denote by S(R2) the Schwart,;o; c\a.;;ll and by S'(R2 ) the dual space of tempered distributions. A functional Banach space E is defined by the property S(R2) C E C S' (R2) where the two embeddings are continuous ones. In general S(R2) is not dense in E and we let Eo denoLe the closure in FJ of the space of testing functions. The space E* is the dual space of Eo and not of E in general. The norm in E* is denoted by II . II If E' = BV, then Eo is defined by f E L2 and V' f E L l . The generalized RO F models we have in mind are based all the energy •.
K( u) = Il u ll E + A l l f - u ll�
,
( 1 2)
The optimal decomposition f = u + v is the one which minimizes K(1.) . This op timal decomposition exists and is unique whenever the Banach space E has the following property: for every sequence Uj E E, j E N, such that l I uj li E < 1 and lIuj - ulb ---> 0, j ---> 00 , we have llullE < 1 . With these notations Theorem 3 is still valid and v is the argument of inf{ II f - v 1 l 2 ; IIvll. < 2\ }.
The Besov space Z = iJi'oo is close to nv as will be proved below. Moreover this Besov space is isomorphic to a trivial sequence space and this isomorphism is given by the wavelet expansion. Let us consider now the variant of the ROF algorithm where the BV norm is replaced by the norm in Z. It will then be proved that the u component is obtained by a variant on the standard wavelet shrinkage.
The spaee BV can be defined by the existence of a constant C such that for every y E R2 we have ( 13) II!(- - Y) - f O ll l < Gly l The lower bound of these constants G is a norm which is equivalent to the usual BV norm of f. Similarly the homogeneous Besov space Z = st, oo i� dcfincd by the existence of a constant G such that for every y R2 we have
E
11f( - - y ) + f( · + y ) - 2f O l h < Glyl
(1 4 )
The embedding BV c Z is obvious and should be compared to the well known fact that a Lipschitz function always belongs to the Zygmund class, the converse not being true. It is therefore surprising that for a large eolleetion of functions the two norms are equivalent. Here is the full story. Let SN denote the collection of all step functions with N levels. In other terms .f = CI X E1 + . . . + CNXEN where Eb · · · , EN are Borel sets and ct , · . . , CN are N real coefficients. We then have
ALI HADDAD AND
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YVES MEYER
THEOREM 4. There exist two positive constants Co and C1 such that for every f E SN we have Co llfll z < I l f l l BV < C1 N I l f li z ( 15) The proof of Theorem 4 relies on a deep theorem by Gerard Bourdaud [8] , [9] , [35] . If A E R, we define O;,(t) = (t - A)+ . We then have
THEOREM 5. There exists a constant Co such that, for every f E Z and A E R, we have (16) I I B;, U) ll z < Co llfll z Before returning to Theorem 4, let us restate Theorem 5 under the following form: COROLLARY 4. If b > a > 0, we write I = [a, b] and define a function TI (t) of the real variable t by TI (t) = ° if t < a, TI (t) = t - a if a < t < b, and finally TI (t ) = b - a if t > b. Then we have Ih (f) lI z < 2 Co llfll z
Indeed TI (t) case:
=
(1 7)
Ba (t) - (h (t). Then the proof of Theorem 4 begins with a simpler
LEMMA 3. Let E Then we have
C
R2 be a Borel set and let X E the indicator function of E. 1
ll z < I l xE ll BV < I I XE ll z xe 1 1 2
(18)
The proof of ( 18) relies on an obvious remark: If a, b, and c all belong to {O, I } , then we have la - bl < l a - 2b + ci ( 19) The proof is trivial and left to the reader. Returning to ( 18), we apply (19) to a = X E (X + y), b = Xe(x ) , and C = X E (X - y). This yields the right-hand estimate in (18) . The left-hand estimate is true for any function f E BV. We now treat the general case in Theorem 4. Without loosing generality, we can assume Cl < C2 < . . . < CN . The corollary of Theorem 5 is applied with IN = [CN-l, CN] . We then obtain I I ( CN - CN-l) XEN li z < Co llfll z Since Ilxe IIBV < Il xE ll z , it implies II (CN - CN - l )XEN I I BV < Co llfll z
We then consider Ilk = Ek U · · · U EN 5 is applied to h = [Ck-l > Ck] (with Co
= =
=
L:� (Ck - Ck-l) xO k
(21)
{x; f(x) > cd. The corollary of Theorem ° by convention) and we obtain
I I ( Ck - Ck - l ) XO JBV < Co llfll z
But f
(20)
(22)
and thc triangle inequality ends the proof.
Since the BV norm and the Besov norm are equivalent ones for simple functions, it is tempting to replace the standard ROF energy by K(u)
=
lIuli z + A l l v ll �
(23)
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VAlUATIONAL METHODS IN IMAGE PROCESSING
where v = f - u as before. A main difference with the standard ROF algorithm is coming from the fact that the Banach space Z is not contained in L2 (R2). Instead Z is contained in the Lorentz space L2,00. A typical example of a function in Z is Ix l -l which is not square integrable. Good news are coming. The space Z admits a simple wavelet characterization. Let (1/;1 , 1/; , '1j;3 ) be three functions in the Schwartz class such 2 that the eolledion 'lj!j,1c = 2j'lj!(2jx - k ) , j E Z, k E Z2, 1/; E {1j;! , 1/;2 , 1/;3 } , is an orthonormal basis of L2 . We then have
LEMMA 4. A function f E L2,00 belongs to Z coefficients c(j, k) = J f1/;j ,1c dx satisfy sup
(L
j EZ IcEZ2
Ic(j, k) l )
=
=
.si'oo
if and only if its wavelet
C < 00
Similarly a tempered distribution f belongs to the dual space Z· only if its wavelet coefficients c(j, k) satisfy
(24) =
B- l , 1 if and 00
L sup Ic(j, k) 1 = 0' < 00 .J k
(25)
This is proved in [26] and an excellent introduction to Besov spaces can be found in [36]. We remind the reader that Z· is not the dual space of Z but rather the dual of the closure in Z of the spaee of testing functions. The rtorm of f in Bi ' oo and the infimum of C in (24) are equivalent norms and a similar remark applies to 13001,1 and C'. In other words the two Besov spaces Bi'oo and Bool,l are identified to simple sequence spaces. This is not the ease for the space BV of functions of bounded variation. We have by [15] and [26] ;
THEOREM 6. The wavelet coefficients c(j, k) of a function f E BV belong to
weak-II . More precisely there exists a constant 0 which only depends on the wavelet basis such that for every positive A, we have o 3 # { (j, k) E Z ; Ic(j, k) 1 > A } < >: lIflIBV
(26)
However ( 26) does not characterize RV and (26) does not even imply f We now turn to the variant of the ROF algorithm. We write Z the new energy is defined by K (u) = lI u liz + A ll f - u ll�
=
E
Bi ' oo .
B�'oo and (2 7)
The optimal decomposition f = u + v is the one which minimizes K( u). It exists and is unique. For simplifying the following discussion the norm in Z will not be defined by (14) but instead by
lI u li z
=
sup
L
jEZ kEZ'
ICu (j, k ) 1
(28)
which is an equivalent norm. Here Cu (j, k) are the wavelet coefficients of u . With these notations K ( u) lIuli z + Ao-(u) where O'(u) 2::j 2:: 1: icf (j, k) - cu(j, k) 1 2 . This variant on the ROF algorithm leads to the following algorithm. =
=
ALI HADDAD AND YVES MEYER
284
y (j k) which mimimi7.es (29) sup ( L ly (j, k) l ) + L L i cf (j , k) - y (j, k W k JEZ kEZ2 .i Then the wavelet coeffi.ci!mt� of are cu (j, k) = y(j, k), (j, k) E Z3. This optimiza We want to find the sequence
,
u
tion problem can be solved by the following procedure. Let a positive number 7) be
j,
given and let us assume that for each by a limited budget defined by
the sequence
y(j, k), k E Z2, is constricted
L k Ez2 I y(j, k) 1 < 7). We are asked to minimize CT(U)
within this budget limitation. This problem is named P(ry) . The minimum of can be decoupled into
is denoted by w(7) . Problem P(7)
a
CT(U)
:;equence of problems.
We are given a sequence Then we want to minimize L k IX k - Yk 1 2 under
Each one of them is standard and reads as follows :
Xk , k E Z, and a positive number 7). the constraint L k I Ykl < 7) . This is a variant on the standard wavelet shrinkage. Indeed Yk and ;C k should have the same sign and satisfy 0 $ I Ykl < I X kl. These two over T/ E
requirements say that the wavelet coefficients are shrunk towards step consists in minimizing 7)
+ AW(7)
[0, ) (0
O.
The second
and is left to the reader.
depends on the wavelet basi:;. It is not t r ans l ation and rotation invariant.
This approach can be questioned. When defined by (24), the norm in
.si'oo
.
7. Analysis of textured images As it was already said, the f into a sum
ROF
algorithm is aimed at decomposing an image
u + v where u represents the objects contained in f while v models the
textured components. A collection of images will be tested to confirm t hi s working hypothesis. These images are explicitely given as a
9
and a texture
h. More
precisely we consider a sum
simplest sketch we can figure while component . For instance
swn
h
between a cartoon image
f = 9+h
where 9 is the
is an oscillating function modeling a textured
hex) = m(Nx)x(x)
which is 27f-periodic in one variable and has
where a
m(x)
is a continuous function
vanishing mean while
X
is the in
9. We write iN (x) = g(x) + x(x)m(Nx) and N will b e arbitrarily large. Can the ROF algorithm be trusted ? Does the ROF algorithm yield a decomposition iN = UN + VN where the sketch UN is close to the original sketch 9 ? More precisely we expect UN = 9 + EN , VN = h - EN where the L2 norm of the error term EN tends to O. dicator function of a rectifiable domain. This example of textures will be given a
systematic treatment in Section
This
fairy tale
is untrue, since the
ROF
algorithm applied to f = 9 does not
g back but instead a new function g. A more precise statement will be given in Lemma 9. The best to be expected in the general setting is UN = 9 + EN, VN = II, + 9 9 - EN with EN -> O. This is true, as Theorem 7 will tell us. get
-
Theorem
7 will apply to the RO F algorithm, but also to more general contexts.
We begin with a Hilbert space
II . II.- If
are given a dense subspace V denoted by
x
H and II · 11 will denote the corresponding norm. We c H together with a norm which is finite on V and
tt. V, then
lower semi-continuous on
H:
I l xli . lim
J - OO
=
00 . Let us assume that the norm
I l x - xj ll = 0
II . II . is (30)
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VARIATIONAL METHODS IN IMAGE PROCESSING
Let us denote by F c V the closed convex set defined by II . I I . < 1 and let Pp : H >-> F be the orthogonal projection; PF (:);) = z i� thc point in F which minimizes liz - I II. We know from Theorem 1 that the decomposition given by the ROF algorithm is x = y + z, z = PF(X) . Writing RF = I - Pp, we have Rp(x) = y. The main theorem of this section is the following: TlIEOREM 7. For every x, x' E H we have
II Rdx') - RF(x) 1 I
< 13(11 1: 11 + II x' ll h/l l x' - x II .
(31)
Before proving Theorem 7, let us return to the ROF algorithm. Then F will bc dcfined by I I · II . < A where II . II . is the dual norm as in Section 6. Theorem 7 reads COROLLARY 5. We consider the ROF model 'with a given vahw of the param eter A. Let it and h be two functions in L2(R2). Let h = U, i + Vj be the ROF decomposition of Ij , j = 1 , 2 . Then we have (32)
\Ve now assume that h is a sum h = it + It hetween a cartoon image II and a textured component h satisfying II h ll . < c . We then have ,
(33)
In other words the ROF algorithm does not perform what could have been dreamed, since the cartoon component II is not preserved. Instead it is modified into U I . This being said, the RO F algorithm acts in a consistent way: when it is applicd to h it yields Jl = U\ + VI and when it is applied to 12 = it + h it yields 12 = U2 + V where 'U = 1l.} + O( y'C) , V2 = V I + h - O( y'€) , the errors being 2 2 measured in L2 . The relevance of (31) comes from the fact that in many applications the norm which controls x' - x is much weaker than the norm which is used in the left-hand side of (31). It is indeed the ca.<;e for the ROF algorithm. Theorem 7 is not in teresting when I l x' - x I I . > 1 . Indeed the mapping PF is a contraction and RF is Lipschi t� Wc have II RF (x') - RF(x) 1 1 < 2 1 1x - x' il which improves on (31 ) . We will prove the optimality of the square root in (31 ) . The counter-example which will be given below concerns the standard ROF algorithm. One cannot replace RF by PF in (:n ) . Otherwise we would have Il x' - xii < Cyfl l x' - x II . whenever Ilx I I < 1 , I lx' ll < 1 , which is not true. A trivial counter-example is givcn by the usual ROF algorithm applied to II = 0 and h ex) = cos(w . x)
.
Finally the weight given by II x ll + I lx' ll in the right-hand side of (31) cannot be erased. The simplest counter-example does not concern the standard ROF al gorithm but the wavelet shrinkage. It is given by H = 12(N) and IIxll* = 11 1: 11 00 = sup I x nl . Here PF (x ) = i is defined by in = Xn if I xn l < 1 while in = signxn if not. In other words RF(X) = x is defllled by xn = 0 if I xn l < 1 and xn = Xn - signxn if not. This operator RF is a shrinkage which pulls the coefficients back to O. We now check (31) on the twu sequences (xn ) and (x�.) defined by x� = 1 + € if 1 < n < N and x� = 0 if n > N + 1 . \\'hen Xn = 1 if 1 < n < N and In = 0 if n > N + 1, we
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have I lx' - x ll oo = f while II RF(x') - Rdx) 11 = EJN. The constant 13 is obviously not optimal. The proof of Theorem 7 begins with a standard lemma
LEMMA 5 . Let F c H be a closed convex set containing O. Let Xo E II, Zo = PF (xo ) , Yo = Xo - PF (xo ) , and d = Ilyo l l · If X o = YI + Zl where Zl E F and IlyI iI < d + Ell xo ll , then we have ,,2
II Yl - Yo II < 2 E + 2 " xo l The proof is standard and left to the reader.
(34)
The set F is defined by II . 11* < 1 in the following discussion and the other notations of Lemma 5 are kept. We then have LEM MA 6 . If 0 < 1 + 1], then we have
f
< 1, 0 < 1] < 1 , Xo = Yl + Zl , II Yl l 1 < d + E l l x o ll, Il zI i I . <
(35) II Yl - yo ll < 9JE + 17 II Xo II For proving (35), we define 2\ = ( 1 + 1]) - l ZI which ensures Ilzl ll. < 1 and l ) 21 E F. We then define iit = Xo - Zl . By trivial algebra we have Yl = ( 1 + 17 Yl + 1](1 + 1]) - 1 Xo and the triangle inequality implies 1] 1]d 10 + 1] " + (36) < IlyI iI < d + i3 l 1x ol l , 13 = 1 + 1] (1 + 1] ) ll xo ll 1 + 1] On the other hand, the definition of Yl and the triangle inequality give _
1]
(37) ( 2 + 1]) II xo II II:ih - yI il < 1 + 1] Finally Lemma 5 is applied to the suboptimal decomposition X o = fit + Zl . This is the non-trivial ingredient of the proof. It yields lIih - Yo II < 2 Combining (37) with (38) yields (35) .
132
13 + 2" Il xoll
We now prove Theorem 7 under the following form
LEMMA 7. If x , x' E H, x = Y + z, z = PF (x) , x' = Y' + z' , ' d Ily li. = Ily' l l , and Il x' - x II . < 1, then - =---'"-"" ' I l y - yll < 1 3Jll x' x l I . ( lI x ll + II x' ll l
(38)
' z
= PF (x' ) , d =
(39)
In Lemma 7, F is defined as in Theorem 7. We have 0 E F which implies ; + 0 is challenging the optimal Ilyll < Il x l l · Indeed the trivial decomposition x = r decomposition x = Y + z . We also have IIY'II < Il x' ll . The proof of Lemma 8 starts with the following fact (40) ( ll x ll + II·T' II ) Id' - dl < 1 ,, where 10 = II x - x'II*.
:
For proving (40), one writes x = x' + x - x' = y ' + z' + x - x' = y" + z" with z" = (1 + E)- l ( z' + X - x') and Y" = Y' + l�E (z ' + X - x'). Since II zll . < 1 ,
2 87
VARIATIONAL M ETHODS IN IMAGE PROCESSING
the triangle inequality yields I l z" l l . < 1. It implies Il y ll < I l y" ll since x = y + z i� optimal. The definition of y" and the triangle inequality yield (41)
Since y' = x' - z' and I l y' l i ::; IIx 'lI, we have liz' + x - x' il < IIx ll + II x' l l . Combining this estimate with lI yl l < lI y"l I and (48) , we obtain
:
d < d' + x ll + II x' lI) (lI 1 t But x and x' are playing symmetric roles. We also have d' ::; d + and (40) follows from these two inequalities.
(42)
l�< ( lI x ll + I I x' l I )
We now return to x = y + z and write x = y' + w where w = z' + x - x' . Then IIwll. < 1 + t and lIill < d + l�« lI x ll + II x' l I ) by (47). Lemma 6 yields (43)
lIy' - yll < 1 3(l I x ll + II x' I ! ) JE
as announced. 8. Optimality in Theorem 7 Theorem 7 is optimal in the following sense. The square root in the RHS of (32) cannot be replaced by an exponent larger than 1/2. For proving this remark, let us denote by D the unit disk (centered at 0 with radius 1) and denote by Dc the disk centered at 0, with radius 1 + f . Consider h = XD, h = XD, . Then we have 1112 - h ll . ::; f. This follows from a more general observation given by the following lemma. LEMMA 8. If g is a radial function and if her) = II g li. = II h lloo
; I: sg(s) ds,
then (44)
The proof of Lemma 8 is straightforward. It suffices to write g(r) = Ix (�h) + :y (�h) which follows from the definition of h. Then g = divH, H = (�h, � h) and IIHII"" = I I h l loo. This implies IIg l i . < I I h ll oo • The lower bound i� much easier. The norm of a radial function fer) in BV(R2 ) is simply 21r 1000 I f' (r) l r dr. A lower bound of the dual norm II g li . is given by 21r sup I fr�c g(r) f(r) drl where this supremum is computed on the collection of all testing functions satisfying 21r 1000 I f' (r) I r dr < 1 . This ends the proof of Lemma 8. Lemma 8 implies the following result LEMMA 9. Let XR denote the indicator function of the disk I xl < R. We then have I I xR I I . = R/2 . The ROP decomposition of xR is given by XR = UR + VR where UR = 0 if )'R < 1 and UR = ( 1 - >.1 ) X R, Vn = >.1 XR if )' R > 1 . The first assertion of Lemma 9 is given by Lemma 8. The second one follows from the characterization of the ROP decomposition given by Theorem 1 . We have II VR II . = ft and J tLR vR dx = lI uRIIBvllvn ll . which ends the proof. We return to the optimality claim. Assuming ). = 2, the ROP decomposition of h and 12 are 11 = Ul + VI , 12 = U2 + V2 where UI = ! 11 , U2 = (1 - 2(1�<» ) 12-
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ALI HADDAD AND YVES MEYER
It implies U2 u] 2(1�') h + annulus 1 < Ixl < 1 + E o Finally -
-
2�i-e:) x. where x. is the indicator function of the IIu2 u 1 11 2 > J;< which implies the optimality. -
9. Examples of textures We still consider the original Rudin-Osher-Fatemi algorithm. Here and in what follows, II II . denotes the norm in the Banach space G which is the dual of W1,1. The ROP decomposition yields u = 0 (no object) whenever Ilfll. < A. We now would like to better understand the meaning of this condition. More precisely we aim at showing the following .
PROPOSITION 1 . Oscillating patterns have a small II II. norm. .
This is still a vague statement since we did not define what an oscillating pattcrn is and the smallness condition in Proposition 2 doe� not mean anything. Proposition 3 will take care of these issues. For the time being, let us introduce the following defiJlition:
DEFINITION 5 . Let J.l be a non-negative Borel measure. We say that J.l is a Guy David measur'e if there exists a constant G such that lor every disk D(xo, R) centered at .TO with radius R we have J.l(D(xo , R)) < GR
(-1 5)
For instancc let r be a rectifiable curve in the plane and let a denote the arc length on r . Then a is a Guy David measure if and only if r i� Ahlfors regular. Guy David proved that this condition is equivalent to the boundedness of the Cauchy integral operator acting on L2(r; da). If It is a signed Borel measure, we say that J.l is a Guy David measure if and only if the measure 1 11 1 satisfies (45) and the optimal G in (45) will then be denoted by 111111D. Understanding the dual norm II . II . is much easier if one takes in account the following theorem
THEOREM 8. A non-negative Borel mmsure J-L belongs to G if and only if J.l is a Guy David mmsure.
Theorem 8 does not hold in dimensions larger than 2. Before proving Theorem 8 let us stress its meaning. We need to prove the following estimate
I
J f dl1 l < Gllfll Bv ll J.l II D
(46)
whenever f belongs to the Sobolev space W I,I . The LHS of (46) does not have a meaning if f E B V. For instance the integral J f(X l , 0) dX l is meaningle�s if I E BV. Indeed a function in B V does not have a trace on a line and is only defined almost everywhere. Vie now return to the proof. One way is obvious. It suffices to test (53) on a function fR defined by fR (X) 1 on the disk D(xo, R) with fR (X) = 0 outside 2 R�lxl in between. Then IIf ll the disk D(xo, 2R) together wi th IR(X) R Bv = CR and J IR dJ-L > J.l(D(xo . R) , since J.l is non-negative. This ends the easy part of the proof. =
=
VARIATIONAL METHODS
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The other direction is more involved. It suffices to treat the case where I(x) > O. For proving (46) we replace p, by p, * cf>. where cf>. (x) = �cf>(:). The function cf> is smooth with compact support and satisfies J cf> = 1 . This permits to extend (46) to any function I E BV and what is even more important it permits to use weak limits in BV. More precisely we can approach a function I E BV with com pactly supported functions in BV Next these functions are replaced by simple functions, as in Proposition 1. Let u� now denote by I a compactly supported simple function. Then I(x) = L� Cj Xj (x) where 0 < Co < Cl < . . < CN . Here Xj is the indicator function of a set Ej which is a finite union of squares. We write OJ = Ej U . . U EN. Finally 1 I /11 B v = E�- l (cj+l - cj )lj where Ij is the length of the boundary anj . Finally J f dp, = E�- l cjp,(Ej). But Ej = OJ \ nj H which implies p,(Ej) = ",(nj) - p,(nj+l)' Using Abel's transformation we obtain N · J I dp, = Eo (Cj - cj -dp,(nj ) . Here C-l = O. It then suffices to use the following geometrical estimate p,(n) :::; 11p,llvH1 (afl) which implies ",(nj ) < Clj llp,llv and J I d", < CII/IIBv l I ",lIv. This is Theorem 8. For proving our estimate we write fl as the finite union of its connected components nrn and we denote by Lm the length of anm. Then Om is contained in a disk Dm of radius Lm. It suffices to use (45) for Dm to prove the geometrical estimate. .
.
Theorem 8 is not valid in other dimensions. Indeed a domain 0, H2 (aO) :::; 1 can be unbounded. ,
C
R3 whith
Guy David measures are needed in understanding the pointwise multipliers of BV. DEFINITION 6. We say that a measumble 17J,nction m(x) is a tiplier 01 BV if, for every f E BV, the junction g(x) = m(x)f(x) BV.
pointwise mul also belongs to
The closed graph theorem yields the continuity of the mapping M m(x)f(x). We then have for every f E BV,
:
f(x) f-)
(47) II m(x) f(x) II B v < CII /IIBV It is readily seen that ( 54) implies m(x) E £DO . If m(x) E Loo, it suffices to prove (54) when I E Wl , l . Indeed if I E BV, there exists a sequence Ij E W1,1 such that II fj llBV < II / l1Bv and Iim llf - 1; 112 = 0 which ends the proof of our remark. If I E Wl,l we have "il(m!) = V f + fVm. Since m E L=, we have m"ilf E L l and we are led to estimating Ilf"ilmlll. Since f E W I ,I implies I f I E W I , I , everything ends with the computation of J l"ilmlf dx when f is a non-negative function in m
WI, I . Then Theorem 8 implies the following
measumble function m(x) is a pointwise multiplier of BV if and only il m E LOO(R2) and "ilm is a Guy David measure. THEOREM 9. A
For instance let us consider the indicator function xn of a domain 0, with a rectifiable boundary. Then Xn is a poinwise multiplier of BV if and only if an is a Guy David curve. We then apply Theorem 9 to texture analysis. The following corollary preludes t.he analysis of some textures which are present in some SPOT images.
290
ALI HADDAD AND
YVES MEYER
COROLLARY 6 . Let us define the multiplier norm of m by Ilmll M = Ilmlloo +
II'V'mIiD. Then there exists a constant Co such that Il m(x)f(x) l l . < Co ll ml lM ll f li. for' every f E L2.
(48)
Let us apply this corollary to some textured components found in a few SPOT images. In what follows a is a small positive number. We begin with a real-valued function w{x) = W(Xl ' X2 ) E LOO(R2 ) which is a-periodic in X l : W{Xl + a , X2) = W{Xl' X2 ) ' We also assume Je: wet, X2 ) dt = 0 identically in X2. The latter assumption can be questioned and will be suppressed in a moment. This periodic function w{x) is modeling a textured component of a SPOT image. However this textured component is delimated by a boundary. It implies that our texture model will be given by the product m{x)w(x) where m{x) is the indicator function of a bounded set D, or more generally m belongs to LOO and 'V'm is a Guy David measure. Then Bernstein's inequalities yield Il wll. :::; ;al l wlloo. Therefore the product m{x)w(x) is a texture in a sense given by the following estimate:
PROPOSITION 2. With the preceding notations we have
II m{x)w(x) lI. < Coa llw lloo ( lI m lloo + lI'V' m I lD ) (49) We now treat a full image f{x) which is a sum f{x) = fo {x) + m(x)w{x) (50) where fo E L 2 and m(x), w{x) are as above. We apply the ROF algorithm to fo and obtain
fa = Uo + Vo
(51)
We then keep the same value of the tuning parameter A and we apply the ROF algorithm to f. Theorem 8 yields f = u + v where l Iu - uo l1 2 < CVa and Ilv - (vo + w(x)m(x)) 112 < Cfo. Let us now compare two images f and fa which only differ by the textured component m(x)w{x) . The ROF algorithm splits fa into a sum Uo +vo where va is taking care of the textured components which were already present. If we add m(x)w(x) to fa , the ROF algorihm will incorporate it inside the texture as long as the mean value of the oscillating pattern is 0 (we here neglect an error term which is O{ fo)) . What would happen if the mean value of w is not zero (w is still a periodic in xJ) ? We will see that the ROF algorithm does not incorporate the whole of m( x}w into the texture component. Indeed we have w = Wo + Wt where Wo fulfills the vanishing mean condition and WI does not depend on Xl. Then the product wlm can be incorporated into fo and we have f = fa +mwl + mwo. We can use the preceding discussion where fa and mWl are glued together. When the ROF algorithm is applied to the full image f (x) = fo (x) + w(x)m(x), it yields f = u + v where vex) '7 Vo + wo (x)m(x) + R(x). This error term R(x) is coming from the new 'object' wlm which is now combined with fo and which does not stay inside u but spoils v.
10. A new algorithm by Stanley Osher and Luminita Vese This new algorithm was implicit in [26]. It will be named the Osher-Vese algorithm in hommage to the remarkable scientific work achieved by Stanley Osher and Luminita Vese in [37] and [38] (see also [34]). One should also consult [3] , [4], and [20j. The Osher-Vese algorithm is a variant of the standard ROF algorithm.
VARIATIONAL METHODS IN IMAGE PROCESSING
291
The notations of the preceding sections are kept : G denotes the dual space of W i, 1 and the norm in G is • . Here the given image does not need to be a function in but can also be a tempered distribution belonging to G. We aim at splitting this image into a sum u + v where the two components and v have the same meaning as in the standard ROF algorithm: the objects which are present in the given image f should belong to and the textured components a.nd the noise are incorporated in The functional we want to minimize is now 8 (u) + An optimal decomposition exists but is no longer unique in some instances. Counter-examples will be described below. The infimum of over all the decompositions .f = will be denoted by e(f) . The role of the tuning parameter is different, as Lemma 1 1 will tell us. We begin with an obvious remark:
11 · 1 1
L2
u
u
v.
=
e(u)
Il u llBv A ll v ll
• .
u+v
A
LEMMA 10. For every function f E BV we have
Il fll� < Ilfll.llfllBv 1 1 11111. < 2J1T' II fll 2 < 41r IIf llBv
and (52)
The second estimate follows from the isoperimetric inequality and a simple duality argument yields the first one. The following observdtion is a trivial conse quence of Lemma 10. LEMMA 1 1 . If 0
f is u = 0,
11
= f.
< A < 41r,
then the Osher- Vese decomposition of any function ,
This should be compared with the similar statement for the ROF algorithm: =} u = 0, v = f. The proof of Lemma 1 1 is straightforward. We have III ull . . This obviously implies 41r, the energy of the trivial decomposition If of f given by f = 0 + f is smaller than It implies u = O. If = 41r, U = 0 is still a solution but this solution is not unique in general. A counter-example will be given after the following definition.
IIfll. < 21>IIfli. < lI u li. + III - u ll. < 4� lI u llBv + A l l fll . < i;: ll u linv + A lii - u ll. - A < e(u). D EFINITION
-
A
7. A function .f E BV is extremal if ll f ll� =
1 1 /11. II /I I Bv
An example of an extremal function is the indicator function of a disk. Other examples are given and studied in an important contribution by E. Giusti [21] and two exciting papers by G. Bellettini, V. Caselles a.nd M. Novaga [6] , [7]. For example let C be a compact and convex set with a C2 boundary. Then E. Giusti proved that the indicator function of C is an extremal function if and only if the curvature " at every point of the boundary does not exceed P(C)/ICI . Here P(C) is the length of the boundary of C. Let us write a = and c = b= If = ci a , then all the decompositions f = + v", = af + ( 1 - a ) f a E (0, 1) have the This shows that the Osher-Vese decomposition is not unique same energy in general. Are there other functions f for which the uniqueness fails ? We now turn to a larger class of functions and prove some exciting properties of the 06her-Vese algorithm. These functions which will be named "cartoon images" have a trivial Osher-Vese decomposition. Thc existence of cartoon images explains the higher performances of the Osher-Vese algorithm, when compared with the ROF algorithm. We begin with defining cartoon images.
u'"
8(u",).
D EFINITION
1 1/11 .,
11/ 1 1 2,
Ilf llB v . A
,
8. A function f E BV is a cartoon image if there exists another
function g E BV such that
f (x )g(x) dx =
II f llBV
and
I l g li. = 1
(53)
ALI HADDAD AND YVES MEYER
29�
In other words a function f is a cartoon image if there exists a second function 9 E BV, 9 =I- 0, such that J f(x)g(x) d:r; = IIfI lB V l I gl I It then suffices to mUltiply 9 by a suitable constant. •.
If f is a cartoon image, the pair (f, g) is extremal in the sense given in [26] and we will say that 9 is dual to f. More precisely, let us apply the standard ROF algorithm to H given image f E BV. If I l f l i. > :/:r, the ROF algorithm yields f = u+ v where I l v l l = A and .r u(x)v(x) dx = Ilu l lBv II v I!.. Since v = f - u E BV, it implies that u is a cartoon image. Conversely if u is a cartoon image and if v is dual to u, then f = u + A 7J it; the ROF decomposition of f. We can conclude: .
3. A function u is a cartoon image if and only if there exists a function J of bounded variation and a tuning parameter A such that the RO F algorithm yields f = u + v. PROPOSITION
Here are some other examples and counter-examples. Let n be a bounded domain with a C3 -boundary. Then the indicator function XI! of n is a plain image. Indeed we consider the normal vector vex) at x E 8n and extend it to a C2 -vector field H (x) such that IIHlloo = 1 . We then define g= divH ami we have Il gl l. < 1 . But ( 54) It implies I Ig li . = 1 and f is a cartoon image. Similarly a piecewise mnstant func tion with jump diHcontinuities across C3-boundaries is a cartoon image. Here are two counter-examples. The indicator function of a polygon cannot be a cartoon image, as it is proved in [26] . A second (;Qunter-example is given by f(x) = exp ( - lx I 2 ) Then J fg dx = IIfllBV and I Igli. = 1 imply g(x) = I x l - l . This 9 cannot belong to BV. However the radial function :p defined by ip(x) = 1 when I x l < 1 , cp(x) = 1:1:1-2 if I xl > 1, is a cartoon image. We now have: .
10. Let 9 be a canoon image and let 9 be dual to g. If A > AO = IIjj I!BV, then the corresponding Osher- Vese decomposition of 9 is the trivial decom position given by u = g, v = o. THEOREM
We have AO > 27f by Lemma 10 and I lgll. = 1 . Therefore Theorem 10 is consis tent with Lemma 11. Theorem 10 implies uniqueness. We argue by contradiction. Let 9 = u + v be an optimal decomposition minimizing K (n) . We have
I lgll B v =
gg dx = J 1Lg dx +
J vjj dx
=
( 55)
It + h
Ih l < lI u llBV and I h l ::; II v ll. l Ig llll v by Lemma 2. It implies Il g llBv < lI u llBV + II v ll. lljjllBv < lI u linv + A l l v l l . unless v = O. The trivial decomposition
But 9
= 9 + 0 is winning against 9 = 'u + v.
Let us now treat a more involved situation where f = 9 + h with being a cartoon image. Let 9 be dual to 9 and AO = IIgI lB V·
I lh ll.
< E, 9
VAlUATIONAL METHODS IN IMAGE PROCESSING
293
THEOREM 11. If A > Ao, then the Osher- Vese algorithm yields a decomposition f = u + v 1JJheTf� 71. is close to 9 in L2 . More precisely 2€A (56) 1111, - gil. < A _ A ) 1 1 11, - 911 2 < G(A ) vC o
Here we do not know whether the optimal decomposition is unique. The con stant G(A) will be made explicit in the proof. We first write
1=
J f (x)g (.'E) dx = II + 12 = IlgllBV + J g (x) h (x) dx
We also have
1=
u(x)g (x) dx +
v ( x )g(x ) dx =
h
+ 14
(57)
(58)
We used the fact that 9 is plain to obtain h = IIgll BV. Then Lemma 2 yields the following bounds 12 > - ll gII Rv llhl l . and 13 + 14 < IlglI . llull Bv + IlglIBv llvl1 . Therefore
IIgll. 1 1911LJV - llgIIBv llhll. < I < IlglI . llullBv + 11911 Bvllvll. It suffiees to oLerve that Ilg li. = 1 to obtain I lgllBV - Go l l h l l . < I l u l l BV + Go llvll. But we also have
..
(59 )
(60)
(61) since f = 11, + v is assumed to be an optimal decomposition. Since A > Go) it suffices to combine (60) and (61) to obtain II vll. < ��gO('. We have u - 9 = h - v and Il u - g il . < 2 t A ACO by the triangle inequality. On the other hand f = 11, + v is an optimal Osher-Vese decomposition. It implies lIullB v < I lglIBV + Af which yields Ilu - gllBV < 2 1!911BV + >.c. Then J Ill, - 912 dx < Ilu - 911BVIlu - gil. < 2€ A ->CO (2 11 911BV + fA) . This ends the proof of the second assertion. Theorem 1 1 is raising the problem of the converse implication. Is it true that cartoon images are the only functions for which the Osher-Vese algorithm yields a trivial answer ? Theorem 1 1 is a first step which should be completed by more mathematics and more numerical experiments. We would like to thank Jerome Gilles for providing us with numerous examples. We express our gratitude to the referee ; his constructive remarkH were a valuable help.
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(1993) 413-422. G. Bourdaud and M.E.D. Kateb. Fonctions qui op erent sur les espace.' de Besov. Math. Ann. 303 (1995) 653-675. A. B raides Approximation of free discontinuity pro blem•. Lecture Notes in Mathematics 1694 Springer (1991). E. Candes and D. Donoho. Ridgelets: a key to higher-dimensional intermittency? Phil. 'Trans. R. Soc. Lond . A 1760 ( 1999) 2495-2509. E. Candes and D. Donoho. New tight frames uf cu" velels and op tim al representation of ob jects with piecewise C2 singulari ties. Communications on Pure and A pplied Mathematics, Vol . LVII, (2004) 0219-0226. J.-F. Cardoso. Blind signal separation: statistical p,inC'iples. Proceedings of the IEEE Spe cial issue on blind identification and estimation 9 (10): 2009-2025, October 1 99B. A. ChamboUe and P.L. Lions. Image recovery via total variation minimization and related problems. Numer. Math. 76 (1997) 167-178. A. Cohen, R. DeVore, P. Petrushev and H. Xu. Nonlinear approximation and the space BV(R2 ) . American Journal of Mathematics 121 (1999) 587-628. G . David. Singular sets of minimizers for the Mumford-Shah functional. B irkhiiuser (2004). R. DeVore an d B. J. Lucier. Fast wavelet techniques JOT' near optimal image compression. 1992 IEEE Military Communications Conference O ctober 1 1-14, (1992). D. Donoho and I. Johnstone. Wavelet shrinkage: Asymptopia? J.R.Statist. Soc. B 57 (1995) 301-369. D. Gabor. Theory of communicati on. J. lEE, 93 (1946) 429-457. J. G ill"". Ph. D. J une 22 (2006) available from Jerome . Gillesllietca . fro E. Giusti. On the equation of surfaces of prescribed mean curvature, Existence and unique ness without boundary conditions. Invent. Math. 46 (1978) 1 11-137. D. Hubel. DiscUT'so de inv es ti dura de Doctor Honoris Causa. (1997) Universi d ad Alltonoma 10
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[23] A. Hyviirinen , J Karhunen, and E. Oja. Independent Component Analysis. John Wiley & Sons (2001). [24] S . M allat . A Wavelet Tour of Signal Processing Academic Press (1998). [25] D. Marr. Vision, A computational investigation into the human representation and process ing of visual information. W.H. Freeman "'lid Co (1982). [26] Y. M eyer . Osc'iliating patterns in image processing and in s ome nonlinear evolution equa tions. (L ewis Memorial Lectures) AMS (2001 ) . [27] J-M. Morel and S. Solimini. Variational methods in image segmentation. Birkhiiuser, Boston (1995). [28] D. Mumford. Book review on [39] . Bulletin of the American Mathematical Society 33 n.2, April 1996. [29] D. Mumford and J. Shah. Boundary deteelion by w'inimizing functlOnals. Proc. IEEE Conf. Compo Vis. Pattern Recognition (1985). [30] D. Mumford and J. Shah. Optimal representations by piecewise smooth functions and asso ciated variational problems. Comm. Pure Applied Mathematics 42 (5) (1989) 577-685. [31] B. A. Olshausen and D. J . Field. l!)mergence of stmple-cell receptive field properties by learn ing a sparse code for natural images. N ature 381 (1996) 607-609. [32] S. Osher and L. Rudin. Total variation based image restoration with free local constraints. In Proc. IEEE ICIP, vol I, pages 31-35, Aus tin (Texas) USA, Nov. 1994. [33] S. Osher, L. Rudin and E. Fatemi. Nonlinear total variation based noise removal algorithms. Physica D 60 (1992) 259-268. [34] S. Osher, A. Sole, and L. Vese. Image decomposition and restoration using total variation minimization and H - 1 -norm. Multiscale M odeling and Simulation 1 (3): (2003) 349-370. [35] P. Oswald. On the boundedness of the mllpping f t---> I f I in Besov spaces. Comment. Univ. Carolinae 33 (1992) 57-66. [36] J . Peetre. New thoughts on Besov Spaces. Duke Univ. M ath Series (1976). [37] L. Vese and S. Osher. Modeling textures with total variation minimization and oscillating patterns in image processing. Journal of Scientific Computi ng 19 (2003) 553-572. .
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VAlUATIONAL METHODS IN IMAGE PROCESSING
[38]
L. Vese and S . Osher. lmage denoi ing and decomposition with total variation minimization s
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DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALIFORNIA, Los ANGELES, CALIFORNIA 90095-1555 E-mail address: ahaddadlDmath . ucla. edu CMLA, ENS-CACHAN, 94235 CACHAN CEDEX, FRANCE E-mail add.·ess: ymeyer
,
Contempor:ny Mathema.tics Volume 446, 2007
Null hypersurfaces with finite curvature flux and a breakdown criterion in General Relativity Sergiu Klainerman
In honor of Haim Brezis, for his luminous energy, inspiration
and
friendship
1. Introduction ,
I report on recent work in collaboration with Igor Rodnianski concerning a geomet criterion for breakdown of solutions (M, g) of the vacuum Einstein equations,
ric
Rct,l3(g)
=
(1)
O.
Here Ro,13 denotes the Ricci curvature of the I 3 + 1 dimensional Lorentzian manifold (M, g). The main result discusser! here is stated and proved in [Kl-Ro6] ; the proof depends however on the results and methods of [KI-RolJ , [KI-Ro2J , [KI-Ro3] [KI-Ro4] which establish a lower bound for the radius of injectivity of null hy persurfaces with finite curvature flux as well as [KI-Ro5] in which we cunstruct a Kirchoff-Sobolev type parametrix for solutions to covariant wave equations. Assume that a part of space-time MJ C M is foliated by the level hypersurfaces �t of a time function t , monotonically increasing towards future in the interval I C JR, with lapse n and second fundamental form k defined by, k(X, Y)
=
n=
g(DxT, Y) ,
( _ g(Dt, Dt») -
1 /2
(2)
where T is the future unit normal to �t, D is the space-time covariant derivative associated with g, and X, Y are t:mgent to L:t . Let � o be a fixed leaf of the t foliation, corresponding to t = to E I, which we consider the initial slice. We assume that the space-time region M r is globally hyperbolic, i.e. every causal curve from a point p E MJ intersects La at precisely one point. Assume also that the initial slice verifies the assumption. 1991 Mathematics
Subject Classification.
The author is partially supported by
NSF
35J10
grant DMS-0070696. ©2007 American Mathema.tical Society
297
298
SERGlU KLAINERMAN
A 1. There exists a finite covering of I:u by a finite number of charts U such that for any fixed chart, the induccd metric g verifies
\;Ix E U with 6.0 a fixed positive number.
(3)
Though our work in [KI-Ro6] covers only the second of the following two situations below, it applies in principle to both. (1) The surfaces
2: t
are asymptotically flat and maximal. trk = o .
(2) The surfaces 2:t are compact, of Yamabe type - 1 , and of constant, nega tive mean curvature. They form what is called a (CMC) foliation . trk
=
t<0
t,
We may assume in what follows that the region MJ corresponds to the time interval I = [to t.) with t. < O. Without loss of generality we shall identify the entire space time M with MI. We can also assume that the initial hypersurface 2:0 corresponds to t o = - 1 . ,
Given a t-foliation of M and p E M we can define a point-wise norm III(p) I of any space-time tensor II via the decomposition, o x = -X T + X, X E TM X E T2:t . ,
We denote by I I II (t ) I I Lp the LP norm of II on 2:t . More precisely, IIII(t) II Lp
=
r
Jr;,
IIIIPdvg
with dVg the volume element of the metric g of Et . The main result. I want to report is the following, THEOREM 1 . 1 (Klainerman-Rodnianski) . Let (M, g) be a globally hyperbolic devel opment oj 2:0 foliated by the level hypersurfaces of a time junctzon t < 0, verifying
conditions (1) or (2) above, such that 2:0 corresponds to the level surface t = to. Assume that Eo verifies Al. Then the first time T* < 0 of a breakdovm is charac terized by the condition
lim sup t�T...
"
( l I k(t ) ll u
",
+ I I V' log n(t ) IILoo )
=
00.
(4)
More precisely the space-time together with the foliation 2:t can be extended beyond any value t. < 0 for whichl,
sup IIk (t) I I L� + 1 1 V' log n(t ) II L=
t E[to .t.) l For simplicity we
use
below the same constant
Ao
=
6.0 < 00.
as in (3).
(5)
299
CURVATURE
Condition (5) can be reformulated in terms of the deformation tensor of the future unit. normal T, 7r = (T)7r = LTg. By a simple calculation, expressed relative to an orthonormal frame eo = T, e I , e2 , e3 , we find, (6) 7r00 = 0, 7rOi n- I V ·n -
,
,
Thus condition (5) can be interpreted as the requirement that T is an approximate Killing vectorfield in the following sense,
A2. There exists a constant .6.0 such that, sup 1 1 7r(t) IILoo t E lto,t.)
�
( 7)
.6.0 ,
In addition to the constant .6.0 in AI, A2 the constant Ro, which bounds the £2 norm of the spacetime curvature tensor R on �o, plays an essential role, (8) To prove the theorem we have to show that if assumptions Al and A2 are satisfied then the space-time M J , I = [to, t ) t. < ° can be extended beyond t We want to emphasize that theorem 1 . 1 is a large data result; indeed one need not make any smallness assumptions on the constants .6. 0 and Ro . .
,
• .
Our theorem is connected and partially motivated by the following three eartier breakdown criteria results:
1 . The first is a result of M. Andersson, [And] , who showed that a breakdown can be tied to the condition that lim sup II R (t) II Loo
=
00 .
(9 )
It is clear that condition (4) is formally weaker than (9) as it requires one degree les� of differentiability. Moreover a condition on the boundedness of the £00 norm of R covers all the dynamical degree of freedom of the equations. Indeed, once we know that I I R(t) II L� is finite, one can find bounds for n, Vn and k purely by elliptic estimates.
2. Our result can be also compared to the well known Beale-Kato-Majda criterion for breakdown of solutions of the incompressible Euler equation atv +
(v · V)v = -Vp,
div v = 0,
with smooth initial data at t = to. A routine application of the energy estimates shows that a solution v blows up if and only if t. •
to
(10)
The Beale-Kato-Majda criterion improves the blow up criterion by replacing it with the following condition on the vorticity w = curl v :
1tot• IIw(t) IIL�dt = 00.
(11)
ROO
SERGIU KLAINERMAN
Similarly, in the case of the Einstein equations energy estimates, expresHed relative to a special system of coordinates ( such as wave coordinates) , show that breakdown does not occur unless
1
t•
to
lI ag (t) IIL "" dt =
00 .
This condition however is not geometric as it depends on the choice of a full coor dinate system. Ohl;erve that both the spatial derivatives of the lapse 'Iln and the components of the second fundamental form, kij = - � n- l 8t gij , can be interpreted as components of ago Note however that after prescribing k and 'Iln we are still left with many more degrees of freedom in determining ago
3. Finally, the result whose proof is closest in spirit to ours and which has played the main motivating role in developing our approach, is the proof of global regUlarity of solutions of the Yang-Mills equations in JR3+1 by Eardley and Moncrief, see [EMI] . To explain the connection of their result to ours we review below its main ideas. Recall that the curvature t,cnsor Fa;3 8aA{J - a{:J A + [An ' A{J] of a Yang Mills connection Aadx"' , with values in the Lie algebra Bu(N) il; a critical point of the Yang-Mills functional =
Y M[F]
=
and verifies the wave equation,
",
r Tr ( * F J'R.3 1 1
1\
F)
(12) where
=
mJlvD �A) D!,A) denotes the covariant wave operator, D�A) = aJl + [AJl ' .] . Thus. with 0 the usual O (A)
D' Alembertian in IR3+ 1 , OrA)
•
[Aa , F]) + [A", auF + [Ao, F]] OF + [A, DF] + [a A F] + [A , [A, F]j .
ao (aaF
+
,
Since the Minkowski space-time metric m = -dt2 + t5ij dxidxj is static (in particular n = 1 and k = 0) the energy of F associated with the energy-momentum tensor Q[Fj,,;3 = Fa).,F;3)., + *F"A *F/u and vectorfield T = at iH conserved. In particular, the flux of energy Fp through the null boundary N- (p) ofthe domain of dependence .:1- (p) of an arbitrary point p can be bounded by the energy of the initial data which we denote by 10, We assume that smooth data for F is prescribed at t = 0 and restrict .:1- (p) and N- (p) to t > O. We recall that the flux has the form, Fp =
(
N- (p)
)1/2 Q [F] (L, T)
with L the null geodesic generator of N- (p) normalized by < L, T >= 1 . The proof of the global regularity of solutions of the Yang-Mills equations is based on the boundedness of the flux Fp < 10 < 00. Here is a summary of the main steps.
301
CURVATURE
(1) Rewrite (12) in the form OF = F * F - (O(A) - O)F. Using the explicit representation, in IR3+1, of solutions to the inhomogeneous wave equation, we deduce, for all points p, with t > 0, F (O) (p ; 00) + (47r) -1
F (p)
(47r)-1
.,. - 1
N- (p;Ju)
(O(A)
r- 1 F * F
(13)
O)F.
N- (p;�o) Here N- (p, 00) I'f�pre�cnt� the portion of the null cone N- (p) included in the time slab [t(p) - 00, t(p)], with t(p) the value of the time parameter at p. Also r is the distance, in euclidean sense, to the vertex p and F(O) (p; 00 ) represents the homogeneous solution to the wave equation whose initial data at t = t(p) - 00 coincide with those of F. (2) Consider the second term on the right hand side of (13). Using the explicit form of the nonlinear term F * F one notices that at least one factor of the product can be estimated by the flux fp through the null hypenmrface N- (p) . Denoting I F I = La)3 I F"'I1 I , we have by a simple estimate2, I (F (P)
�
F ( O) (p; 150) 1
:s
fp (
�
1N- (p;6o)
1/ F r 2 ) 2 I I II L� (J- (p;6oiJ
/ fp I I F I 0 :s � 2 l r,�(J-(p,8o» ' where II F I IL= (J- (p,c5o» denotes the sup- norm of I F I for all points in the domain of dependence J- (p) of p in the slab [t(p) oo, t (p)] and to = t(p) > 0 denot es the value of t at p. Therefore, we deduce the following, �
LEMMA 1.2. If 0�/ 2 . fp is sufficiently small, then for any t > 0 ( 14) II F ( t) IIL= :s II F (t OO) IIL� + II DF (t oo ) IILoo . �
�
(3) Arguing recursively and using the standard local existence theorem for the Yang-Mills system one can find bounds for all components of the curvature tensor F(p) depending oIlly on the fact that fp is unifonnly bounded and the initial data F (O) is smooth. (4) One can show that ( 14) remains true even as we take into consideration the presence ofthe third term in (1:')) . This i� done by choosing a specified gauge condition for A called the Cronstrom gauge. That is, one asslImes that the connection I-form A satisfies (x y)QAa = 0, where XCi are the space-time coordinates of p and yO those of a point q E N- (p). One can use t.his condition to derive uniform estimates of 8A in terms of F . (5) In [KI-Ma] the global regularity result wa.'i reproved by strengthening the classical local existence result to A E III (IRa) and E E L2 (]R3 ) , which iH at the same regularity level as the energy norm. That required, instead of the pointwise estimates ( 14), a new generation of L4 type estimates, called bilinear. The premise of the [KI-Ma] approach was the fact that, once we have a local existence result which depends only on the energy norm of the initial data, global existence can be easily derived by a. simplc continuation argument. �
2
and in what follows c > 0 is a universal constant . Here
we denote
by A ;S
B any inequality of the form
A < cB, where
302
SERGJU KLAINERMAN
In what follows we give a short summary of how the mains ideas in the proof of the Eardley-Moncrief result for Yang-Mills can be adapted to General Relativity.
(1) One can easily show that the curvature tensor R of a 3 + 1 dimensional vacuum spacetime (M, g), see ( 1 ) , verifies a wave equat.ion of the form, ( 15)
DgR = R * R where Dg denotes the covariant wave operator Dg = D"D" . (2) Recall that the Bel-Robinson energy-momentum tensor has the form
Q[R] a{3'Y" = R",x'YIL R; J' + * R",x'Y1' * R; t· and verifies, D � Q,,{3'Y� = O. It can thus bc used to derive energy and flux estimates for the curvature tensor R. As opposed to the case of the
Yang-Mills theory, however, in General Relativity the background metric is dynamic and thus does not admit , in general, Killing fields (and in particular a time-like Killing field) . This means that we can not associate conserved quantities to a divergence free Bel-Robinson tensor. It is at this point where we need crucially our bounded deformation tensor condition A2. Indeed that condition suffices to derive bounds for both energy and flux associated to the curvature tensor R. Using the Bel-Robinson energy momentum tensor Q the energy associated to a slice 2:t is defined by the integral
Q [R](T, T, T, T),
E(t) =
(16)
while the flux, through the null boundary N- (P) of the domain of depen dence (or causal past) :7- (p) of a point p, is given by the integral
F- (p) =
(
1
N-
(p)
Q[R] (L, T, T, T), ) '
( 17)
where L is the null geodesic generator of N- (p) normalized at the vertex p by < L, T >= 1 . As in the case of the Yang-Mills equations, it is precisely the bound edness of the flux of curvature that plays a crucial role in our analysis. In General Relativity the flux takes on even more fundamental role as it is also needed to control the geometry of the very ohject it is defined on, i.e. the boundary of the causal past of p. This boundary, unlike in the case of Minkowski space, are not determined a-priori but depend in fact on the space-time we are trying t.o control. (3) In the construction of a parametrix for ( 1 5) we cannot, in any meaningful way, approximate Dg by the flat D'Alembertian D. To deduce a for mula analogous to (14) one might try to proceed by the geometric optics construction of parametrices for Dg, as developed in [Fried] _ Such an approach would require additional bounds on the background geometry, determined by the metric g, incompatible with the limited assumption A2 and the implied finiteness of the curvature flux. Vie rely instead on a geometric version, which we develop in [KI-Ro5], of the Kirchoff-Sobolev formula, similar to t,hat used by Sobolev in [Sob) and Y. C. Bruhat in [Brl , see also [M] . Roughly, this can be obtained by applying to (15)
303
CURVATURE
the measure Ao(u), where u is an optical function3 whose level set u = 0 coincides with N- (p) and A is a 4-covariant 4-contravariant tensor de fined as a solution of a transport equation along N- (p) with appropriate (blowing-up) initial data at the vertex p. After a careful integration by parts we arrive at the following analogue of the formula ( 14) :
R(p) = R° (P; 00)
A . (R * R) +
Err · R, r iN - (p;"o)
( 18)
N- (p;oo) where N- (p; 0o) denotes the portion of the null boundary N- (p) in the time interval [t (p) - <1"0, t (p)] and the error term Err depends only on the extrinsic geometry of N- (Pi 00) . The term RO (p; 00) is completely determined by the initial data on the hypersurface L:t(p)-60 ' As in the flat case 4 , one can prove bounds for the sup-norm of RO(p; <1"0) which depend only on uniform bounds for R and its first covariant. derivatives at values of t' < t (p) < t - o. (4) As in the case of Yang-Mills, the structure of the term R* R allows us to estimate one of the curvature terms by the flux of curvature: I
r
iN- (p;.o)
-
A . (R * R) I
;S F- (P ) · II R IIL"" (N- (p;oo» ' II A II£2(N� (p;oo)) (19) <
provided that,
'"
(20)
Neglecting, for a moment, the third integral in (18) one may thus expect to prove a result analogous to that of Lemma 1.2.
THEOREM 1 .3. There exists a sufficiently small 00 > 0 and a large constant C, depending only on �o in assumptions Al and A2 as well as no in (8) such that for all to < t < t., II R (t) II L� ;S C sup (II R (t) IIL� + IIDR (t) IIL� ) ' t
(21)
(5) The proof of (19) depends on verifying (20). In addition, to estimate the third term in (18) , we need to provide estimates for tangential derivatives of A and other geometric quantities associated to the null hypersurfaces N- (p). In particular, it requires showing that N- (P) remains a smooth (not merely Lipschitz) hypersurface in the t.ime slab (t(p) - 00 , t (p)] for some with dependent only on the constants �o and no . Thus to prove the desired theorem we have to show that all geometric quantities, arising in the parametrix construction, can be estimated only in t.erms of the flux of the curvat.ure F; along N- (P) and our main assumption AI. Yet, to start with, it is not even clear t.hat we can provide a lower bound for the radius of injectivity of N- (p). In other words the congruence of null geodesics, initiating at p, may not be controllable5 only in terms of the
3That is gafJ 8a u8{3'u = O. 4This is by no means obvious as we need to rely once more on the Kirchoff-Sobolev formula. 5Different null geodesics of �he congruence may intersect . or the congruence itself may have conjugate points, arbitrarily close to p.
304
SERGIU KLAINEltMAN
curvature flux. Typically, in fact, lower bounds for the radius uf conju gacy of a null hypersurfaee in a Lorentzian manifold are only available in terms of the sup-Ilorm of the curvature tensor R along the hypersurface, while the problem of short, intersecting, null geodesics appears not to be fully understood even in that context. The situation is similar to that in Riemannian geometry, exemplified by the Cheeger's theorem, where pointwise bounds on sectional curvature are sufficient to c:ontrol the ra dius of conjugacy but to prevent the occurrence of short geodesic loops one needs t.o assume in addition an upper bound on the diameter and a lower bound on the volume of the manifold. In a sequence of papers, [KI- RoIJ-[KI-Ro3] ' we have been able to prove a lower bound, depending essentially only on the curvature flux, for the radius of conjugacy of null hypersurfaces6 in a Lorentzian spacetime which verifies the Einstein vacuum equations. The methods used in these articles can be adapted to provide all the desired estimates, except a lower bound on the " size" of intersecting null geodesics which needs a separate argument. The lower bound on the radius of injectivity of the null hypersurfaces N- (p), needed here, is discussed in [KI-Ro4] (6) As in the case of Yang-Mills equations one call use the result of Lemma (1.3), together with the classical local existence result for the Einstein equations to show that solutions can be extended as long the bounds on (T)1f hold true. ( 7) Finally we observe that our construction is gauge invariant, as it does not require a gauge condition such as that of Eardley-Moncrief, but depends instead on an appropriate tensoriaJ definition of the transport term A along N- (p) . This in fact suggest a possibility of reproving the Eardley Moncrief result itself with the help of a carefully constructed cuvariant Hadamard parametrix for 0 A . ThiH was in fad implemented in [KI-Ro5]. The proof of Theorem 1 . 1 contains the following main steps.
1.
Curvature L2-bound . One shows, based on condition A2, that the flux
uf curvature Ro, through the null boundary N- (P) of the c:ausal past .:r- (p) of any point p to the future of �o, must remain bounded by a universaJ constant. This follows quite eaHily from energy estimates obtained with the help of the Bel Robinson tensor Q defined above.
Higher derivatives bounds. One shows how to derive £2 bounds for higher covariant derivatives of R by using condition A2 as well as an auxiliary condition on the £00 norm of the curvature tensor R. The proof, though quite involved, is rather standard. It is based on higher energy estimates and L2 elliptic 2.
theory along the leaves �t of the time foliation.
3.
Curvature too-bound Tills is the main step in the proof. One shows how to
transform the L2 bound of the curvature flux cstabliHhed in Step 1 into a pointwise bound. This require a geometric, approximate, representation formula for solutions to the temmrial equation (15) . The representation formula ( 18) , with a precise 6together
with many other estim&LeS
of various geometric quantities associated
to N- (p).
CURVATURE
305
expression for the error term Err, was developed in [K l Ro5] In applying formula (18) to prove Theorem 1 .3 one needs to have good control on the regularity of the null hypersurfaces N-(p). More precisely we need, -
.
Sa. Lower bound for the radius of injectivity of the null hypersurfaces N- (p) . Null hypersurfaces can develop singularities, called caustics, even in flat space. The pre::;ence of caustics is detected by an important invariant, called the null second fundamental form, x(X, Y) = g(DxL, Y)
(22)
Here X, Y are arbitrary vectorfields tangent to the null hypersurface while L is the null geodesic generator of the hypersurface. According to the Jacobi identity we have,
D LX + X · X = R( ·, L , ·, L )
(23)
The trace trX of X verifies the well known Raychadhouri equations, L(trX) +
�
trx2 + Ixl 2 = °
(24)
where X is the traceless part of X , see [KI-Rol] for precise definitions. From (24) it is ea�;y to see that if trx is negative at some point then it must become infinite along the null geodesic passing through that point, independently on the size of the curvature tensor. This corresponds to a conjugate point. Th avoid conjugate points along N- (p) we need to show that trx remains close to its flat value ; where s is the afine parameter such that s(p) = 0, for a uniform interval s E [0, 8] , with 8 independent of p. To do this we need to be able to get a uniform bound for the integrals along past null geodesics from p, J: Ix12. Observe that equation (23) does not allow one to obtain such an estimate since it seems to require pointwise bounds for R. In the next section we shall discuss our main ideas for avoiding this difficulty. ,
3b. Higher derivative estimates for },f- (p) . It is not enough to control trx in the uniform norm we also need to control up to 2 derivatives of X tangential to N- (p) . This can be done by using an extension of the ideas discussed in the next section. To finish this section we would like to point out possible refinements of our main Theorem 1.1. We expect that one should be able to replace the pointwise condition A2 with the integral condition, t,
to
(25)
Moreover it may be possible to improve the result even further by eliminating the term V' log n in (4) or (25) and requiring instead only a pointwise bound on n.
306
SERGIU KLAINERMAN
2. Lower bound for the radius of injectivity of the null hypersurfaces N- (p) Null hypersurfaces in a Lorentzian space can be viewed as level hypersurfaces of an optical function u, i.e. a solution to the Eikonal equations (3 gOl 8",u8fju = O. (26)
Let L = -g",{30aU Ob be the corresponding null generator vectorfield and s its affine parameter, i.e. L (s) = 1 , slso = 0 where So C � is a given, initial, two dimensional compact surface which we assume to be diffeomorphic to the standard sphere. The level surfaces S. of s generates the geodesic foliation on 1l. We denote by V' the covariant differentiation on S. and by V'L the projection to Ss of the covariant derivative with respect to L. At every point of p E 1l we denote by L to be the unique null vector which is orthogonal to the Ss sphere passing through p and such that geL, L) = -2. We introduce the total curvature flux7 along 1l to be the integral, (27) with a, (3, p, . . null components of the curvature tensor R. More precisely, for any vector-fields X, Y on 1l tangent to the 2-surfaces Ss .
a(X, Y)
=
R(X, L, Y, L),
1 (3 (X) = R(X, L, L, L) , 2
p=
1
-R(L, L, L, L) . . . 4 (28)
The geometry of 1l depends in particular on the null second fundamental form x(X, Y) =<
DxL , Y >
(29)
with X, Y arbitrary vector-fields tangent to the s-foliation. We denote by trx the trace of X, i.e. trx = oabXab where Xab are the components of X relative to an orthonormal frame (ea )a=1 ,2 on Ss . In view of the Einstein equations we have, d trx + � (trx) 2 = - lxl 2 (30) dS 2 with Xab = Xab - � trxoab the traceless part of X.
2.1. Conjugate points. Even in Minkowski space one cannot adequately control the geometry of null hypersurfaces without a uniform bound on, at least, trX. Indeed one can show by an explicit calculation that if the LOC norm of trx is allowed to become arbitrarily large bad singularities, called caustics, occur. To avoid them we are led to ask whether we can bound the LOO norm of trx simply in terms of an initial data norm8 Io and total curvature flux Ro . At first glance this seems impossible. Indeed, by integrating (30), in order to control lltrxliLoo we need 7The justification
for this quantity can be found in IC-KJ in connection to the Bell Robinson tensor and the Bianchi identities. 8Which contains informations about So.
307
CURVATURE
to control uniformly the integrals Ir IxP, along all null geodesic generators r of 'H. On the other hand X verifies a transport equation of the form, d " 1 trx . X = -a. X ds + 2 •
(3 1)
where a, the null curvature component defined by (28), is only L2 integrable on 'H. Thus, unless there is a miraculous gain of a spatial derivative ( along the surfaces of the s- foliation) of the integrals of a along the null geodesic generators r, we have no chance to control the Lco norm of trX' Fortunately this cancellation occurs and it is best seen by observing that X verifies a Codazzi type equation of the form, div X
=
-13 + V'trx + . . .
�
(32)
with ,13 as defined above. One can show that div X defines an elliptic system on Ss and therefore we expect that X behaves like V-I ( -13 + � V'trX) with V - I a pseudo-differential operator of order - 1 . We are thus led to control, uniformly, the integrals
12
=
r
I V- 1 . V'trxI2 .
(33)
Consider first h . Can we estimate it in terms of the curvature component 111311 £2(1-1) ? The idea is to interpret this as as a restriction problem. This makes
sense dimensionally and it corresponds to a trace type inequality of the type IIUIIL2(r) ;S IIV'Ullu('H), applied to the tensor U = V-1j3. Unfortunately it is well known that this type of trace theorems, just as the sharp Sobolev embedding into Lco, fail to be true. Vve might be able to overcome this difficulty if we could write 13 = V'LQ with Q a tensor verifying9 11V'2Q llu('H) ;S Ro. Indeed such a sharp trace theorem holds true in flat space 1 0. The breakthrough, which allows us to make use of an appropriately adapted version of the sharp trace theorem men tioned above, is provided by the Bianchi identities which, expressed relative to our null pair L, 1:., takes the form of a system of first order equations connecting the L derivatives of the null components a, 13, p, 0" . . . to their spatial derivatives V'. In particular we have, ignoring the quadratic and higher order termsll , div 13
=
L(p) + . . . , curl 13 = -L(O") + . . .
Once more this is an elliptic Hodge system in 13 and we can write, formally V-Ij3 = V-2 L (p, -0") + . . . = V'L (V- 2 (p, -0")) + [V- 2 , V'L] (p, -u) + . . . ,
(3 4)
with V-2 a pseudo-differential operator of order -2. Ignoring for the moment the commutator [V-2, V'L](p, -0") and the other error terms, we have indeed
Q = V-2(p, -0-),
and
( 35 )
one can check, with the help of L2 elliptic estimates, that indeed be bounded by the curvature flux Ro. 9We denote by 'V ('V, 'V L) all first derivatives 011 11 and by V2Q all second derivatives. lOThat is the estimate ( Jr I 'V LQ I2) � ::s 11'V 2 QIIL2("H ) + IIQI IL2(1t) holds true. l lIt t urns out in fact that the ignored terms are not so easy to treat as they required various renormalizations and delicate estimate•. =
308
SERGIU KLAINERMAN
This circle of ideas seem to take care of the integral lt in (33). Unfortunately we hit another serious obstacle with [2 . The problem is that the operator 1)- 1 V is a nonlocal operator of order zero and therefore does not map L'X) into Lex;. To overcome this difficulty we were forced to try to prove a stronger estimate for trx. The idea is to try to prove the boundedness of trx not only in Loo but rather iII a Bf)sOV space of type B� 1 (55 ) which both imbeds in LOO (5.) and is stahle relative to operators of order zero. This simple, unavoidable fact, forcf)s us to work with spaces defined by Littlewood-Paley projections which adds a lot of technical difficultiefP . Moreover the standard LP- theory, based on Fourier transform, would require the use of local coordinates on 1{. The most natural coordinates, those transported by the hamiltonian flow generated by L, are not sufficiently regular. Thus we are forced to rely on an invariant, geometric, version of the LP-theory which we develop, together with an a.ppropriate paradifferential calculus, in [Kl-Ro2] by following a heat flow approach. An informal introduction to the sharp trace type tbeOTf)mS in Besov spaces needed in our work and the geometric LP theory on which they rely is given in [KI-Ro3] . .
,
2.2. Cut Locus points. The circle of ideas discussed above will only give us a lower bound for the radius of conjugacy of the null hypersurfaces N- (p). It is however possible that the radius of conjuga.cy of the null congruence is bounded fwm below and yet there are paRt null geodesics from a point p int.ersecting again at points arbitrarily close to p. Indeed, this can happen on a Lorentzian manifold with a thin neck, i.e. which looks locally like M = §I X �2 X R Clearly OIl such a flat Lorentzian manifold M there can be no conjugate points for the congruence of past or future null geodesics from a point and yet there are plenty of distinct null geodesi(',l:; from a point p in M which intersect on a time scale proportional to the size of the neck. There can he thus no lower bounds on the null radius of injectivity expressed only in terms of bounds for the curvaturf) tensor R. This problem occurs, of course, also in Riemannian geomet.ry where we can control the radius of conjugacy in terms of uniform bounds for the curvature tensor, yet, in order to control the radius of injectivity we need to make other geometric assumptions such as, in the case of compact Riemannian manifolds, lower bounds on volume and upper bounds for its diameter. It should thus come as no surprise that we also need, in addition to bounds for the curvature flux, other assumptions on the geometry of solutions to the Einstein equations in order to ensure control on the null cut-locus of points in Ivl and obtain lower bounds for the null radius of injectivity. This is discussed in [KI-R04]
12[t requires,
in particular, a
sharp
trace theorem in
I:lesov spaces
in a rough background.
CURVATt:RE
309
3. Gauge Invariant Yang-Mills theory The methods developed in connection with theorem 1 . 1 can be used to give a straightforward gauge invariant proof of the global existence result of Eardley Moncrief mentioned above. Assume that A is a Yang-Mills connection in the 4dimensional Minkowski space, i.e., it verifies the equations
D A1 F0(:1 = 0.
f
( 36)
where F denotes the curvature of the connection.
As mentioned earlier the approach of Eardley-Moncrief was based on the fundamen tal solution for a scalar wave equation in Minkowski space ( Kirehoff formula) and made UHe of Cronstrom gauges. In [KI-Ro5] we present a gauge independent proof which we review below. The main new ingredient is the use of a gauge covariant first order Kirchoff-Sobolev parametrix. Recall that F verifies the gauge invariant wave equation,
(37) The proof is based on the following Kirchoff-Sobolev parametrix formula, see [KI-Ro5],
47r < F(p), J >
-
+
where
Io
=
2
r
r
1N - (p;hol
1N - (p;ho)
l
< W, [F, Fj > _
< A(A ) W, F > +Io
r « 1N- (p)nB, (p) -80
r 2 1N- (p/>o)
< [FLL , W] , F >
(38)
W, DTF > - < DTA, F »
(39)
depends only on initial conditions on L":t(p)- 80 ' Here J is an arbitrary (} valued anti-symmetric 2-tensor on ]R3+ 1 , W is a g valued 2-form on ]R3+1 verifying the gauge invariant transport equation 13
DL W + r-1W
( rW) l r=O = J,
(40) < , > denotes the positive definite invariant scalar product on the Lie algebra and b. (A) denotes the tangential gauge invariant Laplacian. The first term in (38) 1=
=
0,
N- (p;"o)
< W, [F, F] > ,
can be estimated precisely as before in terms of the ilux fp and the sup norm of F,
III
<
The second and third terms
II
=
r
IN- (p; 80)
6�/ 2 fp II F IIL�(.1- (p,ool) '
< [FLL' W] , F > ,
III =
can be estimated by,
N - (p;lio)
< A(A)W, F >
I II I � fp II W llu(.1- (p,80 ) ) II F II LOC (.1- (1',80 )) , III < II F II LOO (.7- (p, 80) II b. (A ) W II £ 1 (.1 - (1' 80) , �
l�Here
L
=
8,· - 8t, DL
=
ar - at
+ [AL, ,I and
"
=
Iyl.
,
SERGIU KLAINERMAN
310
We estimate II W II L2(.:1- (p,6o)) and 1It:,(A) W IIL1(.:1- Cp,6o ) using the transport equa tion (40). In fact we show the following14,
II W I! ,,2 C.:1 - (p,6o ) II W II L' (.:1- (P,50)
< <5�/2 IJI < <5�/2I J IFp
( 41)
'"
(42)
�
The second estimate is particularly dangerous since it seems to depend on deriva tives of F. Indeed, introducing W = rW, we rewrite (40) in the form,
D�A) W = 0,
W lr=o
=
J.
Commuting the transport equation D�A) W = 0 with r2t:,CA) we obtain,
D�A \r 2t:, (A) W)
We also have the equation
=
r2 [Ft , viA ) W) + r2 V�A) [At , WI .
-) DL(A ) (rVa(A) W
= r [FLa, WI ·
\Ve combine these equations into the system:
D�A ) W = 0,
D�A ) (rViA)W) = r [F La, WI , DL(A) (1'2 t:, ( A ) W) = 2r2 [FLa , Va( >') W] + r2 [Va(A) FLa , W] . .
( 43)
Observe indeed that the last term on the right hand side of (43) contains covariant derivatives of F. Thus estimate (42) seems at first glance impossible. To avoid this problem we need to take into account the fact that the integral along a null geodesic of r2 [ViA) Ft, W] compensates for the loss of derivative. This can only be done by using once more the Bianchi identities.
References [And] [Br] [C-K] [EMl] [EM2] [Fried] [HE] [HKM]
M. Andersson, Regularity for Lorentz metrics under curvature bounds,arXiv:gr qc/020907 vI, Sept 20, 2 002 . Y. Choquet-Bruhat, Theoreme d'Existence pour certains systemes d 'equations aux derivees partielles nonlineaires., Acta Math. 88 (1952), 141-225. D. Christodoulou, S. Klainerman, The global nunlinear stability of the Minkowski space, Princeton Math. Series 4 1 , 1993. D. Eardley, V. Moncrief, The global exi.,tcnce of Yang-Mills-Higgs fields m 4dimensional Minkowslci space. I. L ocal existence and smoothness properties. Comm. Math. Phys.83 (1982) , no. 2, 171-191. The global existence of Yang-Mills-Higgs fields in 4D. Eardley, V. Moncrief, dimensional Minkowski space. II. Completion of proof. Comm. Mat h. Phys.83 (1982), no. 2, 193-212. H.G. Friedlander The Wuve Equat>on on a Curved Space-time, Cambridge University Press, 1976. Hawking, S. W. & Ellis, G. F. R. The Large Scale Structure of Space-time, Cambridge: Cambridge University Press, 1973 Hughes, T. J. R., T. Kato and J. E. Marsden Well-posed quasi-linear second-v'rder hy perbolic systems with applications to nonlinear elastodynam'ics and general relativity, Arch. Rational lvIech. Anal. 63, 1977, 273-394
fact the estimate for � (A)W is a lot more complicated divergence, but we should ignore this here. 14ln
as
it leads
to a
logarithmic
CURVATURE
(KIJ (KI-Ma) (KI-Ro1] (KI-Ro2]
(KI Ro3] -
(KI-Ro4]
311
S. Klainerman. PDE as a unified subject Special Volume GAFA 2000, 279-315 S. Klainerman and M. Machedon, Finite Energy Solutions for the Yang Mil ls Equa tions in JRl+3, Annals of Math., Vol. 142, (1995), 39-119. S. Klainerman and I. Rodnianski, Causal g eometry of Einstein- Vacuum space times with finite curvature flux Inventiones Math. 2005, vol 159 , No 3, pgs. 437-529. S. K lainerman and I. Rodnianski, A geometric approach to Littlewood-Paley theory, to appear in GAFA(Georn. and Punet. Anal) S. Klainerman and 1. Rodnianski, Sharp trace theorems for null hypersurfaces on Einst ein metrics with finite curvature flux, to appear in GAFA S. Klainerman and I. Rodnianski, Lower bounds for the radws of injectivity of null -
Rodnianski, A Kirchoff- So balev pammetrix for the wave equa, tions in a curved .pace-time. preprint. S. Klainerman and I. Rodnianski, A breakdown criterion in General Relativity preprint. V. Moncrief, Personal communication. S. Sobolev, Methodes nouvelle a 1-esoudre Ie probleme de Cauc hy pour les equations lineaires hyperboliques n orm ales Matematicheskii Sbornik, vol 1 (43) 1936, 31 -79. hypersurfaccs, preprint
lKI-Ro5) [KI-Ro6] [M] [Sob]
S.
Klainerman and
1.
,
DEPARTMENT OF
E-mail address:
MATHEMATICS,
PRlNCETON UNIVERSITY, PIUNCETO:-l NJ
serilDmath . princeton. adu
08544
Cuntemporary Mathema.tics
Volume 446: 2007
Some Liouville theorems and applications YanYan Li Dedicated to
Haim Brezis
,
with high respect and friendship
We give expo"ition of a Liouville t heorem established in [6] which is a novel extension of the classical Lionville theorem for harmonic fuuctions. To illustrat.e some ideas of the proof of the Liouville theorem, we present a new proof of the classical Liouville theorem for harmonic functions. Applications to gradient estimates of the Liouville theorem, as well as that of ear lier ones in [5], can be found in [6, 7] ami [10]. ABSTRACT.
The Laplacian operator Cl. is invariant under rigid motions: For any function u on JR.n and for any rigid motion T : JR.n -> JR." ,
Cl.(u 0 T )
=
(Cl.u) 0 T.
The following theorem is classical : (1)
u E C2, Cl.u = 0 and u > 0 in JR.n imply t.hat u
constant .
In this note we present a Liouville theorem in [6J which is a fully nonlinear version of the classical Liouville theorem ( 1 ) . Let u be a positive function in JR." , and let 1/J : JR.nU{ oo} JR.""U{ oo} be a Mobius transformation, i.e. a transformation generated by translations, multiplications by nonzero constants and the inversion x -> xllxl2. Set ->
1l.p : =
I J", I
On
y.-2
(u 0 1/;) ,
where J", is the Jacobian of 1/;. It is proved in [3J that an operator H(u, 'Vu, 'V 2 u) is confurmally invariant, i.e. H (ut/J , 'Vu", , 'V2u",)
=
lIeu, 'Vu, 'V2u) 0 1/; holds for all positive
u
and all Miibius 'Ij;,
if and only if H is of the form
H (u, 'Vu, 'V2u)
=
f(,\(AU))
1991 Mathematics Subject Classification. 35J60, 35J70, fi3A30. Key words and phrases. Liouville theorem, conformally invariant, elliptic, comparison prin ciple, gradient estimates. Partially supported by NSF grant DMS-0101118.
313
314
YA NYAN LI
where
I is the n x n identity matrix, ,\ (AU) = (AI ( AU), . . . , An (AU» denotes the eigenval ues of AU, and f is a function which is symmetric in A = (AI , . . . , An ) . Due to the above characterizing conformal invariance property, AU has been called in the literature the conformal Hessian of 'U , Since i= 1
Liouville theorem (1) is equivalent to
1£ E C2, A(AU) E Ofl and
(2) where
fl :=
u
> 0 in lR" imply that
1£
constant,
n
{,\ I L Ai > O}. i=1
Let (3 )
f
c
lRn be an open convex symmetric cone with vertex at the origin
satisfying
(4) Examples of such f include those given by elementary symmetric functions, For 1 < k S n, let I
be the k-th elementary symmetric function and let fk := {,\ E Rn I O'I (A) , ' " , O'k (..\) > a}, whidl is equal to the connected component of {..\ E lRn I <7k(..\) > a} containing the positive cone f satisfies (3) and (4). For an open subset 0 of Rn , consider n,
..\(A")
(5 )
E
of,
in 0.
The following definition of viscosity super and sub solutions of (5) has been given in [6] . 1 . A positive continuous function u in 0 is a viscosity subsolution [resp. supersolution] of (5) when the following holds: if Xo E 0, 'lj; E C2 (0), (1£ - 'lj;) (xo) = 0 and 1£ - 'I/J < a near Xo then "\(A
[resp. if (u - 'lj;) (xo) = a and 1£ - 'lj; 2: a near Xo then A(A"'(xo» E rl. We say that u is a viscosity solution of (5) if it is both a viscosity �upersolution
and a viscosity subsolution.
1 . If a positive u in (,' 1 . 1 (0) satisfies A(A") E of a,e, in 0 , then it is a viscosity solution of (5). REMARK
Here is the Liouville theorem.
SOME LIOUVILLE THEOREMS AND APPLICATIONS
31.5
THEOREM 1 . ([6]) For' n > 3, let r satisfy (3) and (4), and let u be a positive locally Lipschitz viscosity solution of (6)
A(AU) E ar
Then u
_
'u(o) in JRn .
REMARK 2 . It was proved by Chang, Gursky and Yang in [1] that positive
C l ,l (JR4) solutions to A(AU.) E ar2 are constants. Aobing Li proved in [2] that positive C1,l (JR3 ) solutions to A(AU) E ar2 are constants, and, for all k and n, positive C3 (jRn ) solutions to A(AtL) E ark are constants. The latter, result for c3 (JRn) solutions is independently established by Sheng, Trudinger and Wang in [9] . Our proof is completely different.
In order to illustrate some of the ideas of our proof of Theorem 1 in [6] , we give a new proof of the classical Liouville theorem (1). We will derive (1) by using a .::l : Let
n be a bounded open subset of jRn, 2 > and v E C (f!) n 2, containing the origin O . Assume that u E Cl2oc(n \ {O}) satisfy and .::lu < 0, u > v, in n \ {OJ .::lv > ° in n, and on em. u>v Then inf (u - v) > O.
Weak Comparison Principle for
n\{o}
It is easy to see from t.his proof of the Liouville theorem ( 1 ) that the following Comparison Principle for locally Lipschitz viscosity solutions of (5) , established in [5 , 6] , is sufficient for a proof of Theorem 1 . PROPOSITION l . Let f! be a bounde� open subset of ]Rn containing the origin 0, and let u E cf�� (f! \ {O}) and v E CO, l (n). A ssume that u and v are respectively positive viscosity supe" 'soI1dion and subsolution of (5), and
u>v>0
on an.
Then
inf (u - v» D\{O}
the
O.
For the proof of Proposition 1 and Theorem 1 , see [5, 6] . In this note, we give
Proof of Liouville theorem (1) based on the Weak Comparison Principle for .::l . Let
1 vex) : = - [min u(y) J l x I 2-n, 2 Ivl= l Since Ul and Vl are still harmonic functions, an application of the Weak Comparison Principle for .::l on f! :=the unit ball yields
(7)
lim inf Ivln-2u(y) > O. Iy l�oo
316
YANYAN LI
LEMMA 1. For every x E Rn, there exists '\0 (x)
,\n -2 >.. 2 ( y - x) ) < u (V) ux,>, (y) : = I y _ ;"(; I n 2 u (x + I v 2 xl _
>
'if
0 s'uch that
0 < >.. < >"0 (x ) Iv ,
-
xl
> >.. ,
Proof. The proof is essentially the same as that of lemma 2.1 in [8] . Without loss
of generality we may take x = 0, and we use u), to denote 'uo , >, . By the positivity and the Lipschitz regularity of u, there exists TO > 0 such that '1'
7, - 2
"
'if
n-2
u( r, B) < s 2 1L ( s , B) ,
The above is equivalent to u>, (y ) <
(8)
u(y) ,
'I'
0<
o<
< s < '1'0, B E §,,- l .
>.. < I v l < '1'0 ,
\Ve know from (7) that , for some constant c > 0, u(y) > cly I2- n ,
C
Let >"0 : = (
Then
(9)
maxl zl<,-o u (z)
)n
1
2,
'if
A u),(y) < (I y°1 )71- 2 ( maxo u (z» < cl yl2 -n < u( V) , Izl<"
It follows from (8) and (9) thaL
u>,(y) < 1),(Y) ,
Lemma 1 is established.
'if
0 < A < Ao , Iyl > '1'0'
0 < ,\ < AO, Iyl > ,\ , o
Because of Lemma 1, we may deane, for any x E Rn and any 0 < 8 < 1, that A,, (X )
:=
SUP{11 > 0 I ux,)' (Y) < (1
+ 8)'u(y),
LEMMA 2 , For any x E Rn and any ° <
'if
Ii
0 < ,\ < 11, Iy - xl
> >..} E (0, 00] .
< 1 , '\,, (x) = 00 .
Proof. \Ve prove it by contradiction. Suppose the contrary, then, for some :r: E Rn
and some 0 < 8 < 1, A,, (X) < 00. We may assume, without loss of generality, that x = 0, and we use U>, and '\" to denote respectively 1£0 ,>, and '\,, (0) . Since the harmonicity is invariant under conformal transformations and multiplication by eOIlstants, and since -
-
an applicatioIl of (7) yields, using the fact that (1£), », inf [(1 + 8)U.\6 (V ) O
-
( 1 + 8) u (V)
-
u >'.
1£,
u(V)] > O.
Namely, for some constant c > 0, ( 1 0)
-
(y ) > clvI 2 - n ,
317
SOME LIOUVILLE THEOREMS AND APPLICATIONS
By the uniform continuity of u on the ball {z I Izl < .A8}, there exists E > 0 such that for all A6 < A < A6 + € , and for all Iyl > .A, we have ( 1 + o)u(y) - U5.. (Y) + [U5., (y) - u>. (y)] -A26 Y 2y A 2u( > cly l2 -n _ I Y I 2 -n I An-2 u( ) An)1 6 Iv l 2 I v l2 This violates the definition of A6. Lemma 2 is established. ( 1 + o)u(y) - u>. (y)
>
_
>
-
By Lemma 2, Ao
00
,
for all 0 < 0 < 1 . Namely,
(1 + o)u(y) > ux,>. (Y),
'r/ 0 < 0" < 1 , x E ]Rn , Iy - x l >
Sending 0" to 0 in the above leads to
c l I 2 -n . 2 v
).
>
o
O.
n E x V ]R , Iy - xl > ). > O. Liouville theorem (1) is established.
u(y) > u r,>. (Y) ,
This easily implies 'U
u(O).
o References
[lJ S.Y.A. Chang, M. G ursky and P. Yang, A prior estimate [or a class of nonlinear equations on 4-manifolds, Journal D'Analyse Journal Mathematique 87 (2002), 151-186. [2J A. Li, Liouville type theorem for some degenerate conformally invariant fully nonlinear equa tion, in preparation. [3J A . Li and Y.Y. Li, On some conformally invariant fully non li near equations, COfllm. Pure App!. Math. 56 (2003), 14IG-1464. [1J A. Li and Y.Y. Li, On some conformally invariant fully lIonlinear equations, Part II: Liouville, Harnack and Yamabe, Acta Math. 195 (200.5 ), 117-154. [5J Y.Y. Li, Degenerate conforrnally invariant fully nonlinear elliptic equations , arXiv:math.AP /0.';04598 vI 29 Apr 2005; v2 24 May 2005; final version, to appe". I n Arch. Rational Mech. Anal. [6J Y.Y. Li, Local gradient estimates of solutions to some conform ally invariant fully nonlinear equations, arXiv:math.AP /0605559 vI 20 May; v2 7 Jul 2006. [7J Y. Y. Li, Lucal gradient estimatrn of solutions to some conformally invariant fully nonlinear equat ions, C . R. Math. Acad. Sci. Paris, Ser. J 343 (2006), 249-252. [8J Y.Y. Li and L. Zhang, Liouville type theorems and Harnack type inequalitiesfor semilinear elliptic equations, Journal d'Analyse Mathematique 90 (2003), 27-87. [9J W. Sheng, N.S. 'I'rudillger and X.J. Wang, The Yamahe problem for higher order cllrvatures, arXiv:math.DG/0505463 vl 23 May 2005. [10J X.J. Wang, Apriori estimates and existence for a class of fully noulinear elliptic equations in con formal geometry, Chill. Ann. Math. 27B (2) (2006), 169-178.
OF MATHEMATICS, N J 08854, U SA
DEPARTMENT AWAY,
RUTGERS UNIVERSITY,
E-mail address: yyli0math. rutgers . edu
110
FHI>LI:'IGHUYSEN
ROAD,
PISCAT
Contemporary l\/Iathematics Volume 446, 2007
Analysis on Faddeev Knots and S kyrme Solitons: Recent Progress and Open Problems Fanghua Lin and Yisong Yang
It is
,
our great pleasure to dedicate this article to Haim Brezis
for his leadership, encouragement, and suppa,·t over many years.
We report some recent progress and discuss some related unsolved problems concerning the existence of the energy-minimizing configurations in the Faddeev quantum field theory model giving rise to knotted solitons and in the Skyrme model modeling elementary particles. Many issues related to the corresponding evolutionary systems for the Faddeev knots and Skyrme solitons, however, remain untouched. These are rather unusual and challenging systems of nonlinear partial differential equations of hyperbolic type. We also describe some simple but fundamental mathematical issues concerning these ABSTRACT.
models.
1. Introduction The area of quantum field theory is fascinating for analysts because it involves all types of analysis problems at both the classical levels and the quantum levels of diverse subjects of fundamental importance. According to quantum field theory, elementary particles and their interactions are elegantly described by continuous fields defined over spacetime and hosted within a vector bundle. Spacetime sym metry is related to special relativity and gravity and vector bundle symmetry is related to all other fundamental inter-particle forces. There are many interest ing and important mathematical issues in quantum field theory worthy of pursuit, which provide challenges as well as opportunities for mathematicians. For example, topological solitons known as domain walls, vortices, monopoles, and instantons in their simplest cases are often related to integrable systems and the construction of these solutions leads to further development of the solution methods including in verse scattering method [1 , 34, 48, 60, 95], the Backlund transformations [1, 72], 2000 Mathematics Subject Classification. 35J20, 35Q51, 58Z05, 81T45, 81V35. Key words and phrases. Quantum field theory, topological invariants, Faddeev model,
Skyrme model, calculus of variations, compactness, minimization, growth laws. The first author was supported in part by NSF Grant DMS 0201443. The second author was supported in part by NSF Grant DMS--0406446. ©2007
319
American Mathema.tical Society
320
FAKGHUA LIN AND YISONG YANG
the Penrose twistor method [8, 86] and ADHM [6], duality [45], group represen tations [46] ; when the systems encountered are not integrable, functional analysis methods are needed in order to understand the structure of the solutions [41, 93J. In this survey article, we report some progress in the mathematical understand ing of two import.ant but closely related quantum field theory models, the SkYrIne model and the Faddeev model, and we also discuss some unsolved problems. The central idea embedded in t.he Skyrme model [76, 77, 78, 79] is to use con tinuously extended , topologically characterized, relativistically invariant, locally concentrated, soliton-like fields to model elementary particles. There are two ways in order to achieve such a construdion in space dimension greater than one: (i ) introduce gauge fields, or (ii) introduce higher-order derivative terms, in the field theory Lagrangian. The first category models are various Yang-Mills gauge field theory models which allow static solitons characterized topologically by winding numbers, hOUlOt.Opy classes, and the Chern indices, and there often exists a beau t.iful self-dual reduction which gives rise to solutions as absolute energy minimizers among various topological classes. The second category models are the Skyrme type models where, although the field configurations arc ela.�sified by homotopy classes, no self-dual reduction exists for nontrivial solutions, and one has to study the minimization problcm for the original energy functional. It is interesting to not.e that , at the quantized level, the Skyrme model gives rise to two types of particles which are essential for nuclear interactions, namely, the mesons which are quantum fluctuations around t.he topologically trivial field configuration, and the baryons which are effectively realized as (topologically nontrivial) solitons. In this way, the baryon-meson and baryon-baryon scat.tering [40, 89] come naturally into the pic ture. More recently, the Skyrme model and its various variations have been applied to many areas including the quantum Hall effect [28, 65], Bose-Einstein conden sates [12, 70j, and cosmology [17, 61, 74] . In the classical Skyrme model, the field configuratioIlS are topologically represented by the homotopy group 71'3 (83) and the fundamental mathematical question is to ask whether there is a static Skyrme en ergy minimizer among each homotopy class. For some introductory reviews on the Skyrme model, see [35, 40, 94] . The Faddeev model [29, 30] was also proposed out. of the same motivation as that of Skynne [79], namely, to model elementary heavy particles by topological solitons, and may be viewed as a constrained or refined Skyrme model [22J . Al though, at a first glance, it seems that the Fadrleev model is a variant of the Skyrme model, it brings about a highly nontrivial twist and allows a stunningly new struc ture: the existenr:e of knotted solitons. It may be interesting to recall that, for mathematicians, knot theory has long been a theory of classification of knots by means of combinatorics and topology. For example, Tait [81] enumerated knots in terms of t.he crossing number of a plane projection; Alexander [31 discovered a knot invariant, known as the Alexander polynomial, arising in 3-dimensional ho mology; Jones [42, 43] found a new knot invariant, known as the Joncs polynomial, which enabled several conjectures of Tait to be proved [62, 63] ; based on a heuristic quantum-field theory argument, Witten [90J derived from the Chern-Simons action a family of knot invariants including the Jones invariant; finally came the Vassiliev invariants [85] which cover the Alexander polynomial and the .Jones polynomial and lay a general framework for the study of the combinatorial aspects of knots. In the Faddeev model, the field configurations are represented by the homotopy group
FADDEEV
7r3(82),
KNOTS AND SKYRME SOLITONS
the set of the Hopf classes from
83
to
82 ,
321
which is identical to the set of
integers, and the knotted solitons, or the Faddeev knots, are energy-minimizing field configurations among the Hopf classes. Using computer simulation, Faddeev
[32, 33]
and Niemi
first produced a ring-shaped (unknotted) Hopf charge (class)
one soliton. Shortly after the seminal work of Faddeev and Niemi, a more exten sive computer investigation was conducted by Battye and Sutcliffe
=
up t.o
=
[13, 14, 15]
who performed fully three-dimensional, highly convincing, computations for the
Q
Q
1
Q
8
= 1 , 2, 3, 4, 5, the energy-minimizing solitons are ring-shap ed and higher charges cause greater distortion, and for Q = 6, 7, 8, the solitons become knotted or linked. In particular, the trefoil knot appears at Q = 7. Thus, as in the case solution configurations of the Hopf charge that, for
from
Q
and found
of the Skyrme model, we are facing again a topologically constrained minimization problem and our ultimate question is to find out whether there is a static Faddeev
energy minimizer among each topological class defined by a nontrivial Hopf index (charge) . For a review, see
[31] .
In this survey article, we report some recent progress on the understanding of the minimization problems for the Skyrme and Faddeev models. In the next section, we review the related problems for the sine-Gordon model , the sigma model, and the Yang-Mills theory, and we emphasize the mathematical differences between these classical models and the Skyrme and Faddeev models we are facing.
3,
In Section
we review the Skyrme and Faddeev models and we introduce both the time
dependent governing equations and the static minimization problems associated with these models. In Section
4,
we present a series of existence results concerning
these static minimization problems. We emphasize the novelty and effectiveness of our techniques developed in treating these problems. In Section
5,
we review our
work on two-dimensional static Skyrme model which is of independent interest and serves as another illustrative comparison of the concentration-compactness principle and our method . Throughout this article, as we discuss various problems and technical iH..'iues, we also present some unsolved problems worthy of future study. In section
6,
we review other mathematical directions and developments concerning
the Skyrme and Faddeev models. In particular, we comment on some additional unsolved problems. In Section
7,
we conclude the article with a summary.
2. Topological solitons, energy lower bounds, and minimization (1 + 1)
dimensional sine-Gordon model, which is also the common origin of both the In order to describe the Skyrme model, we start with the classical
Skyrme and Faddeev models and has often been used as an illustrative labora is governed by a map cP from the Minkowski space
tory mathematical model in particle physics.
time RI, l of the time coordinate
of mesons of mass
'"
> 0
The effective meson field theory
Xo = t and space coordinate Xl = X with the circle 81 . Rewriting ¢ in terms of its two compo
(+-), into the unit nents, ¢o and cP l , we can represent these components by an as cPo(t, x) = cos a(t, x), ¢l (t, x) = sin a(t, x) . The relativistic signature
governing the evolution of the field
(2.1)
£=
�
a
angular variable a Lagrangian density
is
0,., ao'" a
- ",2(1 - cos a) ,
so that its corresponding Euler-Lagrange equation is of the form
(2.2)
FANGHUA LIN AND YISONG YANG
322
This equation, usually called the sine-Gordon equation, gained most of its popular
ity a bit later, after the discovery of the inverse scattering transformation method in the mid- 1960s. The equation (2.2) itself was known and investigated much earlier in
differential geometry with the problem of isometric embeddings of the hyperbolic planes (with negative constant Gaussian curvatures) into the Euclidian 3-space.
Though Skyrme did not seem to be aware of these results in geometry, he did
manage to obtain one-soliton solutions represented by the functions (2.3 )
a(x)
=
4 tan- 1 (e± I«
x-xo) ,
called the 21r-kinks. Since (2 . 1 ) is relativistic (or Lorentzian invariant) , we may use
a Lorentzian boost to switch on the time dependence of the solutions via replacing
(x - xo )
in (2.3) by
(x
-
xo
-
vt)/J1
-
v2 •
Besides these traveling wave solutions,
which Skyrme identified with mesonic excitations of small amplitudes, Skyrme also viewed these 21r-kink solitons as "particles" and he paid attention to the conserved normalized current (2.4) which automatically satisfies the conservation law phasized that this conserved current is
not
8J.tJJ.t = O.
It should be em
a Noether current, meaning it is not a
(2.1); it is not a conserved quantity by the field evolution either, meaning it does not follow from the equation of motion consequence of the symmet.ry of the Lagrangian
(2.2). Rather, it is a topological current which reflects the topological st.ructure of
the model, which will be discussed below.
First, note that quantization around the trivial solution
expected perturbative
meson particles
a = 0 gives rise to the
of mass '" as in the standard Klein-Gordon
model. However, in addition to the above well-known structure, the sine-Gordon
model also has a structure that allows the existence of non-perturbative heavier particles described by localized solutions such as those of the Skyrme 27r-kinks.
In order to see this more transparently, we recall that the associated Hamiltonian (energy) density of (2. 1 ) may be written as
1{
(2.5)
It is seen that the model
=
1 2
has
(a; + a;) + ,,2 ( 1 - cos a). count ably many potential wells which give us the
ground states a N = 2N1r,
(2.6)
N E Z,
and finite-energy nontrivial solutions interpolate between two distinct ground
Without loss of generality, we can malized boundary condition
consider
aCt, - (0) = 2n1r,
(2.7)
states.
the equation (2 . 2) subject to the nor
a(t, (0 ) = 2 ( n + N)1r.
field 4>(t, x) = eia(t,x) winds around the circle S1 as x goes from the left end to the right end of the space axis, and is directly relat.ed to the conserved current (2.4) by the formula It is clear that the integer N counts the number
(2 .8)
Q(a:)(t) =
where p =
JO
of times
J p(t, x) dx = 1: JO (t, x)dx = � 2
is the "charge" density which is seen
( 2 . 2 ) satisfying ( 2 . 7) or ( 2 . 8) ill called
an
to
N-soliton.
the
(a(t, 00) - a (t, -oo» be topological.
A
= N,
solution of
323
FADDEEV KNOTS AND SKYR.ME SOLITONS
It is well known [67] that (2.2) is integrable and one can use the inverse scat tering or the Biicklund transformation method to construct all N-soliton solutions of the equation [72] . An important characteristic of an integrable system is that it possesses infinitely many conservation laws, and our topological one, (2.8) , is just one of them. To evaluate the mass (energy ) of an N-soliton, we rewrite (2.5) as (2.9)
•
Integrating (2.9) over the full space line and using (2.7) , we obtain the topological energy lower bound (2. 10)
E=
00
- 00
for an N-soliton solution, where the sign is chosen to preserve the positiveness of the energy. For static solutions, the minimal bound is saturated if the equation O:x
(2. 1 1 )
± 2;,; sin
� =0
is satisfied, which is a self-dual or anti-self-dual reduction (in the sense of Bogo mol'nyi [19]) of the second-order static sine-Gordon equation O:xx
(2.12)
2 .
- ;,; sin 0: = O.
In other words, the solutions of (2. 1 1) are automatically the solutions of (2.12). We can also establish the converse. In fact, let 0: be a solution of (2. 12) satisfying the boundary condition (2 .7) and set p± = O:x ± 2;,; sin �. We can check that p± satisfies the separable equation
(x) a p± P±. x = ±K cos
(2.13)
2
whose solutions are represented as the integral (2. 14)
x Xu
cos a Cy l dy 2
=
p± (xo )R± (x) .
Using the boundary condition (2.7) , we see that either R+ (x) -+ 00 or R- (x) -+ 00 0 as as x -+ 00 . Since it follows from the boundary condition (2.7) that p± (x) x -+ 00 , we conclude from (2.14) that P+(xo) = 0 or P- (xo) = 0, which implies p+ = 0 or P- O. That is, a solution of (2.12) subject to the boundary condition (2.7) must be a solution of the self-dual or anti-self-dual equation (2. 1 1 ) . In this sense, (2.11) and (2.12) are equivalent . The equation (2.1 1) is elementary. Since a nontrivial solution of (2 . 1 1 ) ean never attain a value of the form 2m1l' (m E Z) at any point in (-00, 00), we see that, modulo translations of the form a f-> 2k1l' + 0: (k E Z) , the only static solutions are those one-soliton solutions of Skyrme given in (2.3), whose mass is the topological energy minimum determined in (2.10) with N = ±1, -+
_
(2.15) We have seen that the sine-Gordon model does not aJlow static N-soliton solu tions when the topologieal soliton charge N is not unit and higher-charge solitons
324
FANGHUA LIN AND YISONG YANG
must be nonstationary
[67] . From this,
we derive a valuable lesson that the topolog
ically cOIlstrained minimization problem for the static sine-Gordon model, namely, the innocently simple problem (2. 16)
EN = inf E(a) =
has no solution if
N i= 0, ± l .
J
OO
-
1 - a; + 1\;2( 1 _ cos a) dx Q ( a) 2
oc
The next example is the classical
0(3)
=
N ,
sigma model which is also the second
common origin of the Sky!lIIe and Faddeev models. The field configuration is a spin
¢ defined over the (2 + I )-dimensional Minkowski spacetime ]R2,1 and taking range in the unit sphere, S2 , of ]R3, namely, ¢ = ( 4)1 , ¢z , 4>3 ) , ¢I + ¢� + ¢� = l eW = l . The dynamics of the field
L
(2.17)
=
� Oi'¢ . oi' ¢.
Note that there is no self-potential term because it is prohibited in two spatial di
Di'Dl-lcj) = O¢,
mensions in order to have a static and nontrivial stable solution. With the notation the Euler-Lagrange equations of ( 2 . 17) are
O¢ - (¢ · O¢)1> = O.
(2.18)
The Hamiltonian density of (2. 1 7) is
'H
( 2 . 1 9)
=
� ((OO¢)2 + (Ol¢)2 + (02 ¢)2 ) .
Finite-energy condition implies that 1>(t, · ) goes to
a
constant unit vect.or, which
¢(t, . ) a continuous map from S2 to S2 . Hence ¢(t, . ) represents a homotopy class in 1r (S2) = Z and is thus characterized by an integer N(t). This integer 2 N (t) is also the Brouwer degree deg(¢(t, . » , of ¢(t, ·) which measures the number of times SZ being covered by itself under the map ¢ (t, . ) . In fact, the integer N(t) makes
is again a topological "charge" . To see this, note that there is again an associated conserved topological current,
(2.20)
JI-I =
so that the total charge through the expression
Ji' ,
� El-l v,,/ ¢ 8
Q (¢)(t)
.
given by
(a,A> /\ o,¢),
/-t, v, 'Y
= 0, 1, 2,
may be calculated using the charge density p =
JO
1 r €jk ¢ ' (Oj ¢ /\ ok
various generalizations have been studied extensively
[73, 80]
and one expects
to have global weak solutions for arbitrary initial data of finite energy and the development of singularities in finite time from some smooth initial data in dimensions. Here we focus on static solutions. Hence (2.18) becomes
(2.22) The total energy is
(2.23)
�¢ - (¢ . [�¢] )¢ = 0. E (¢) = -1 2
{ (Ol<1» 2 + (0 <1» 2 } dx 2 ]R'
(3 + 1 )
FADDEEV KNOTS AND SKYRME SOLITONS
3 25
so that (2.22) is the Euler-Lagrange equation of (2.23) . The basic problem is to find an energy-minimizing configuration among each topological class, i.e. , to find a solution of thc problem (2.24)
EN
=
inf{E ( ¢) 1 E (¢) < 00, Q(¢) = N } ,
given N E Z, which is an N-soliton. This problem was solved by Belavin and Polyakov [16] for arbitrary N. In fact their result is known to be the earliest and simplest complete integration of a physically realistic field Lheory model and immediately triggered a broad spcctrum of activities in other related areas in field theory. In such a solution, one does not engage in a direct minimization analysis of (2.24). Instead, one explores a similar structure as in the sine-Gordon model which reduces (2.12) to (2. 1 1 ) , i.e., the self-dual or Bogomol'nyi structure. To see this, we note that ¢ . ¢ = 1 implies ¢ . (OJ ,p) = 0 and (e/J /\ OFf;) . (4) /\ Oje/J) = ( [e/J /\ Oj ¢] /\ e/J) . (Oj e/J) = (Oje/J) 2 (j = 1, 2) (here no summation is made over repeated indices) . Therefore, we get the expansion
Integrating the above equality over R2 and using (2.2:�) and (2.21), we have the Lopological energy lower bound (2.26)
E(e/J) > 47l" IQ (e/J) I.
Of course, Lhe lower bound (2.26) is saturated if and only if e/J satisfies the equation, (2.27)
J, k
=1 2 ,
for Q(,p) = ± I Q(,p) I . Like before, it is straightforward to examine that (2.27) implies (2.22). Thus, as in the sine Gordon model, (2.27) may be viewed as a. first intcgral of (2.22). Note that, again, the most interesting fact about (2.27) is that the solutions of (2.27) are global minimizers of the energy over the designated topological classes. Although (2.27) is still a nonlinear equation, it can be made linear by using a stereographic projection, and thus, completely and explicitly solved. In view of (2.26) , these obtained N-soliton solutions again have their soliton mass or minimum energy directly related to their topology, (2.28) It is seen that solitons of greater topological charges are naturally more massive. The immediate higher-dimensional extension of the above N-soliton construc t,ion for the sigma model is of course the well-known resolution [2, 67] of the classical Yang-Mills instantons realizing any prescribed topological charge (now the second Chern number) . The key ingredient in thiR context is again not by direct minimization but through a similar self-dual or Bogomol'nyi reduction. For the reduced self-dual system of the Yang-Mills theory analogous to (2.27) for the sigma model, Atiyah, Drinfeld, Hitchin, Manin, and Ward [6, 8, 64, 86] showed that one can obtain all possible solutions [7] using algebraic geometry. It is in teresting to mention that, although there is no study on the direct minimization for the Yang-Mills instanton problem or the Yang-Mills-Higgs monopole problem, there are min-max variational studies concerning these problems and the existence of nonminimal solutions [75, 83] has been established.
326
FANGHt:A LIN AND YISONG YAN G
3. The Skyrme and Faddeev models
We first consider the classical Skyrme model. Let U be a map from the (3+ 1) dimensional spacetime of signature ( + --) into the isospin group SU(2). We introduce the Lie algebra su(2)-valued connection one-form A by A ut d U = Al'dxlL (t = xO, x = (xj ) , j = 1 , 2 , 3 ) Using [. , . j to denote the Lie bracket or matrix commutator on 8u(2) and Tr the matrix trace, we can express the normalized Lagrangian density of the Skyrme model as (3.1 ) Note that the second term in the Lagrangian density, namely, Tr ( [AIL' Av) [AI' , A") ) , was introduced by Skyrme. Although this term leads to a lot of complications due to nonlinearity, its presence allows the model to have stable topological solitons. The Euler-Lagrange equation� of (3.1) are ( 3. 2 ) which represent a difficult system of hyperbolic nonlinear equations, although the structure of the connection A implies that it is a flat connection (with vani�hing curvature) . That is, F = d A + A 1\ _4 = 0, or in component form, (3.3) Such a property is often associated with integrable �ystems. The Hamiltonian density of (3.1) is 1 1 (3.4) 1{ = 2 Tr A� + L A� - -2 Tr L [Ao, Aj]2 + L [Aj , AkJ2 . -
=
.
- -
l<j <3
l<j<3
l�j
Finite-energy condition implies that the map U(t, x) approaches a constant group Hence U(t, · ) may be identified with a map element at spatial infinity Ixl = from JEt3 {oo } = S3 to SU(2) which is topologically the3 3-sphere S3 as welL In other words, U(t, · ) represents a homotopy class in 7r3 (S ) = Z, which is given by a topological integer, say Q(U)(t). In fact, as in the sigma model ease, this integer is again a topological charge which is induced from a conserved topological current JIL where (3.5 ) so that the topological index Q(U)(t) is expressed the total charge associated with JIL , (3.6) Now consider the static case, Ao = O. Following the idea of [19), Manton and Ruback [58] obtain 1 1 ' kl 1{ Tr (Aj = ) 2 ± EJ [A Ad (3.7) k, ± EJ Tr ( AjA k A£) , 2 which suggests the following topological lower bound for the energy, 1{dx > 247r2 IQ(U) I , E(U) = (3.8) 00.
U
as
-
' k[
R"
2
FADDEEV KNOTS AND SKYRME SOLITONS
327
Of course, the above lower bound is attained when U satisfies the equation
1 'kt (3.9) Aj ± {.J [Ak ' Ai] = O. 2 Manton and Ruback [58] argue that the only solution of (3.9) is trivial, Aj = 0, or the map U must be a constant matrix. As a consequence, one cannot produce physically interesting solutions by exploring the self-dual or Bogomol'nyi structure of the Skyrme model. Thus we encuunter an analysis problem. In order to see the problem more transparently, we follow [76) to represent the group element U flS ( 3.1 0)
where O'j (j = 1, 2, 3) are the Pauli spin matrices and the fields rp4 and rpj are all real valued functions. Then we have uut = (rp� + rpJ ) I2 , which implies that ¢ = ( rpj , ¢!4) lies in the unit sphere S3 , which happens to be the topological manifold of SU(2). In terms of the map rp : ]R3 S3, the static Skyrme energy after dropping the unimportant constant coefficients becomes ->
(3.11)
E( rp ) =
1
( L laj rp l 2 + L laj¢ 1\ ak¢1 2 ) dx, 1It3 1<j<3 1<j
so that the topological charge associated with the map ¢! : ]R3 -> S3 is given by 1 (3.12) Q (¢) = IItS det (rp, fh rp , �rp, (h¢)( x ) dx , 27[ 2 which is the tupological degree of rp when viewed as a map from S3 = ]R3 U { oo} into itself. The topological solitons in the Skyrme model are then the solutions of the constrained minimization problem EN = inf{E(¢» I E(¢» < 00, Q(¢» = N } .
(3.13)
The Faddeev model has exactly the same mathematical structure except that, instead of S3, we have S2 as the image space (that is, the mesonic field has only 3 components) , which is the coset manifold SU(2)jU(1). To see the relation between the Skyrme model and the Faddeev model more clearly, we follow the reformulation of eho and his coworkers [10, 22) for the Faddeev model. Introduce the angular variable w and the spin vector n = (nj) (j = 1 , 2, 3) , we represent the group element U E SU(2 ) as
(3 . 14)
VT(t, x) = exp
w
(t , x) O'jnj ( t, x) 2i
The condition UUt = h implies that n = (nj ) lies in S2 Inserting (3. 1 4) into the following slightly renormalized Skyrme Lagrangian density, we have 1 I £ Tr ( AI'A ' ) + Tr ( [AI' , A v ] [A I' , A V ] ) 32 -
-!
(3. 15)
328
FANGHUA LIN AND YISONG YANG
The Euler-Lagrange equations of the action (3.15) are D/.J)J1-w - sin w(DJ1-n . DJ1-n)
+! sin w (8 w8vn - DvwDJ1-n) · (DJ1-wfJVn - 8Vw81' n) J1+ sin2 � DI' « o wDVn - oVwDl'n) · Dv n) J1-
.
w
w (3. 16) - sin2 '2 sin '2 (Dl' n A Dvn) . (Dl'n A oVn) = 0, DJ1- sin2
� [n
f\
DJ1-n] +
� sin2 � ( [DvwDVw] [n
f\
81'n] - [81'w8V w] [n f\ 8vn] )
(3. 17)
Note that it is consistent to set w = 1r in the above equations, which reduces the action density (3.15) and the equations of motion (3. 16) and (3. 17) into 1 1 . £, = DJ1-n D/'n - (DJ1-n A Dvn) . (Dl'n f\ 8Vn) , (3.18) 4 2 (3.19)
which are exactly the Lagrangian density and the equations of motion of the Fad deev model [29 , 30, 31] . Note also that (3. 19) is the Euler-Lagrange equation of (3.18) . Hence thc Faddeev model is indeed a submodel of the Skyrme model. We remark t.hat there has been no mathematical study on the well-poscdness of either the system of equations (3. 1 6) and (3.17) or the equation (3. 19), both hyperbolic, nor any study on the behavior of the time-dependent solutions of these important equations. Since n is parallel to 8" n A ovn, it is seen that Fl'v (n)FJW(n) = (D 'n A Dvn) . l l f\ BVn) whcre Fl'v (n) = n · (DJ1-n f\ 8vn). Thus (D 'n the Lagrangian density of the Faddeev model is also written [29, 30, 31] £,
(3.20)
=
� Dl'n . Dl'n - ! FI'., (n)FI'V (n).
Let AI' Cn) be such that FJ1-v (n) = DI' Av(n) - DvAJ1-(n) . The conserved topological current J I' is given by (3.21 )
Hence, as before, we have the topological total charge Q(n) (t) expressed as (3.22)
Q(n)(t) = 11<3
JO (t, x) dx =
3 1 21r 2
rJJR3 fjklFjk (n)At(n) dx.
The above quantity, in fact, has a more familiar identity. To see this, we recall that finite-energy condition implies that the map n(t, ·) approaches a constant vector as Ixl --> 00. Hence we may view nU, ') as a continuous map from ]R!3 U {(X)} = 83 into S2 so that it can be represented by a topological integer, say N(t) , in the homotopy group 1r3(S2) = Z. This integer is the Hopf index and it was Whitehead [88] who found that such an index has the integral representation given by the formula (a. 22) . What has been illustrated above is that the Hopf index of the map n(t, ·) is also a naturally defined "charge" induced from a conserved topological current as those in the other quantum field theory models.
FADDEEV KNOTS AND SKYRME SOLITONS
�2!l
In the static case, the (conveniently renormalized) Faddeev energy takes the form
E (n)
(3.23)
=
r IR3
F�l(n)
dx. ..
The energy-minimizing maps in each topological class tons called the Faddeev knots
[13, 14, 15, 32J.
Q(n)
=
N
are knotted soli
Hence we need to study the
following topologically constrained minimization problem (3.24)
EN
for a prescribed integer
N
= 1 , 2, · · ·
,8
= inf { R (n)
N.
=
The computer simulations in
N }, [13, 14, 15, 32]
are for
for which the problem (3.24) is truncated over a finite large box
which "approximates" the full space In his rCM
I E(n) < x, Q(n )
]R3 .
2002 address in Beijing,
Faddeev
[31] proposed the above minimiza
tion problem to the mathematicians and noted that the main difficulty involved is the lack of compactness. In fact, this difficulty is not isolated and arises also in models as discussed in Section 2. Vve have seen that even for the one-dimensional
the general existence problem for topological solitons in other quantum field theory
sine-Gordon model, the associated minimization problem (2. 16) has no solution
when the topological charge
N
is different from
0, ±l.
For the two-dimensional
minimization is not possible and one needs to pursue a self-dual or Bogomol'nyi re
sigma model and four-dimensional Yang-Mills theory, we have seen that direct duction in order to obtain absolute minimizers for the energy functionals. We note also that, in addition to this self-dual structure, another common feature of the above two soluble models is that they are both conformally invariant field theories. Many people have noted that conformal invariance is often crucial to solvability. \Vhen conformal invariance becomes invalid, the complete solvability �mch as that associated with the sigma model and the Yang-Mills instanton model may not be available. For example, except in the critical phase
[41J
between two types of su
perconductivity, people have not been able to establish for the Ginzburg-Landau theory on
]R2
the existence of an energy minimizer realizing a given quantized flux
(the first Chern number), and a similar situation happens for the Chern-Simons theory
[23, 93J ; except
in the BPS limit
[19, 66, 83] '
people have not been able to
establish the existence of a Yang-Mills-Higgs monopole of any monopole number (the winding number); although there have been some works on the existence of energy-minimizing unit-charge Skyrme solitons problematic unfortunately. !
[24, 25, 71, 56] '
the proofs are
Thus, in view of the above picture with regard to
the role played by conformal invariance in the solvability of quantum field theory models, it should not come as a great surpriHe to see that Aratyn, Ferreira, and Zimmerman
[4]
were able to obtain in toroidal coordinates a wide family of
ex
plicit solutions realizing all possible integer values of the Hopf invariant for their
1t
curiously defined field theory model with a static Hamiltonian density of the for =
2: 1
which is conform ally invariant over
]Ra.
Howeyer, the
original Faddeev energy (3.23) is not conformally invariant. Therefore we indeed encounter a difficult problem here.
l Esteban has recently published an errat.1Im [27J.
FANGHUA LIN AND YISONG YANG
330
4. Progress toward an existence theory In this section, we report our existence results obtained for the static Skyrme model and the Faddeev model. For convenience, we begin with the Faddeev model. As in Section 3, we use E(n) and Q(n) to denote the static Faddeev energy 82 . First recall that the lower and the Hopf charge of a finite-energy map n : �3 bound ---+
(4. 1 )
where C > 0 is a universal constant, was derived a long time ago by Vakulenko and Kapitanski [84] . This lower bound ensures the existence of an integer No =f. 0 such that For such No , we have
THEOREM 4.1. The fundamental problem (3.24) with N = No has a solution. Besides, we have
THEOREM 4.2. The problem (3.24) has a solution for N
=
±1 .
Furthermore, we have
THEOREM 4.3. There exists an infinite subset § of the set IZ of all integers so that for any N E § the problem (3. 24) has a solution.
Although we do not know how big the set § is or whether § =f. IZ, we can make the above statement more precise as follows.
THEOREM 4.4. For any N E IZ, there is a decomposition Ns E IZ, s = 1 . . . . (4.2) ,
so that the following sub-additivity relation
. ,
£,
(4. 3)
holds. In fact, all the integers Nl , ' " , Ne in (4 . 2) may be cho.sen to be members of the set §. Besides, the sublinear growth (upper) bound (4.4)
EN < CIN I3/4 .
is valid. Here, in (4.4), C is a universal positive constant.
The above result may be interpreted physically as follows: If EN is viewed as the mass of a particle of charge N and EN, , . . . , EN, are the masses of constituent particles Of substances, then (4.2) is a charge conservation law and (4.3) says that the mass of the composite particle is greater than Of equal to the sum of the masses of its constituents or substances because possible extra energy may be needed for the constituents or substances to form a bound state and, as a result, the composite particle may look "heavier" . In fact, at this moment, it may be amusing or even relevant to compare the above statement with the familiar quark model in nuclear physics. It is known that the mass of a hadron gets considerable contributions from quark kinetic energy and from potential energy due to strong interactions. For hadrons made of the light quark types, the quark mass is only a small contribution to the total hadron mass. For example, the mass of a proton is about 0.938 GeV je but the sum of the masses of two up quarks and one down quark is merely a total of
FADDEEV KNOTS AND SKYRME SOLITONS
33 1
0.02 GeVIe-? (proton is made of two up quarks, each of electric charge +2/3, and one down quark, of electric charge -1/3; the mass of an up quark is about, 0.004 GeVIc2 and the mass of a down quark is about 0.008 GeVIc2 ) and we see clearly the law of (electric) charge conservation and the mass inequality. Actually, an even closer physical situation analogous to our result concerning the energy splitting process expressed by (4.2) and (4.3) is nuclear fission. For details, see [52] . Therefore, we may call (4.3) (together with (1.2)) "the Substantial Inequality" , which will be seen to be a crucial technical ingredient of our method. Comparing (4.1) and (4.4), we see that the sharp sublinear growth estimate EN � IN13/4 holds asymptotically for a large Hopf charge I N I . It will be seen that the upper bound (4.4) is another crucial technical ingredient. Significant difficulties arise when we attempt to gain further knowledge about the set § stated in Theorem 4.3 because a minimizing sequence of the problem (3.24) may fail to "concentrate" in IR3 . On the other hand, when we consider the problem over a bounded contractible domain, a more sat.isfactory re::mlt is valid because, technically, a bounded domain prohibits the minimizing sequence to "float" away and "concentration" is trivially guaranteed: THEOREM
4 .5. IJet n be a bounded contractible domain in IR3 and consider the
admissible set of all the field configurations which assume a (;Onstant value on the boundary of and outside n . Then, over such an admissible set, the problem (3.24) has a solution for any N E Z.
This theorem ensures the existence of the knotted solutions of respective Hopf charges obtained in [13, 14, 15, 32] where the full space IR3 is replaced by a large box in order to carry out computer simulations. In the following, we discuss some structural technical ingredients of our analytic methods. To put our approach in context, we compare our methods with the familiar concentration-compactness principle. We then show the importance of the 3/4-power upper bound and the substantial inequality in establishing various existence results just stated. For clarity, we divide our discussion into several subtitled paragraphs. Concentration-compactness and substantial inequality. When one considers the minimization problem (3.24) , one naturally encounters the three alternatives in the concentration-compactness principle due to Lions [54] for the minimizing sequence, namely, (i) compactness (concentration of energy up to translations) ; (ii) vanitihing (energy density is fiattened to zero everywhere) ; (iii) dichotomy (energy splitting into floating chunks) . In order to achieve convergence (compactness) , one usually needs to rule out (ii) and (iii) to arrive at (i) . For our problem, however, it is impossible to rule out (iii) completely. Indeed, we are in a situation where we have to accept (iii) (splitting) and achieve something less than (i) (concentration) . More precisely, we show that the energy at the worst would split into "topologically concentrated" floating chunks characterized by (4.2) and (4.3), which we referred to as the Substantial Inequality. We shall see later that this important inequality will allow us to obtain existence (hence convergence) indirectly. The 3/4-power upper bound and knotted solitons. We note that a profound implication of (4.4) is indeed the existence of knotted solitons at sufficiently high
332
FANGHl-"A LIN AND YISONG YANG
Hopf cha.rges. To see this, we show that, for a large value of Q, a Fadd88v eIlergy minimizer prefers to appear as a clustered configuration (a knotted soliton) realizing the topology designated by Q than appear as a field configuration with widely separated fmergy lumps of a simpler topology (a multisoliton of a sum of unknot ted solitons) realizing the same topology. Such a result may be illustrated most easily by showing that, if N > 0 is sufficiently large, a Faddeev energy minimizer with the Hopf invariant Q = N can never be represented as a multisoliton of I,he sum of N widely separated solitons, each of a Hopf charge Q = 1 (an unknot). If the above described multisolitons were allowed, then, away from the local concentration regions of these unknots, the field configurations gave negligible eonl,ributions to the total energy. Hence, approximately, we would have EN � N E1 , which contradicts (4.4) for large N . Therefore, unlike vortices, monopoles, instantons, and cosmic strings, which do not mind to stay apart at least at the BPS limit, the Faddeev knots prefer to stay together in a clul;tered structure. In other words, the Faddeev knots like to stay knotted. Existence theorems obtained by the substantial inequality and the 3/4power upper bound. To prove Theorem 4.1, we write the decomposition No = Nl + . . . + N£ , N. E Z, s = 1 , · · · , f so that ENo > ENl + . . . + EN, (splitting into £ fioating chunks) > £ENo . Hence £ = 1 (no splitting) and concentration-compactness is achieved . In other words, Theorem 4.1 is proved. In order to prove Theorem 4 . 3, we suppose otherwise t hat 3 is finite. Set N° = max{ N E §}. Again, since for any N E N, there is a decomposition N = Nl + . . . + Ni , lV, E 3, s = 1 , . . . , £, we have N < £No . On the other hand, we also have BN > EN, + . . . + ENI 2 fENo . Combining these two inequa.lities, we have EN > (ENo /NO)N, which contradicts the upper bound (4.4) . This proves Theorem 4.3. Proof of the 3/4-power upper bound. We first recall the following fa.c t: if u E C1(IR3, 52) is such that u ( x ) =constant for 1:[;1 �mfficiently large and that v : 52 52 is a smooth miip of degree deg(v) , then the Hopf invariant of u = v 0 u : IR3 52 satisfies -->
-->
(4.5)
Q(u.)
=
(deg(v))2Q(u).
We begin by considering the case N = n2, for a positive integer n. We decom pose the upper hemisphere 5i as Hi = Ui l B(i) U D. Here B(i) 's are mutually disjoint geodesic balls of radius T � 1/ vn inside 5i . We define a Lipschitz map v : 52 --) 52 ilS follows: vex) = (0, 0, 1) for all x E 52 \ ur J B(i), and on each B(i), v is such that VlaE(i) = (0, 0, 1), v(B(i)) covers 52 exactly onee, and v : B(i) -> 52 is orientation-preserving. In other words, the degree of the map from B(i) onto 52 is exactly 1 . We can further require that I I V v liu(s2) < cvn for a positive constant c independ8nt of n. We then construct a map h : IR3 52 such that h is a constant outside the ball Bfo" I I V h II L� (Il!') � c/ vn for a constant c independent of n, and that Q(h) = 1 . Let 11. = V 0 h E B. Then (4.5) give� Q(lI,) = n2 = (deg v)2 = N . On the other hand, I I V11.II LOC (Il!.3) < (;2 and 11.( x ) is a constant for x outside the ball B..;n. Hence --+
(4.6) For the general case, we have n2 < N < (n + 1)2 for some positive integer n. We observe that k = N - n2 < (n + 1)2 - n2 = 2n + 1 . Let ho : Bl 32 be a smooth map with ho laE! = (0, 0, 1 ) and Q (ho) = 1 . Take k points X l , · . . , X., E R3 -->
FADDEEV KNOTS AND SKYRME SOLITONS
333
such that I Xi l » -Iii and that I Xi - xj l » 1 + -Iii for all i , j = I, · · · , k, i i= j . We then define u : ][(3 S2 as follows: u(x) = 1L(:]; ) = ( v 0 h) (x) for x E B..;dO), u(x) = ho (x - Xi) for x E B1 (Xi ) , i = 1 , · · · , k, and u(x) = (0, 0, 1) otherwise. Here u is constructed a.� in the case N = n2 before. It is obvious that u is a Lipschitz map from ][(3 into S2 with u(x) = (0, 0, 1 ) for I :r. l large. Besides, . -;
Q(11)
(4.7)
=
Q(u) + k
=
n2 + k = N.
Moreover, E(11) = E(u) + kE(hu) < C1 (-lii) 3 < C2N:l/4 • We have thus proved (4.4) in the case that N is positive. For negative N, one simply needs to change orientation.
Attainability of the energy E1 • We now show, with the help of a sharpened
estimate of the universal constant in (4. 1 ) and an appropriate estimat.e of EI , that the substantial inequality method may be exploited further to prove Theorem 11.2. That is, ±1 E S. First, we note that the method of Vakulenko and Kapitanski [84] may be refined by using the Sobolev inequality with optimal constants [9, 69, 82] to obtain the following sharpened energy lower estimate [87, 52] (4.8)
Besides, using the Hopf map as a test map, it. can be shown [87, 52] that the energy EI has the upper estimate ( 4.9)
Now the proof of 1 E § can be done as follows. Suppose that EI is not attainable. Then, in the minimization process for EI , concentration doe� not occur and there holds the nontrivial energy splitting in view of the substantial inequality (4.8), (4.10) (4. 1 1)
Ej 1
>
ENl + . . . + ENt ,
N1 + . . · + N[,
Ns E Z \ {O},
8 = 1 " " , £,
e > 2. Since each EN, > 0 in view of (4.8), we see from (4. 10) and the fact EI = E-l that N. i= ± 1 for s = 1, , " , f. In view of (4.1 1 ) , one of the integers,
with
must be an odd number. Assume that Nl is odd. Then INl l > 3. Of course, I N2 1 > 2. Therefore (4.9) and (4.8) lead us to (1.12)
whidl is a contradiction and the proof follows. For details, we refer to [52] . \Vc now turn to the Skyrme model. As remarked before, the technical struc ture of the Skyrme model is similar to that of the Faddeev model. In particular, the existence of an absolute energy minimizer of lhe SkYI'Ine model of unit baryon number cannot be proved via a direct application of the concentration-compactness principle as was originally conceived in [24, 25] but can only be proved indirectly via a use of the substantial inequality. Here we present a correct proof for the exis tence of a Skyrme soliton in the class of unit baryon number (topological charge), which is in fact a by-product of our method for the Faddeev knot problem. It can be shown that all the existence results (except the (3/4)-growth law) parallel to those in Theorems 4.2, 4.4, and 4 . 5 hold for the problem (3. 13) and we skip their corresponding statements. We sketch below the proof of Theorem 4.2.
334
FANGHCA LIN AND YISONG YANG
Proof of attainability of El in the Skyrme model by the substantial inequality. Recall that there is a decomposition 1 = Nl + . . + Nf, N. E 1L, o9 = 1 . . € so that .
,
.
"
(4. 13)
We assert that this decomposition must he trivial. That is, e = 1 and Nl = ±l. In fact, if this is not trivial, then there is an Ns (1 < s < €) so that I Ns l > 2. Hence (4. 13) says that El > EN
(4.14)
•
.
On the other hand, however, Esteban [24] has obtained the estimates (4. 15) where I S3 1 is the volume of the unit 3-sphere. Inserting (4. 15) into (4.14), we arrive at a contradiction. We should note that (4.15) states that El is in fact the least positive static energy of the Skyrme model, or using the similar notation as that for the Faddeev model, we have No = 1 . 5. Skyrme model in two dimensions and compactness Recently, there has been some interest in formulating a Skyrme theory over a (2 + I)-dimensional spacetime, following the original idea of Skyrme. In such a lower-dimensional field theory, in addition to the usual Skyrme term, one must impose a potential term in order to stabilize the solitons. Hence, parallel to the classical minimization problem of the static Skyrme energy over the spatial do main �3 , we encounter the minimization problem over the spatial domain �2, for which the static Skyrme energy now contains an additional potential tcrm. Thill is the problem of the existencc of two-dimensional Skyrmions. In this context, interestingly, the technical difficulties (in three dimensions) in the Esteban paper [24] may all be overcome to yield a complete proof of an existcnce theorem for the two-dimensional Skyrmions following the ideas given in [24) by directly using the method of concentration-compactness [54) . \Ve emphasize that such an approach works only for the two-dimensional Skyrme model. Below we review this model and its analysis. In particular, we eornpare the method of concentration-compactness and the substantial inequality method in some detail. In normalized form, the two-dimensional static Skyrme energy functional gov S2 is defined by (cf. [5) and references erning a conliguration map n : :lR.3 therein) --->
(5.1)
E(n)
=
1
2
]R2
where k = (0, 0, 1) is the north pole of 82 in :lR.3 , and A , /-t are positive coupling constants which should not be confused with the spacetime variable indices used in the earlier sections. Note that, sometimes in literature, the potential term in (5. 1 ) is chosen to be of a lower power, j.t(1 k . n) , which makes the potential energy of a stereographic projection take infinite value. In order to maintain a finite value for the potential of a stereographic projection, we observe the above (common) convention for the choice of the potential density. However, our general analysis is not affected by such a convenient, definitive, choice. -
FADDEEV KNOTS AND SKYRME SOLITONS
335
Finite-energy condition implies that 0 tends to k as I x l 00 . Therefore o may be viewed as a map from S2 to itself which defines a homotopy class in 7I"2(S2) = Z, who�e integer representative is the Brouwer degree of 0 with the integral representation --->
(5.2)
Q ( o) =
1
471"
1l!.2
o · (010 II 020) dx.
,
Like before, we are interested in the basic minimization problem
(5. 3 )
EN = inf{E(o) I E(o) <
00 ,
Q(o) = N }
,
where N E Z. Below is our main existence result for two-dimensional Skyrmions obtained in [51] by the method of concentration-compactness [54] as in Esteban
[24, 25].
THEOREM
5. 1 . If the coupling constants >. and /1 satisftJ
(5.4)
then the minimization problem (5. 3) has a solution for N = ± l . Moreover, E1 < EN for all lNl > 2 if >'/1 < 12.
Roughly speaking, the condition (5.4) guarantees E1 < EN for all INI > 3. Note also that the condition (5.4) is crucial for the exclusion of dichotomy (the alternative (iii)) in t.he concentration-compactness principle. We show how to use our substantial inequality method to improve (5.4) signif icantly. First, we observe that the following substantial inequality holds for any >., J.I > 0:
5.2. Let N be a nonzero integer and {OJ } a minimizing sequence of the problem (5.3). Then either (i) holds (hence a subsequence of {OJ } converges weakly to a solution of (5. 3)) or there are nonzero integers N1 and N2 such that THEOREM
(5. 5) A s a consequence, if § denotes the subset of Z \ {O } for which every member N E § makes (5. 3) solvable, then § of 0. In particular, for any N E Z:: \ {OJ, there are integers N1 , . . . , Nt E § such that
(5.6)
N = N1 + . . . + Ne
and EN > E!V! + . . + EN£ ' .
Next we recall that we established [51] a topological lower bound which states that there is a positive constant C(>', J.I, N) (i.e., the constant only depends on the coupling parameters >. and J.I and the nonzero integer N) such that E(o) > 41T1 deg(ol l + c(>., /1, Q(o)) (Q(o) of 0). In particular, we have
(5.7)
EN > 441V1 ,
N of O.
If for N = 1 the compactness (the alternative (i)) for a minimizing sequence of (5.3) does not occur, then by Theorem 5 .2 there are nonzero integers N1 and N2 so that 1 = N1 + Nz and (5.8)
Since EN, > 0 and EN2 > 0, we �ee from (5.8) that Nl of ±1 and N2 of ± l . However, one of the Nl and N2 must be odd. Assume N1 is odd. Then I Nl l > 3 .
336
FANGHUA LIN AND YJSONG YANG
1N2 1 > 2.
Since N2 must be even, so get
Inserting these facts into
(5.8)
and
(5.7),
we
El > 47r (3 + 2).
(5.9)
On the other hand, llsing the stereographical projection as a test map, we may get an upper estimate for
Er
of the form
1 47r 1 + 2
(5. 10)
It i� clear that the inequalities
(5.9)
and
(5. 10)
are compatible only when
In other words, energy splitting (dichotomy (iii»
cannot occur when
(5. 1 1 ) Under this condition, the minimization problem
(5.3)
has
a
AIL > 192.
A, f.L
satisfy
solution for N
which gives a three-fold improvement upon the statement of Theorem
5. 1 .
See
for details.
= 1,
[52]
6. Other directions and developments In this section, we review some directions and developments not covered in the previous sections and we comment on some other unsolved problems. In the first two Hubsections, we are concerned about the Skyrme modeL \Ve return to the Faddeev model in the third subsection. vVe then disc1lsS HOrne additional mathematical issues on the Skyrme and Faddeev models in the last subsection.
6.1. Radially symmetric solutions in the Skyrme model.
We first con
sider radially symmetric solutions of the Skyrme model, also called the hedgehogs. In terms of the is given hy
group element and the radial variable
SU(2)
(3. 14)
in which
the Lagrangian density
(6. 1)
£(f)
=
nj (x)
(3. 15)
�2 (Dt !)2
_
=
xj /1'
and
becomes
1 ( D f) 2 r 2
wet, x )
H(f) =
� (Dd) 2
+
= 2/(1', t). Under this ansatz,
sin2 / _
1.2
with the associated Hamiltonian
(6 .2)
� (Or f)2 + Si�: /
1+
Si;:/ + (Orf)2
Finite-energy condition implies that the hedgehog angular variable the bounda.ry condition
lim fer, t) = (N + k)7r, T_O
(6 .3)
for some integers N and
k.
Q (I) (t)
21 1 "" 7r
(3.6)
satisfy
kiT
takes the reduced explicit form
0
Or/(r, t) sin2 /(7', t) d1'
- (1(0, t) - f(oo, t» = N, 7r
=
/ must
r CjklDjf(r, t) (n · [Dk n J\ Dt n] ) sin2 / (1', t) dx \ 47r JJlI.3
-(0.4)
lim fer, t) r---+ oo
.
On the other hand, we observe that, under the hedgehog
ansatz, the topological charge
-
a hedgehog
l' = l x i ,
FADDEEV KNOTS AND SKYRME SOLITONS
337
in terms of the boundary condition (6.3). The Euler-Lagrange equation of (6. 1 ) is 2
(6 . 5) Itt - frr - - fr r
=
2f . 2 j'; . 2 sm f + 2' sm(2 f) r r rr
-
sin(2f ) r2
sin2 I 2 1+ r
'
,
r
> 0,
which governs the dynamics of hedgehog solitons of the Skyrme model and whose well-posednesH has not yet even been studied. We now consider static hedgehogs only which are the solutions of the following two-point boundary value problem
(r2 + 2 sin2 f ) 1T>' + (1,.) 2 sin(2f) + 2r fr
(6.6)
f(O )
which makes the Skyrme energy (6.7)
E(f)
=
27l'
00
o
sin(2f) 1 +
N7r,
1 (00) 2
Si�: =
sin2 f + r2
f
,
r>
0,
0,
sin2 f dr
finite. Of course, (6.6) is the Euler-Lagrange equation of (6.7) . It can be shown that, for any integer N, (6.6) has a solution which may be obtained as the energy minimizer of (6.7) subject to the boundary condition stated in (6.6). See [26, 92] . In other words, among any topological class in 7l'3(S3 ) , the static Skyrme energy has a critical point which minimizes the energy functional among all radially symmetric field configurations. This result, although simple, is in sharp contrast with what we know for the sine-Gordon model where critical points of the energy of higher topo logical charges do not exist, as illustrated in Section 2. However, it is not known whether any of these critical points would be the absolute energy minimizers for the Skyrme model. In [56, 71] , the authors attempted to establish that the hedgehogs are absolute energy minimizen; at Q = N = ± l . Unfortunately, their proof appears to be problematic: their main idea is to use a procedure called Gelfand's Valley Method and make comparisons through the Steiner symmetrization and rearrange ment inequalities. However, the described minimization procedure leads to some incorrect orthogonality conditions (the expressions (4.5) and (4. 18) in [71]), which in fact may only be asserted as being obtained by maximization, contradicting the original expectation of these authors. More recently, a study of the interaction energy of widely separated Skyrme solitons, hence the configurations are not radially symmetric, was carried out [59] based on asymptotic analysis.
6.2. Geometrized Skyrme model and generalization. Motivated by con structing explicit or approximate static Skyrme solitons, Manton and Ruback [58]
geometrized the Skyrme model so that the energy governs maps from S� (the three sphere of radius R > 0) into the unit sphere S3. Note that, in this context, the classical Skyrme model may be recovered in the R ---7 00 limit. Lieb-Loss [49, 55] and Manton [57] showed that, modulo isometries, the "identity" map u ( x) = xI R is the unique absolute energy minimizer among the topological class defined by N = 1 for R < 1 . Lieb and Loss [49, 55] also showed that the identity map is a local minimizer for R < J2. Manton and Ruback [58] earlier showed that the identity map is unstable for R > J2. In fact, these results can be reformulated [91] for maps between two closed n-dimensional Riemannian manifolds, (M , g) and (N, h), where 9 and h are the metrics on M and N respectively, as follows.
338
FANGHUA LIN AND YISONG YANG
N be a differentiable map and u'h denote the pullback of the Let u : M metric h nnder u. Use dVg to denote the canonical volume element of (M , g) and aj (A) the elementary symmetric polynomials formed from the n eigenvalues of the symmetric matrix A. The generalized static Skyrme energy is then written ---+
(6.8)
E(u ) =
1M {al (g-lu'h) + an_ l (g-l u· h) } dvg•
Replacing the pair S'k and S3 , we consider the situation when (M , g) and (N, h) are homothetic [49, 55, 9 1] . That is, there is a diffeomorphism 'IjJ : M N such that 'IjJ* h = ,.,.2 g for some constant ,.,. > O. Then there holds the general result [91J that, up to isometries, the homothetic map with ,.,. > 1 is the unique minimizer of E among all maps of nontrivial topological degrees. Besides, this homothetic map is a (stable) local minimizer for the energy provided that [91J ,.,.2 + 21 T2( n- l ) (n - l) (4 - n) > 0 , ---+
(6.9)
,.,. 2 ..!. _ ! n 2
+ ,.,.2 (n- l)
� + n2 _ 2 n
(n - 1 )
>
0.
Note that, when n = 3, we have T2 > 1/2, which is the result of Lieb-Loss [49, 55] and Manton [57] ; for n = 4, we have T4 > 1/6; for n = 5, we have 1/2 > ,.,. 6 > 1/12; etc. More recently, Riviere proved in [68] that, for the geometric Skyrme model governing maps from S'k into S3 , there is an Ro E ( V3fi, J2] such that for R < Ro , the identity map x/ R remains to be the unique minimizer. Whether or not one can take Ro to be equal to J2 remains open. We also note that, in the geometric compact context here, there has been no study on the direct minimization problem for the energy functional (6.8).
6.3. Fractional-exponent growth law and knot energy. We noticed that one of the crucial facts that guarantees the existence of energy minimizing Faddeev knots in infinitely many Hopf classes is the sublinear energy growth property EN < C l NI3 /4 . That is, the minimum value of the Faddeev energy for maps from IR3 into 52 with Hopf invariant N grow sublillearly in N when N becomes large. This property implies that certain "particles" with large topological charges may be energetically preferred and that these large particles are prevented from splitting into particles with smaller topological charges. Such a feature may be a clue to the "stability of matter" problem and may also be relevant to the existence and stability of large molecular conformation in polymers and gel electrophoresis of DNA. In these problems, a crucial geometric quantity that measures the "energy" of a physical knot of knot (or link) type K (or simply knot) is the "rope length" L(K), of the knot K. To define it, we consider a uniform tube centered along a space curve r. The "rope length" L cr) of r is the ratio of the arclength of r over the radius of the largest uniform tube centered along r . Then (6. 10)
L (K) = inf{L(r) I r E K}.
A curve r achieving the infimum carries the minimum energy in K and gives rise to an "ideal" or "physically preferred" knot [44, 47], also ealled a tight knot [21]. Clearly, this ideal configuration determines the shortest piece of tube that can be closed to form the knot. Similarly, another crucial quantity that measures the geometric complexity of r is the average number of crossings in planar projections
FADDEEV KNOTS AND SJ
339
of the space curve r denoted by N(r) (say) . The crossing number N(K) of the knot K is defined to be (6. 1 1 )
N (K)
=
inf { N (r ) I r E K},
which is a knot invariant. Naturally one expects the energy and the geometric complexity of the knot K to be closely related. Indeed, the combined results in [20, 21 J lead to the relation G1 N(K)P ::; L(K) < G2 N(K)P, where GI , G2 > 0 are two universal constants and the exponent p satisfies 3/4 < p < 1 so that in truly three-dimensional situations the preferred value of p is sharply at p = 3/4. This relation strikingly resembles the fractional-exponent growth law for the Faddeev knots just discussed and reminds us once more that a sublinear energy growth law with regard to the topological content involved is essential for knotted structures to occur. In our quantum field theory problem, it is the underlying property and struc ture of the homotopy group 1T3 (S2) and the Faddeev energy functional formula that guarantee such sublinear growth. Generally, it seems that such a property may be related to the notion of quantitative homotopy introduced by Gromov [36J . For ex ample, we may consider the Whitehead integral representation of the Hopf invariant and the "associated" knot energy ala Faddeev. More precisely, let u : R4n-1 -+ s2n (n > 1 ) be a differentiable map which approaches a constant sufficiently fast at infinite. Denote by f2 the volume element of s2n and IS2n l = Is'n f2. Then the integral representation of u in the homotopy group 1T4n _ l (s2n), say Q (u) , which is the Hopf invariant of u, is given by 1 (6.13) Q ( u) = I S2n l IR4n-l v 1\ u' (f2) , dv = u* (f2) . We can introduce a generalized Faddeev knot energy for such a map u as follows, (6. 12)
(6.14) For this energy functional, we are able to establish the following generalized sub linear energy growth estimate < G2 I NI (4n- 1 )/4n , < EN _ (6 . 1 5) G1 IN I (4n- 1 )/4n _ where EN - inf{E(u) I E (u) < 00, Q(u) = N} and GI , G2 > 0 are universal constants. It is seen clearly how the dimension number of the space comes into play. We have reported some of our preliminary results along this line of generalizations in [53J and detailed work will appear in a separate forthcoming paper.
6.4. Additional technical issues. In addition, we mention some other in
teresting technical questions which may be worth pursuing. Soholev spaces of mappings. Returning to the original Faddeev model and the minimization problem (3.24) , we have introduced the following Sobolev-space of mappings from R3 into S2 ,
X = {n : R3 -+ S2, E(n) < oo}. It was proved in [50J that for each n E X, dCn' CO » = 0, here f2 is the area form on S2 . Hence, by a result of Bethuel [18], n can be strongly approximated by a sequence nj E Goo (R3, SZ) such that nj -+ n in H l (R3 , S2) as j -+ 00. (6.16)
FANGHUA LIN AND YISONG YANG
340
81)(n) = 0 on �3, where <5 1)(n) = M- 1 n* (11) . ThuH
the HopI invariant
Q(n) =
(6. 1 7)
1)(n)
such that d1) ( n)
= n* (11)
is the adjoint operator of d. For example, one may take
Thi�, in turn, implies that there is a one-form
1 16K2
and
1)(n) 1\ n*(n) r JR3
Q(n) E Z; as wa;; ::;howIl in Theorem 3.3 of [50]. It is not clear, however, if any map n E X can be approximated in Ltoc by a sequence of 2 smooth maps nj E COO (�3 , 8 ) such that E ( nj ) remains uniformly bounded. IL is certainly unknown if n E X can be approximated by a sequence of nj E Coo (�3 , 82 ) n in Hl�c (�:\ 82 ) and E(nj) remains to be uniformly bounded, as such that Ilj 00 . We refer to [37, 38, 39] for some recent works concern ing sHch density j is well defined. Moreover,
->
questions of smooth maps in the spaces of Sobolev maps. ->
Energy splitting.
imizers in every topological class of maps can be easily proved over a bounded For both Faddeev and Skyrme models, the existence of min
contractible smooth domain in
IR3
R,
(in particular, on a ball of radius
Understanding of these minimizers as
R -> 00
say
BR)'
remains to be a difficult question.
In fact, it is even unknown whether the minimum value
(6.18) satisfies
EJ.j = inf{E(n) I n : BR EfS -> EN
as
R ->
->
82 , n laB ,, = canst, Q(n) = N}
00.
We know that, if maps in a minimizing sequence of the Faddeev (or Skyrme) energy functional split as in the Substantial Inequality, then we have EN > + . . . + ENe - It is believable, in thi::; case, that, one would also have EN < EN, + . . . + Hence, EN = EN] +- . + EN, . This is the case for the two-dimensional Skyrme
EN1
EN, .
.
model due to the technical lemma
6. 1
in
[51]
(see also Lemma
7.4
in
[50] ) .
The
validity of this lemma in three dimensions for either the Faddeev or Skyrme model i::; unknown . On the other hand, if one has this lemma for the three-dimensional case, then one can show that E.� -> EN as -> 00 is true. Moreover, if the Skyrme would he valid.
model has only the cla::;s
Regularity.
N = ±1
R
being realized by minimizers, then
EN = IN I El
One of the most fundamental problems that remains open for both
Skyrme and Faddeev models is the regularity of solutions . It may be relatively easier to check the regularity of those specially constructed I:iOlutions (such as the hedgehogs) as described above. It is however a difficult problem for the absolute energy minimizers.
These questions are also closely related to similar questions
in nonlinear elasticity (see
[U]).
Indeed, the minimization problems here may be
compared with similar constrained minimization problems in nonlinear elasticity.
7. Conclusion Quantum field theory renderH to analysts a wide variety of mathematical prob lems of fundamental importance and a subclass of these problems are various con strained minimization problems giving rise to topological soliton::; modeling elemen tary particles and their interactions. As illustrated by the sine-Gordon model, there sometimes may not exist a critical point of the energy functional over a given topo logical class, and the existence of absolute energy minimizers is often a consequence of a self-dual or Bogomol'nyi structure in the problem and a direct minimization
FADDEEV KNOTS AND SKYRME SOLITONS
341
is likely to be impossible as witnessed by the earlier studies ranging from Lhe sine
Gordon model t.o the Yang-- Mills gauge field theory models. In the study of the Skyrme and Faddeev models, there is no useful self-dual or Bogomol'nyi structure.
Consequently, we have to consider the direct minimization problems of the corre
sponding energy functionals. In this situation, the well-established concentration
compactness principle cannot be used directly. Thus, we need 1.0 analy,,;e the be
havior of the splitting field configurations of a minimizing sequence. As a result,
we can establish an important inequality, called the substantial inequality, resem bling the process of a composed particle splitting into finitely many sub-particles during which Lhe total topological charge of the composed particle is exactly di vided into the sum of the topological charges of the sub-parti(:les and the mass
(i.e.,
the energy) of the composed particle is at least equal to the sum of the masses
of its sub-particles, with possible extra uncounted kinetic and bounding energies,
resulting in a topological charge conservation law and a mass or energy subaddiLive inequality, relating the charge and mass of the composed particle to the charges and masses of its sub-particles. As a consequence of this inequalit.y and some suit able energy estimates, it follows that splitting cannot occur at the unit topological charge for boLh the Skyrme model and the Faddeev model. In other words, the existence of a unit-charge Skyrme soliton and Faddeev knot is proved. In the case of the Faddeev model, we have in addition established a fractional-exponent growth
law for the energy minimum in terms of the topological charge. Such a growth law
ensures the existence of the Faddeev knots realizing an infinite class of topological charges and plays an e�Hential role in explaining why knotted structure is preferred
over isolated multiple-soliton configurations. In the Lwo-dirneIlt;ional Skyrme modcl case, although both the concentration-compactness method and the substantial in
equality method work, the latter provide� a much stronger existence result than the former. It is our hope that the described substantial inequality method may be use
ful to other diffieult topologically constrained minimization problems in quantum field theory.
Acknowledgment.
The authors would like to thank Xiaosong Lin for many help
ful conversations and communieations.
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,
.
,
COURANT INSTITUTE OF MATHEMATICAL SCIENCES, NEW YO RK 10021 E-mail address: l1nf«lcims . nyu . edu
NEW
YOUK UNIVERSITY, NEW YORK,
DEPARTMENT OF MATHEMATICS, POLYTECHNIC UNIVERSITY, BROOKLYN, E-mail address: yyang«lmat h . poly. edu
NEW
YORK 1 1 20 1
2007
Contemporary MFlthp.mA.tir:J:! Vulume 446, -
The precise boundary trace of positive solutions of the equation
b.u
=
uq
in the supercritical case.
Moshe Marcus and Laurent Veron
To Haim, with friendship and high esteem.
precise boundary trace of positive solutions of Au uq in a smooth bounded domain !1 C RoN , for q in t.he "uper-crit.ical case q > (N + 1 ) I (N 1 ) . The construction is performed in the framework of the fine topology associated with the Bessel capacity C2 /q , q ' on 8!1. We prove that the boundary trace is a Borel mellsure ( i n general unbounded) ,which is outer regular and essentially absolutely continuous relative to this capac ity. We provide a necessary and sufficient condition for such measures to be the boundary trace of .. pORit.ive solution and prove that the corresponding generalized boundary value problem is well-posed in the class of <7-modcratc AnSTR.AC'I'. We construct the =
-
solutions,
1. Introduction In this paper we present a theory of boundary trace of positive solutions of the
equation (1.1 )
in a bounded domain 0 C ]RN of class C2. A function u is a solution if u E L?ocCO) and the equation hulds in the distribution sense. Sernilinear elliptic equations with absorption, of which (1.1) is one of Lhe rnm;t important, have been intensively studied in the last 30 years. The foundation for these studies can be found in the pioneering work of Brezis starting with his joint research with Benilan in the 70's [2], and followed by a series of works with colleagues and students, up to the present. In the subcritical case, 1 < q < qc = (N + l)/ (N 1), the boundary trace theory and the associated boundary value problem, are well understood. This theory has been developed, in parallel, by two different methods: one based on a probabilistic approach (see [11, 5, 6] , Dynkin's book [3J and the references therein) -
1991 Mathematics Subject Classification.
Key words and phrases. measures.
Primary
31C15.
Nonlinear
elliptic
3fi.J60,
.3 5J67;
Secondary 3 1 B 1 5 , 3 1 B20,
equations, Bessel capacities, fine
©2007 345
American
topology,
Borel
Mathematical Society
346
MOSHE MARCUS A:I!D LAURENT VERON
and the other purely analytic (see [1 3 14, 1 5] ) . In 1997 Le Gall showed that this theory is not appropriate for the supercritical case because, in this case, there may be infinitely many solutions with the same boundary trace. Following this observation, a theory of 'fine' trace was introduced by Dynkin and Kuznetsov [7J. Their results demonstrated that, for q < 2, the fine trace theory is satisfactory in the family of so-called a-moderate solutions. A few years later Mselati [1 7J used this theory and other results of Dynkin [3] , in combination with the Brownian snake method developed by Le Gall [12] , in order to show that, in the case q = 2 all positive solutions are a-moderate. Shortly thereafter Marcus and Veron [16J proved that, for all q > qc and every compact set K c an, the maximal solution of (1.1) vanishing outside K is a-moderate. Their proof was based on the derivation of sharp capacitary estimates for the maximal solution. In continuation, Dynkin [4J used Mselati's (probabilistic) approach and the results of Marcus and Veron [16J to show that, in the case q < 2, all positive solutions are a-moderate. For q > 2 the problem remains open. Our definition of boundary trace is based on the fine topology associated with the Bessel capacity C2/q,q' on Dn, denoted by 'Iq . The prcscntation requires some notation. ,
Notation 1 . 1 . a:
13 > 0 put p(x) dist (x , an) and n" = {x E n ; p (x ) < 13 } , n� = n \ n" , I:" = an{3'
For every x E IR N and every
b: There exists a positive number
Vx E n"o
( 1 .2)
3!
;=
130
such that,
a(x) E an : dist (x, a(x))
=
p(x).
If (as we assume) n is of class C2 and 130 is sufficiently small, the mapping x (p(x) , a(x)) is a C2 diffeomorphism of 0.130 onto (0, 130) x an. e : If Q c an put I:,,(Q ) = { x E I:" : a( x ) E Q}. d: If Q is a 'Iq-open subset of an and u E C (an) we denote by u� the solution of (1.1) in n� with boundary data h = UXE8(A) on I:" . f-t
Recall that a solution u is moderate if lui is dominated by a harmonic function. When this is the case, u possesses a boundary trace (denoted by tru) given by a bounded Borel measure. The boundary trace is attained in the sense of weak con vergence, as in the case of positive harmonic functions (see [13J and the references therein) . If tr u happens to be absolutely continuous relative to Hausdorff (N 1) dimensional measure on an we refer to its density f as the L1 boundary trace of u and write tr u = f (which should be seen as an abbreviation for tr u = jd lHIN -1) . A positive solution 'lL is a-moderate if there exists an increasing sequence of moderate solutions {un } such that Un r u. This notion was introduced by Dynkin and Kuznetsov [7J (see also [9J and [3] ) . If ,1 is a bounded Borel measure on an, the problem -
(1.3)
-Llu + uq = 0 in n, u = /l on an
possesses a (unique) solution if and only if /l vanishes on sets of C2/q,q,-capacity zero, (see [15J and the references therein) . Thc solution is denoted by Uw The set of positive solutions of ( 1 . 1) in n will be denoted by U(n). It is well known that this set is compact in the topology of C (n ) , i.e., relative to local uniform convergence in n.
PRECISE BOUNDARY TRACE
347
Our first result displays a dichotomy which is the ba.'iis for our definition of
boundary trace.
1.1.
Let u be a positive solution of ( 1 . 1 ) and let e either, for every 'rq-open neighborhood Q of �, we have THEOREM
u dS /' j3� 0 iE {3 (Q )
( 1 . 4) or
( 1 .5 )
lim
j3�O
The first case OCCUTS if and only if
E,,(Q)
l Uqp(X)dX = DO ,
( 1 . 6)
e
udS
an . Then,
= DO
lim
ther'e exists a 'rq-open neighborhood Q of
E
such that < DO.
D = (0, ,60 )
x
Q
f01' every 'rq -open neighborhood Q of e . A
point e
holds, and a
E an is called a singular point of u in regular point of u in the second case.
denoted by S(u) and its complement in
an
by R(u).
the first case, i.e. when ( 1 . 4)
The set of singular points is
Our next result provides additional information on the behavior of solutions
near the regular boundary set R ( u).
THEOREM 1 . 2 . The set of regular points R (u ) is 'rq -open and theTe exists a non-negative Borel measure p on an possessing the following properties.
(i) For every cr E R(u) there exist a 'rq-OJlt!1t neighborhood Q of cr and a moderate solution 11! s'lLch that Qc
( 1 . 7)
and ( 1 . 8)
u
�
-> 'UJ
R (u )
p( Q) < -
,
lucally uniformly in n
(ii) It i.� outer regular relative to 'rq •
Based on these results we define the tr Cu
( 1 .9)
=
,
DO ,
(tr w ) xQ
precise boundary trace of u
by
(fJ, S(u » .
Thus a trace is represented by a couple (p, S), where
P is an outer regular measure relative to
an \ S.
= PXQ'
'rq
S C an
is 'rq-closed and
which is 'rq-Iocally finite on R
However, not every couple of this type is a trace,
A necessary and sufficient
condition for such a couple to be a trace is provided in Theorem 5 . 1 6 . The trace can also be represented by a Borel measure
( 1 . 10) for every Borel set (1.11)
v(A)
A
c
=
an. We
put
It(A) DO
'rq .
otherwise,
tr u : = v.
This measure has the following properties: (i) It is outer regular relative to
if A c R(u),
=
v
defined as follows:
.'
MOSHE MARCUS AND LAURENT VERON
348
(ii) It is essentially absolutely contirmous relative to C2/q,q" i.e., for every '!'q-open set Q and every Borel set A such that C2/q,q' (A) = 0, v(Q) = v(Q \ A).
The second property will be denoted by v
j-<
It implies that, if
C2 /q,q"
v( Q \ A) < 00 then v( Q n A) = O. In particular, v is absolutely continuous relative to C2/q,q' on '!'q-open sets on which it is bounded. A positive Borel measure possessing properties (i) and (ti) will be called a q-perfect measure. The space of q-perfect measures will be denoted by Mq (an). \Ve have the following necessary and sufficient condition for existence:
be a poszli1Je Borel measure on an, possibly unbounded. The boundary value problem q -�u + u = 0, u > 0 in n, tr (u) = v on an (1. 12) possesses a solution if and only if v is q-pcrfcct. When this co ndition holds, a solution of (1. 12) is given by ( 1 . 13) v = sup{uvxQ : Q E Fu } , THEOREM
1.3. Let
v
where
Fu : = {Q : Q q-open, v(Q) < oo},
G :=
U Q,
F
=
an \ G
:Fn u
and UF is the maximal solution vanishing on an \ F. Finally we establish the following uniqucnc:;s result,.
1 .4. Let v be a q -perfect measure on an . Then the solution U of problem (1. 12) defined by (1. 13) is O'-moderate and it is the maximal solution with boundary trace v . Furthermore, the solution of (1 . 12) is unique in the class of a-moderate solut'ions. THEOREM
For qc < q < 2 , results similar to those stated in the laBt two theorems, were obtained by Dynkin and Kuznetsov [7] and Kuznetsov [9], based on their definition of finc trace. However, by their results, the prescribed trace is attained only up to equivalence, i.e. , up to a set of capacity zero. By the present results, the solu tion attains precisely the prescribed Lrace and this holds for all values of q in the supercriticaI range. The relation between the Dynkin-Kuznetsov definition (which is used in a probabilistic formulation) and the dcfinition presented here, is not yet clear. The plan of the paper is as follows: Section 2 presents results on the C2/q, q,-fine topology which, for brevity, is called the q-topology. Section 3 deals with the concept of maximal solutions which vanish on the boundary outside a q-closed set. Included here is a sharp estimate for these solutions, ba.<;ed on the capacitary estimates developed by the authors in [16). In particular we prove that the maximal solutions are O"-moderate, This was established in [16] for solutions vanishing on the boundary outside a compact set. Section 4 is devoted to the problem of locali7.ation of solutions in terms of boundary behavior. Localization methods are of crucial importance in the study of trace and the associated boundary value problems. The development of these methods is
PIU"CI SE UOUNDARY TRACE
349
particularly subtle in the supercritical case. Section 5 presents the concept of preci::;e trace and studies it, firstly on the regular boundary set, secondly in the case of a-moderate solutions and finally in the general case. This section contains the proofs of the theorems stated above: Theorem 1 . 1 is a consequence of Theorem 5.7. Theorem 1 .2 ii> a coni>cquence of Theorem 5 . 1 1 . Theorem 1.3 is a consequence of Theorem 5.16 (see the remark following the proof of the latter theorem) . Finally Theorem 1.4 is contained in Theorem 5. 16.
2. The q-fine topology A basic ingredient in our study is the fine topology associated with a Bessel capacity on (N - I)-dimensional smooth manifolds. The theory of fine topology associated with the Bcssel capacity Ca,p in IRN essentially requires 0 < exp < N (see [1, Chapter 6] ) . In this paper we are interested in the une topology associated with the capacity C2/q ,q' in RN - 1 or on the boundary manifold an of a smooth bounded domain n E IRN We assume that q is in the supercritical range for ( 1 . 1 ) , i.e., q > qc = (N + I)/(N - 1 ) . Thus 2q' /q = 2/(q - 1) < N - 1 . We shall refer to the (2/q, q')-fine topology briefly as the q-topology. An important concept related to this topology is the (2/q, q')-quasi topology. We shall refer to it as the q-quasi topology. For definition und details see [1 , Section 6. 1-4] . We say that a HubHet of an is q-open (resp. q-closed) if it is open (resp. closed) in the q-topology on an. The terms q-quasi open and q-quasi closed are underHtood in an analogous manner. Notation 2. 1.
Let A, B be subsets of IR N- 1 or of an.
a: A is q-essentiaHy contained in B, denot.ed A
C2/q,q' ( A \ B)
=
O.
� B, if
b: The sets A , B are q-equivalent, denoted A :£ B , if C2/ q , q' (All.B) = O.
-
c: The q-fine closure of a set A is denoted by A. The q-fine interior of A is denoted by A d: Given E > 0, A< denotes the intersection of IRN- 1 (or an) with the E neighborhood of A in IRN . e: The set of (2/ q, q')-thick (or briefly q-thick) points of A is denoted by bq (A) . The set of (2/q, q')-thin (or briefly q-thin) points of A is denoted by eq(A), (for definition see [1, Def. 6.3.7] ) . Remark. If A c on and B : = an \ A then <> •
(2.1)
A is q-open ·:
Consequently
;,
A
c
eq (B),
B is q-closed -: ;, bq (B)
(2.2) (see [1 , Section 6.4] .) The capacity C2/q,q' possesses the Kellogg property, namely, (2.3)
c
B.
350
MOSHE MARCUS AND LAURENT VERON
(see [1, Cor. 6.3. 17] ) . Therefore (2.4 ) but, in general, bq ( A) does not contain A. The Kellog propert.y and (2.1) implies:
2 . 1 . (i) If Q is a q-open set then Q := e q (an \ Q) is the largest q-open set that is q-equivalent to Q . (ii) If F is a q-closPAi set then F = bq(F) is the smallest q-closed set that is qequivalent to F . P ROPOSITION
•
We collect below several facts concerning the q-fine topology that are used throughout the paper.
2.2. Let qc = (N + 1 ) /(N - 1) < q. i: Every q-closed set is q-quasi closed [1, Prop. 6.4.13] . ii: If E is q-quasi closed then E :!., E [1, Prop. 6.4. 12] . iii: A set E is q-quasi closed if and only if there exists a sequence {Em } of compact ,mbsets of E such that C2/ q • q , ( E \ Em) --; 0 [1, Prop. 6.4.9] . iv: There exists a constant c such that, for every set E,
P ROPOSITION
C2/ q, q ' (E) ::; cC2/q,q' ( E ), see [1, Prop. 6.4.11]. v: If E is q-quasi close.d and F :!., E then F is q-quasi closed. vi: If {Ei} is an increasing sequence of arbitrary sets then -
C2/q,q, (U Ei ) = lim C2/q,q, ( Ei ) . 1. - 00
vii: If { K; } is a decreasing sequence of compact sets then C2/q,q' (nKi) = lim C2/q ,q (Ki ) . '
1, ...... 00
viii: Every Suslin set and, in particular, every Borel set E satisfies C2/q,q' ( E ) = sup{ C2/q,q' (K) : K c E, K compact} = inf{ C2/q,q, (G)
:
E
c G , G open} .
For the last three statements see [1, Sec. 2.3] . Statement (v) is an easy con sequence of [1, Prop. 6.4.9] . However note that this assertion is no longer valid if 'q-quasi closed' is replaced by 'q-closed'. Only the following weaker statement. holds: If E is q-closed and A is a set such that CZ/q , q' ( A ) = 0 then E u A is q-closed.
2.3. Let E be a quasi closed set. An increasing sequence {Em} of compact subsets of E such that C2j q ,q' (E \ Em ) -> 0 is called a q-stratification of E. (i) We say that {Em } is a proper q-stratification of E if m 1 (2 .5 ) C2 jq q ' (Em+ 1 \ Em ) < 2 C2 /q q' ( E ). DEFINITION
,
-
-
,
(ii) Let {Em } be a strictly decreasing sequence of positive numbers converging to zero such that m (2.6) C2/q,q' (Gm+1 \ Gm ) < T Cz/q,q' ( E ), Gm := Uk lE�k .
351
PRECISE BOUNDARY TRACE The sequence {£m } is called a q-pmper sequence. (iii) If V is a q-open set such that C2/q,q' (E \ V) neighborhood of E.
= 0 wc say that V is a q-quasi
Remark. Observe that G := Uk'- EZ' is a q-open neighborhood of E' I in general, only a q-quasi neighborhood of E.
= UEm but,
2.4. Let E be a q-closed set such that C2/q ,q ' (E) > O. Then: (i) Let D be an open set such that C2/q,q' (E \ D) = O . Then E n D is q-quasi closed and consequently there exists a proper q-stratification of E n D, say {Em}. Furthermore, there exists a q-proper sequence {Em } such that LEMMA
G = U:=I (ETn)€m c D and
(2.7) Consequently
UEm = E'
c
0
c
(2.8)
6
/!. D
E
q - q c O c 0 c D.
(ii) If D is a q-open set such that E (2.8) holds.
where 0 ;= U:= I (Em) <m /2 .
/!. D
then there exists a q-open set 0 such that
PROOF. If AI, A2 are two sets such that A l :!., A2 and Al is q-quasi closed then A2 is q-quasi closed, (sec the discussion of the quasi topology in [ 1 , sec. 6.4] ) . Since E n D /!., E and E is q-closed it follows that E n D is q-quasi closed. Let {Em } be a proper q-stratification of E n D and put E' = U�=l ETn . If E' is a closed set the remaining part of assertion (i) is trivial. Therefore we assume that E' is IIot closed and that C2/q,q, (Em+1 \ Em) > O.
To prove the first statement we construct the sequence {{Tn}�=l inductively so that (with Eo = 0 and EO = 1 ) the following conditions are satisfied; (2.9)
(2. 10) (2. 1 1 )
' Em \ (Em-I ) £ m '= C2 /q q ( F:"'m ) < 2C2/q,q, (Em \ Em- I), [;'
1 2 <m - '
,
,
Em < {m-I /2, ,
Choose 0 < E I < 1/2, sufficiently small so that
E;' c D ,
C2/q ,q ' (E2 \ E;I/2)
C2/q,q, (Fm) F:"'m
m
c
D,
>
0,
= 1 2 .... ,
,
> O.
This is possible because our asRumption implies that there exists a compact. subset of E2 \ El of positive capacity. By induction we obtain E' m C E'm m cD m-l-, U Fe",.
(2.12) and consequently
m.
(2. 13) Since Fm C Ern , (2.13) implies that (2. 14)
- UOO G . - UOO k=1 E" k k=1 F'k k , '
-
Grn ;= Urn k = 1 Fe. k= 1 E"k = Urn k .
352
MOSHE MARCUS AND LAURENT VERON
{Em }
The sequence (2. 14) ,
constructed above satisfies
(2.15)
2.6.
Indeed, by
(2.5),
(2. 10) and
k=m+1 < 2 � C2/q,q' (Ek \ Ek l ) < z - m+1C2/q,q' (E). k=m+l ""
Next we show that the Het
(2.13), OO l Fk
is q-quasic1osed. By
(2.16)
/ . 0, .- U"" k=l Ekk 2 -
'
-
lIenee, by (2 . 1 7)
(2.5 ) and (2.10), C2jq , q' (0' \ O:n) <
""
m /2 F < 2 � C2 /Q,q, ( ;k ) - +1Cz/ q,q, (E) . k=m+l
Since O:n is dosed this implies that 0' is quasiclosed. Further any quasiclosed set is equivalent to its fine closure. Since 0 C 0' it follows that 6 C 6' :t, 0' C C. We turn to the proof of (ii) for which we need the following:
Assertion 1. Let D be a q-open set. Then there exists a sequence of relatively open sets {An} SlIch that (2. 18)
The sequence is constructed inductively. Let D� be an open set such that D c D� and A� = D� \ D satisfies C2/q,q, (AD < 1 /4. Let Al be an open set such that A� C Al and C2/ q,q, (A 1 ) :s; 1/2. Assume that we constructed {AU�- 1 and {Adi, - I so that the sets A" are open, (2.18) holds and -
D� = D U A� is open, { D� }�- 1 i!) decreasing, CZ/q , q' (A�) < 2-(k+1), A� C Ak .
(2.19) Let D�
be
an open set such that
D c D;, C D;,_l l C2/q,q, (A�) < 2- ('1 +1) where A� = D� \ D. Then A� C A�_l and consequently A� C An I . Since An I is open, statement (i) -
implies that there exists - q
an
open set An such that q
A� c An c An c An-I, -
-
C2/q , q, (An) <
2-".
This completes the proof of the assertion. Let An and Dn be as in Put Then
Qn
:=
(2.18). By (i) there exists a q-open set Q such that q E c Q c Q C D1 · q
-
Q \ (u� - I (Ak \ Ak+1 ) , Qoo = Q \ (Ur'(A k \ A k+ I ).
Qn is a q-open set and we claim that
PRECISE BOUNDARY a:
353
TRAC),;
b: c:
Since Qn is q-open, a follows from (2. 18) which implies:
Q oo U (U� (Ak \ Ak+1 ) = Qn, C2/q, q' « U� (Ak \ A k+ l ) < 2n - 1 • We verify b, c by induction. Put Ql = Q so that b, c hold for n = 1. If b holds for n = I , . . , j then, q q Qj+l = Qj \ (Aj \ Aj+ 1 ) , E c Qj c Dj , .
which implies b for n = j + 1 . If c holds for n = 1 , .
. .
q Qj+l C Qj \ (Aj \ Aj+l) c D u (ui aqA, ) u Aj+ 1 ) .
so that c holds for n j + 1 . Taking the limit in b as n ==
Taking the limit in c as n
-;
-;
00
, j then, =
Dj+l U (U{. +1 OqAi ) ,
we obtain
q
q E c Qoo c D.
00 we obtain
- q Qoo c D U ( uj"'oqAi)'
However, by the same token,
- q Qoo c D U (Uk'Oq Ai) "tk E N.
Therefore, by (2.18), Q=
t. D.
o
Thus (ii) holds with 0 = Qoo.
LEMMA 2.5. Let E be a q-closed set and let D be a cover of E con.�isting of q open sets. Then, for every ( > 0 there exists an open set 0. such that CZ/q,q' ( OE) < € and E \ 0. is covered by a finite subfamily of D. By [1, Sec. 6.5. 1 1) the (a, p)-fine topology possesses the quasi Lindelof property. Thus there exists a denumerable subfamily of D, say {Dn}, such that PROOF.
0 = U {D : D E D} :!. UDn . Let On be an open set containing Dn such that C2/q,q' (On \ Dn) < (/(2n 3). Let K be a compact subset of E n (Uj'" Dn ) such that C2/q ,q, (E \ K) < (/a. Then {On} is an open cover of K so that there exists a finite sub cover of K, say { 01 ' . . , Od. ,
It follows that
C2jq,q , (E \ U�=I Dn ) < C2jq,q, (E \ K) + L CZjq,q' (On \ Dn) < 2(/ 3. Let 0. be an open subset of an such that E \ U�=lDn c O. and C2/q,q'(0.) < €. This set has the properties stated in the lemma. 0
LEMMA 2.6. (a) Let E be a q-quasi closed set and {Em} a proper q-stratification for E. Then there exists a decreasing sequence of open sets {Qj } such that UEm := E' C Qj for every j E N and
(2 . 20)
q (i) n Qj = E', Qj +l C Qj , (ii) lim C2jq ,q, (Qj) = C2jq,q, (E) .
354
MOSHE MARCUS AND LAURENT VERON
(b) If A is a q-open set, there exists a decreasing sequence of open sets { Am } such that A c nAm =: A' , (2.21) A ,t A' . Furthermore there exists an increasing sequence of closed sets {Fj } such that Fj C A' and (2.22)
(i)
U FJ = A' , -
•
PROOF. (a) Let {em} be a sequence of positive numbers decreasing to zero satisfying (2 .6). Put Then E' = nQj and (2.23 )
q 00 Em <= /2j Q-j C Q'j : = Um=1
C
Qj-l.
Indeed Qj is quasi closed so that OJ ,t Qj . This proves (2. 20) (i ) . If D is a neighborhood of E' then, for every k there exists jk such that U�= 1 (Emy� /21 c D Vj > jk' Therefore, /2 C2/q,q· (Qj \ D ) s C2/q,q. ( Qj \ U�= 1 (Emr� j ) S 2 - k +1C2/q,q. (E) Vj > jk. Hence
C2/q,q' (Qj \ D) --> 0 as j --+ 00 . (2 .24) Let { D;} be a decreasing sequence of open neighborhoods of E' such that C2/q,q' (D i) --+ C2/q,q' ( E'). By (2.24) , for every i there exists j(i) > i such that
(2.25)
C2/q,q. (Qj( i) \ Di) --> 0 as i -+ 00 .
lt follows that
C2/q, q' (E') < lim C2/q , q' ( Qj (i) < lim C2/q,q' (Di) = C2/q,q' (E' ) = C2/q,q' (E ) . This proves (2.20) (ii). (b) Put E = an \ A and let { Em } and {I'm} be as in (a). Then (2.21) holds with Am := an \ Em . In addition, (2.22)(i) with Fj := an \ Qj is a consequence of (2.20 ) (i ) . To verify (2.22) (ii) we observe that, if K is a compact subset of A' then, by (2.24), C2/q,q' (K \ Fj) --> O. Let, {Ki} be an increasing sequence of compact subsets of A' such that
C2/q,q· (Ki) i C2 /q q· (A ) = C2/q,q· (A ) . As in part (a), for every i there exists j(i) > i such that ,
(2.26) It follows that
C2/q,q' ( Ki \ Fj (i)
'
-+
0
as
i --+
00 .
C2/q,q' (A ' ) > lim C2/q,q' ( Fj(i) � lim C2/q,q' ( Ki ) = C2/q,q' ( A') = C2/q,q' ( A ) .
PRECISE BOUNDARY TRACE
355
o
This proves (2.22) (ii).
LEMMA 2.7. Let Q be a q-open set. Then, for every e E Q, there exists a q-open set Q� such that e E Q� c Qe c Q.
PROOF. By definition, every point in Q is a q-thin point of Eo = ao \ Q. Assume that diam Q < 1 and put: Tn = 2 -n ,
Kn = {O' : Tn+1 < 10' - el < Tn },
En := Eo n Kn n B1 (e ) · Thus En is a q-closed set; we denote E = U� oEn. Since e is a q-thin point of E , DC � (T - N +1+2/ ( q- l ) C / (B n E)) q - I < 00 2 q,q' n , � n o which is equivalent to
q l � (T -N+1+2/ Cq- I ) C 2/q,q' (En ) ) - < 00 L....o Let {Em , n}�=1 be a q-proper stratification of En. Let en := {Em,n }�=l be a q-proper sequence (relative to the above stratification) such that (I , n E (0, Tn+2) and C2/q,q' (Vn ) < 2C2/ q ,q' (UEn) where Vn := U� I E:"rr:;;, n Bl (0· Then Vn C Kn- 2 \ Kn +2 ' t; is a q-thin point of the set G = UO"Vn and e rt G. Consequently e rt G. Put 2 / rr Zn : = U�= l E:" :n,, n BI/2 (0 , Fo := Ug"Zw 00
•
11
Since Zn C Vn it follows that t; is a q-thin point of Fo and e rt Fo . Consequently Qo := (Q n B1 /2 (e) ) \ Fo is a q-open subset of Q such that -
-
e E Qo ,
Q o c (Q n Bl/2 (�) ) \ Fo C (Q n B1 / 2 (0 ) \ E c Q .
o
3. Maximal solutions We consider positive solutions of the equation ( 1 . 1 ) with q > qc, in a bounded domain n c ]R N of class C2 A function u E Lioc(n) is a sllbsollltion (resp. supersolution) of the equaLion if -Au + lul q- I u < 0 (resp. > 0) in the distribution sense. If u E Lioc(n) is a !iubsolution of the equation then (by Kato's inequality [8]) A l u l > lul q · Thus l u i is subharmonic and consequently u E L�c(n). If u E Lioc (O) is a solution then u E C2 (0) . An increasing sequence of bounded domains of class C2 , {On}, such that On T 0 and On C 0,,+ ] is called an exhaustive sequence relative to O. PROPOSITION 3.1. Let u be a non-negative function in L�c(n) . (i) If u is a subsolution of ( 1 .1), there exists a minimal solution v dominating u, i. e., u < v < U fOT any solution U ? u. (ii) If u is a supersolution of (1.1), there exists a maximal solution w dominated by u, i. e., V < w < u for any solution V < u. All the inequalities above are a. e . .
MOSHE MA
356
Reus AND LAURENT VERON
PROOF. Let u. = J.u where J, is a smoothing operator and u is extended by zero outside O. Put fi = lim€-+o u. (the limit exists a.e. in 0 and fi u a.e.) . Let ii in L 1 ( 0 ) it follows that (:In, 0iJ' �iJ etc. be as in Notation 1 . 1 . Since u. =
u. I E(j
--+
'ij, 1
--+
in L1 (2:B)
E(j
for a.e.{3 E (0, (30) . Choose a sequence {{3,, } decrea.sing to zero such that the above convergence holds for each surface 2:n := 2:iJ,, ' Put D" : = O�n ' Assuming that u is a subsolution of ( 1 . 1) ill 0, U. is a subsolution of the boundary value problem for (1.1) in Dn with boundary data u. En ' Consequently ii is a subsolution of the boundary value problem for ( 1 . 1 ) in D" with boundary data fil E n E L 1 (2:n). (Here ii q in L}oc ( O) . ) we use the assumption 'U E L�c(O) in order to ensure that u� Let 'Un denote the solution of this boundary value problem in thc L1 senHe: --+
vn = iL on 2:"..
Then Vn E C2(Dn) n L ""' (Dn ) , Vn < lI u IlL�(Dn) and the boundary data is assumed in the L 1 sense. Clearly iL < Vn in Dn, n= 1 ,2 . . . . In particular, 'Un < Vn+l on �n . This implies Vn :s Vn+l in Dn. In addition, by the Keller-Osserman inequality the scquem:e {vn } is eventually bounded in every compact subset of O. Therefore v = lim Vn is the solution with the properties stated in (i). Next assume that u is a snpersolution and let {Dn } be as above. Since u E Lq (Dn) there exists a positive solution Wn of the boundary value problem q u Aw ' D U In ,
-
-
10
Hence U + Wn is superha.rmonic and its boundary trace is precisely ii i . ConseEn quently U + Wn > Zn where Zn is the harmonic function in Dn with boundary data ill . Thus Un := ZM. Wn is the smallest solution of ( 1 . 1 ) in Dn dominating u. This implies that {un } dccrease;; and the limiting solution U is the smallest solution of 0 (1.1) dominating U in O. En
-
PROPOSITION a.2. Let u, v be non-negative, locally bounded functions in O . (i) If tt , V are subso/utions (resp. superso/utions) then max(u, v) is a subsolution (re$p. mine u, v) is a supersolution) . (ii) If u, v are supersolutions then u + v is a superso/ution. (iii) If u is a subsolution and v a superso/ution then (u v)+ is a subsolution. -
PROOF. The first two statements are well known; they can be verified by an application of Kato's inequality. The third statement is verified in a similar way: q 6.(u - v) + = sign + (u - v)6.(u v) > (u - vq)+ > (u - v)� . o
-
Notation 3. 1 . Let u, v be non-negative, locally bounded fUllctions in O. (a) If u is a subsolution, [ul t denotes the smallest solution dominating u. (b) If 'U is a supersolution, [uP denotes the largest solution dominated by 7L. (c) If u, v are subsolutions then u V v : = [m ax(u v)Jt. (d) If u, v are sllpersolutions then u /\ v : = [inf( u, v)] l and u EB v := [u + v) t . (e) If 'IL is a subsolution and v a supersolution then u e v := [(u v) + I t . ,
-
357
PRECISE BOUNDARY TRACE
The following result was proved in [9] (see also [3, Sec. 8.5] ) . PROPOSITION 3.3. (i) Let {ud be a sequence of positive, continuous subsolu tions of (1.1) . Then U := sup Uk is a subsolution. The statement remains valid if subsolution is replaced by supersolution and sup by inf. (i'i) Let T be a family of positive solutions of ( 1 . 1 ) . Suppose that, for every pair UI , U2 E T, ther'e exists v E T such that
max(uI ' U2) < v, resp. min( u I ' U2 ) > v.
Then there e:r:ists a monotone sequence {un} in T such that Un j sup T, resp. Un 1 sup T. Thus sup T (resp. inf T) is a solution. DEFINITION :3.4. A solution u of (1.1) vanishes on a relatively open set Q c 812 if u E C(n U Q) and u 0 on Q. A positive solution u vanishes on a q-open set A c 80 if u = sup{v E U(n) : v < u, v = 0 on some relatively open neighborhood of A } . When this is the case we write u r::; O. =
A
3.5. Lp.t A be a q-open subset of 812 and u I , 1[2 E U(n). (a) If both solutions vanish on A then Ut V U2 r::; O. If U2 r::; 0 and 'Ul ::; U2 then LEMMA
UI
r::;
A
A
O.
A
(b) If u E U (n) and u r::; 0 then there exists an increasing sequence of solutions A {Un} C U(n ) , each a/ which vanishes on a relatively open neighborhood of A (which may depend on n) such that Un j u . (c) If A, A' are q-open sets, A ,!!., A' and u r::; 0 then u r::; O. A
A'
PllOOF. The first assertion follows easily from the definition. Thus the set of solutions {11} described in the definition is closed with respect to the binary operator V . Therefore, by Proposition 3.3, the supremum of this set is the limit of an increasing sequence of elements of thiR set. 0 The last statement is obvious. DEFINITION 3.6. (a) Let u E U (n) and let A denote the union of all q-open sets on which 'U vanishes. Then 812 \ A is called the fine boundary support of u, to be denoted by supp �!1u. (b) For any Borel set E we denote UE = sup{u E U (n) : n � 0, eo = 812 \ E}, Thus UE
=
Uii;'
Ee
3.7. (i) Let A be a q-open subset 0/ an and {un} C U (12) a sequence of solutions vanishing on A. If {un } converges- then U = lim Un vanishes on A . In particular, if E is Borel, UE vanishes outside E. (ii) Let E be a Borel set such that CNq,q' (E) = 0, If U E U(I.1) and U vanishes on every q-open subset of EC = 812 \ E then U = O. In particular, UE = O . (iii) If { An } is a sequence of Borel subsets of 812 such that C2/q,q, (An) 0 then UAn O. LEMM A
->
->
MOSHE MARCUS AND LAURENT VERON
358
(i) Using Lemma 3.5 we find that, in proving the first assertion, we may assume that {un} is increasing. Now we can produce an increasing sequence of solutions {wn} such that, for each n, Wn vanishes on some (open) neighborhood of A and lim Wn = lim Un' By definition lim Wn vanishes on A. Let E be a q-c1osed set. By Lemma 3.5(a) and Proposition 3.3, there exists an increasing sequence of solutions {un} vanishing outside E such that UE = lim Un Therefore UE vanishes outside E. O. The (ii) Let An be open sets such that E C An , An 1 and C2/ q , q' (An) sets An have the same properties and, by assumption, u vanishes in (An)C : = 80. \ An . Therefore, for each n , there exists a solution Wn which vanishes on an open neighborhood Bn of (An)C such that Wn < '11. and Wn --+ u. Hence Wn < UKn 0, the capacitary where Kn = B� is compact and Kn C An· Since C2/q,q' (Kn) estimates of [16J imply that lim UKn = 0 and hence u = O. (iii) By definition UA n = U,4n ' Therefore, in view of Proposition 2.2(iv), it is enough to prove the assertion when each set An is q-c1osed. As before, for each n, there exists a solution Wn which vanishes on an open neighborhood Bn of (An Y such that Wn < UAn and UAn Wn O. Thus w'" < UKn where Kn = B�, is 0 it follows that UKn --> 0, which compact and Kn C An . Since C2/q,q, (Kn ) implies the assertion. 0 PROO F .
-
--t
-
-
-->
-
-->
-
-->
3.8. Let E, F be Borel subsets of an. (i) If E, F are q-closed then UE A UF = UEr, F ' (ii) If E, F are q-closed then LEMMA
(3.1 )
UE < UF UE = UF
-{=;=}} .:
q
[ E C F and C2/ q, q, (F \ E) > O J,
;. E :!.. F.
(iii) If {Fn} is a decreasing sequ.ence of q- closed sets then lim UFn = UF where F = nFn .
(3.2)
(iv) Let A C an be a q-open set and let U E U(0.) . Suppose that u vanishes q-locally in A, i. e., for every point (J E A there exists a q-open set ACT such that (J E Au
c
A,
Then u vanishes on A . In parlicu/ar each solution u E U(0.) vanishes on an \
SUPP�!1 '11. . (i) UE A UF is the largest solution under inf(UE ' Up) and therefore, by Definition 3.6, it is the largest solution which vanishes outside E n F. (ii) Obviously PROOF.
(3 .3)
E :!.. F
;· UE = Up ,
In addition, (3 .4) Indeed, if K is a compact subset of F \ E of positive capacity, then UK > 0 and UK < UF but UK i UE · Therefore Up = UE implies F :!.. E. (iii) If V : = lim UPn then UF < V. If Up < V then C2/q ,q' (suPPlm V \ F) > O. But
359
PRECISE BOUNDARY TRACE
sUPPbn V C Fn so that sUPPbn V c F and consequently V < Up .
(iv) First asl)ume that A is a countable union of q-open sets {An } such that u � 0 An for each n. Then u vanishes on U� Ai for each i. Therefore we may assume that the sequence { An} is increasing. Put Fn = Of! \ An . Then u < Up" and, by (iii) , UFn 1 UF where F = an \ A. Thus u ::; UF , i.e., which is equivalent to u :;t O. We turn to the general case. It is known that the (0, p)-fine topology possesses the quasi-Lindelof property (1)00 [1, Sec. 6.5.1 1) ) . Therefore A is covered, up to a set of capacity zero, by a countable subcover of { AQ' : a E A}. Therefore the previous argument implies that u � 0 0 A
THEOREM 3.9. (a) Let E be a q-closed set. Then, UE
(3.5)
=
inf{ UD : E c D c an, D open}
= sup{UK : K c E, K compact} .
(b) If E, F are two Borel subsets of an then (3.6)
(c) Let E, Fn, n = 1 , 2, . . be Borel subsets of Of! and let u be a positive solution of (1.1). If either C2/ q,q' (Et::.Fn ) --+ 0 or Fn 1 E then .
(3.7)
UPn
--+
UE .
PROOF. (a) Let {Qj} be a I>equence of open sets, decreasing to a set E' ,t E, which satisfies (2.20) . Then Qj ! E' and, by Lemma 3.8 (iii) UOj 1 UE . This implies the first equality in (3.5) . The second equality follows directly from Definition 3.4 (see also Lemma 3.5). -
(b) Let D, D' be open sets such that E n F e D and E \ F e D' and let K be a compact subset of E. Then �
�
-
(3.8)
To verify this inequality, let v be a positive solution such that sUPPbn v C K and let {fJn} be a sequence decreasing to zero such that the following limits exist: D . W = 1I· m vfjD , W, = hm vfj ' . n_oo n
'I,.-CO
n
(See Notation 1.1 for the definition of vf {) Then Since, by [16) UK
v ::;
=
W+
w' ::; UD + UD' .
VK , this inequality implies (3.8). Further (3.8) and (3.5) imply UE <
UFnB + UE\ F .
-
On the other hand, both UFn E and UE\F vanish outside E. Consequently UPnE EB UE\ F vanishes outside E so that -
UE > Upn E EEl UE\F .
This implies (3.6). (c) The previous statement implies, UB <
UF"nE + UB\Fn '
360
MOSHE MARCUS AND LAURENT VERON
If C2/q,q, (E6.Fn) � 0 Lemma 3.7 implies lim UEt.Fn (3.7). If Fn ! E then, by Lemma 3.8, UFn � UE . ,
Notation 3.2. (3 .9 )
=
0 which in turn implies
0
For any Borel set E c an of positive C2/q,q,-capacity put
W+ 2/Q,Q (an) , /-I(an \ E) = O } ,
Vmod ( E)
=
{ul' ; /-I E
VE
=
sup Vmod(E)
THEOREM 3.10. If E is a q-closed set, then UE = VE. (3.10) Thus the maximal solution UE is u-moderate. Furthermore UE satisfies the capac itary estimates established in [16] for compact sets, namely: There exist positive constants cl , C2 depending only on q, N and n such that, for every x E n, 00
m�-=
(3. 11 )
00
m �-CXl where
p(x)
=
dist (x, an) ,
rTn ;=
T m , Bm (x)
Note that, for each point x, Bm (x)
=
=
{ y E an : rm+1 < Ix
0 when
-
y l < rm} .
sup I x - YI < rm +1 < rm < p(x).
y E P;
Therefore the sum is finite for each x E n. Remark. Actually the estimates hold for any Borel set E. Indeed, by definition, UE = UE and C2/q,q, « E n Bm (x) /rm)) C2/q,q, « E n Bm (x» /Tm) . ,...,
Let {Ed be a q-stratification of E. If u E Vmod (E) and J.L = tr u then ul' = sup U/Lk where /-Ik = /-IXEk Hence VE = sup VEk • By [16]' UEk = VEk • These facts and Theorem 3.9(c) imply (3.10). It is known that UEk satisfies the capacitary estimates (3.11). In addition, P ROOF.
C2/q , q , « Ek n Bm (x » /rm» -> C2/q,q, « E n Bm (x » /rm» . Therefore UE satisfies the capacitary estimates.
4. Localization
o
DEFINITION 4.1. Let /-I be a positive bounded Borel measure on an which vanishes on sets of C2 /q,q,-capacity zero. (a) The q-support of /-I (denoted q-supp /-I) is the intersection of all q-closed sets F such that /-I(an \ F) = O. (b) We say that J.L is concentrated on a Borel set E if /-lean \ E) = O. LEMMA 4.2. If /-I is a measure as in the previous definition then,
(4. 1)
PRECISE BOUNDARY TRACR
301
PROOF. Put F = SUpp�O Uw By Lemma 3.8(iv) uJ.L vanishes on an \ F and by Lemma 3.5 there exists an increasing sequence of positive solutions {un } such t.hat each function Un vanishes outside a compact subset of F, say Fn , and u" 1 Uw If S" ;= supp�o un then Sn C Fn and {Sn} increases. Thus {Sn} is an increasing sequence of compact subsets of F and, setting /-tn = /-tXsn' we find Un < uJ.Ln < 'u!-' so that u!-'n r Uw This, in turn, implies (see [ 14] ) /-tn r /-t, q-supp /-t C Un Sn C F. If D is a relatively open set and /-teD) = 0 it is clear that. uJ.L vanishet; un D. Therefore u!-'n vanishes outside S", thus outside q-supp /-t. Consequently up, vanishes 0 outside q-supp /-t, i.e. F � q-supp /-t. q --=
DEFINITION 4.3. Let u be a positive solution and A a Borel subset of an. Put
(4.2)
[U]A ;= Sup{v E U(n) ; v < u,
q
-
supp�o v c A}
and,
[U] A ; = sUp{ [U] F ; F � A, F q-closed}. Thus [U]A = U /\ U , i.e., [ul A is the largest solution under inf(u, UA) . A Recall that, if A is q-open and u E C(an), u� denot.es the solution of ( 1 . 1 ) in n� which equals UX".O (A) on �,o. If lim,o-->o u� exists the limit will be denot.erl by uA. (4.3)
THEOREM 4.4. Let u E U ( n) . (i) If E C an is q-closed then,
rul E = inf { [ulD : E c D c an, D open} . (ii) If E, F are two Borel subsets of an then (4.5) rulE < [uI Fn r; + [uI E\F
(4.4)
and
(4.6)
(iii) Let E, Fn, n = 1 , 2, . . . be Borel subsets of an. If either C2/q,q' (Etl.Fn) or Fn 1 E then
--;
0
(4.7)
PROOF. (i) Let V = {D} be the family of sets in (4.4). By (a.5) (with respect to the family V ) (4.8) Obviously
inf(u, UE )
=
inf ( u, inf UD) DED
=
inf inf(n, UD) > inf [uI D .
D E 'D
D E 'V
[ub, /\ [U] D2 > [U] D, nD2 ' (In fact we have equality but that is not needed here.) Therefore, by Proposition 3.a, the function v ;= inf DE v[u]D is a solution of ( 1.1 ) . Hence (4.8) i mplies rUl E 2: v. The opposite inequality is obvious. (ii) If E is compact (4.5) it; proved in the same way as Theorem 3.9(b) . In general, if {En} is a q stratification of E, [U] En < [U]FnEn + [uI E" \F < [U] FnE + [uI E\F'
362
MOSHE MARCUS AND LAURENT VERON
This inequality and Theorem 3.9(c) imply (4.5). Put A = E and B = F. It follows directly from the definition that, -
-
[[U]A]B < inf(u, UA, UB ) The largest solution dominated by u and vani�hing on AC U Be) is [U]An B . [ [U] A]B < [U]AnB' '
On the other hand
Thus
[U]AnB = [[U]AnB]B < [[ulAJ s .
(iii) By (4.8) rUl E < [ulpnnE + [U] E\Fn '
This proves (4.6).
If C2/q,q, (E!J.Fn) (4.7) .
If Fn 1
�
[U]Pn
<
[U]FnnE + [U]Pn \E·
0 Lemma 3.7 implies lim[u] E�F" ,
=
0 which in turn implies
3.8, UF" UE. If U is a positive solution then inf('u, UE ) = inf(u, inf UF,.) = inf inf(u, UFJ > inf[u]Fn • n n n Since {Fn } decreases w = infn [u] pn is a solution. Hence rUlE > The oppotlite inequality is obvious; hcnee [ltl E = lim [u]Fn ' D LEMMA 4.5. Let u be a positive solution of (1.1) and put E = sUPPbn u, (i) If D is a q-open set such that E C. D then (4.9) [uj D = lim ug = [U]D = U, (3->0 E then, by Lemma
.....
w.
(ii) If A is a q-open subset of an,
-
(4.10)
VQ q-open ; Q
(iii) Finally,
q
C A.
[u] A = 0
(4.11)
U Rj 0 :
PROOF. Case 1 : E is closed. u E C(O U A) and u = 0 on A. E then
Since u vanishes in A ;= an \ E, it follows that If, in addition, D e an is an open neighborhood of
A
).
udS � 0 { J�jl ( D')
so that
(4.12) Since
it follows that (4.1 3 )
U=
1I·
m ufJ
1)
'
If we assume only that D is q-open and E C. D then, for every E > 0, there exists
an open set 0< such that D It follows that
C
0<, E
0,
ufJ
C
0< and
D < U�,, (O;) - ufJ _
E
C2 /q,q' (O�) < where O� .
ill
",I
"fJ
=
0< \ D.
363
PRECISE BOUNDARY TRACE
and lim,�o UE8 (O;) = 0 uniformly with respect to (3. Since limi3�o u�' = u it follows that (4.13) holds. The same argument shows that (4.12) remains valid. Now (4. 13) implies (4.9) . Indeed
u = lim ug < [ulD < u. Hence u = lUll). If Q is a q-open set such that E !!:. Q c Q !!:. D then U = [u]Q < [u] D. Hence u = [u]D. In addition (4. 12) implies (4. 10) in the direction > . Assertion (4.10) in the
opposite direction is a consequence of Lemma 2.7 and Lemma 3.8 (iv). Case 2. We consider the general case when E is q-closed. Let {En} be a stratifi cation of E so that C2/q,q' (E \ En) -+ O. If D i� q-open and E !!:. D then, by the first parL of the pToof, (4.14) By (4.5) (4.1 5 ) Let {Ih} be a sequence decre�ing to zero such that the following limits exist 'UJ
'UJn := l� ([U]E\EJK ,
: = l� ugk ,
Then, by (4. 14) and (4.1 5),
[ulE" Further, by (3.7) ,
<w<
n = 1 , 2, · . . .
[ulEn + w'" < [U] En + UE\ En '
UE\ En -+ O. Hence 'UJ = u. This implie� (4.13) which in turn implies (4.9). To verify (4.10) in the direction ;. we apply (4. 15) with D replaced by Q. We obtain, U� < ([ul En )� + ( [U] E\En ) � By the first part of the proof
[ulEn
-+
[U] E = u,
'
lim ( [ulEJ� = o.
{3-+0
Let {{3d be a sequence decreasing to zero such that the following limits exist:
lim ( [ ] n)� ' U E\ E k ,.., k �oo
Then lim
k-oo
ufj
�k
n =
1 , 2, . . . ,
u ([ lE\ EJQ[J k-oo
< lim
. u� k -oo fJ
lim
k
< UE\En '
Since UE\ En -+ 0 we obtain (4.10) in the direction > . The assertion in the opposite direction is proved as in Ca�e l . This completes the proof of (i) and (ii). Finally we prove (iii). First assume that U � O. If F ii:i a q-closed set such that A q - q q F c A then there exists a q-opcn set Q such that F c Q c A. Therefore, applying (4.9) to v : = [ulF and using (4. 10) we obtaiu k
= lim v� < limu� = O. In view of Definition 4.3 this implies that [U]A = O. v
3fl4
MOSHE MARCUS AND LAURENT VERON -
q
Secondly assume that [u] = O. Then [u]Q = 0 whenever Q C A. If Q is a - q q-open set such that Q c A then [u]Q = 0 and hence u � O. Applying once again Q Lerruna 2 . 7 and Lemma 3.8 (iv) we conclude that U � O. 0 A
A
DEFINITION 4.6. Let u, v be positive solutions of ( 1 . 1 ) in n and let A he a q-open subset of an. We say that u = v on A if u e v and v e u vanish on A (see Notation 3.1). This relation is denoted by u � V . A
THEOREM 4.7. Let u, v E U(n) and let A be a q-open subset of an . Then, u � v ·: ;. lim lu - vl � = 0,
(4.16)
A
-
{3--+0
q
for every q-open set Q s1J.ch that Q C A and
u ::f v
(4. 1 7)
·: :. [u]p = [vIp, q
for every q-closed set F such that F C A .
PROOF. By definition, U � v is equivalent to u e v � 0 and v e u � O. Hence , A
A
by Lemma 1.5 (specifically (4. 11» ,
[u 8 v]p
(4. 18)
for every q-closed set F
� A.
=
0,
A
[v e uJp = 0,
� A, Lemma 4.5 implies that ( ( v - n)+ ) � -> O.
Therefore, if Q
( (u - v)+) �
->
0,
(Recall that u6v is the smallest solution which dominates the subsolution (u - v>+ -l This implies (4.16) in the direction > ; the oppm;it.e direction is a consequence of Lemma 3.7. \Ve turn to the proof of (4. 1 7) . For any two positive solutions u, v we have (4.19)
u + (v - u) + < v + (u - v)+ < v + n e v.
If F is a q-ciosed set and Q a q-open set such that F ( 4.20)
[u]./>' < [vJQ + [u e v]Q .
� Q then,
To verify this inequality we observe that, by (4.19),
[U] F ::; [v] Q + [v]Q ' + [u e v]Q + [u e v] Qc.
The subsolution w := ( [U] F - ( [v]Q + [u e v] Q » + is dominated by the supersolution [v]Qc + [u e vlQc which vanishes on Q . Therefore w vanishes on Q. Since the boundary support of [wi t is contained in F it follows that [wi t 0 so that w O. q - q If U ::f v and F C Q c Q c A then (4.20) and (4.18) imply, [U] F < [v]Q .
Choosing a decreasing sequence of q-open sets {Qn} such that nQn :!., F we obtain [U]F < lim[v]Qn = [V]F ' Similarly, [V] F < [U]F and hence equality. q Next assume that [VIF = [U] F for every q-closed set F C A. If Q is a q-open - q set such that Q c A we have, u 6 v < ( [ulo $ [u] oc ) e [vlQ < [ulQc,
PRECISE BOUNDARY TRACE
365
because [u] Q [v]Q . This implies that u e v vanishes on Q. Since this holds for every Q all above it follows that u e v vanishes on A. Similarly v e u vanishes on D A. =
COROLLARY 4.8. If A is a q-open subset of an, the relation :::; is an equivalence A
relation in U (D.) .
PROOF. This is an immediate consequence of (4. 16).
D
5. The precise boundary trace 5.1. The regular boundary set. We define the regular boundary set of a positive solution of (1.1) and present some conditions for the regularity of a q-open
set.
DEFINITION 5 . 1 . Let u be a positive solution of (1.1). a: Let D e aD. be a q-open set such that C2/q , q, (D) > O . D is pre-regular with respect to u if
(5. 1)
10 [u] '}pdx < 00
VF
t D,
F q-closed.
b: An arbitrary Borel set E is n;gular' if there exists a pre-regular set D such q that E C D. c: A set D e aD. is CJ-regular if it is the union of a countable family of pre-regular sets. d: The union of all q-open regular sets is called the regular' boundary set of 'U , and is denoted by R(u) . The set 5(u) = aD. \ R(u) is called the singular boundary set of u . A point P E R(u) is called a regular boundary point of 'll. ; a point P E 5 (u) is called a singular boundary point of u.
Remark. The property of regularity of a set is preserved under the equivalence relation :t. However note that a point is regular if and only if it has a q-open regular neighborhood. LEMMA 5.2. If D is a q-open pre-regular set then every pO'int � point. Furthermore there exists a q-open regular set Q such that
(5.2)
-
E
D is a TI�gular
e E Q c Q e D.
If F is a regular q-closed set then there exists a regular q-open set Q such that q F e Q, PROOF, By Lemma 2,7, for every e E D , there exists a q-open set Q such that (5.2) holds, Therefore Q is a regular set and e is a regular point . The last assertion is a consequence of Lemma 2.4. D DEFINITION 5.3. Let u be a positive solution of (1.1) and let {Qn} be an increasing sequence of regular q-open sets. If On t. Qn+ l we say that {Qn} is a regular sequence relative to u. If Q is a q-opcn set, {Qn } is a regular sequence relative to u, Qn C Q and Qu := u� 1 Qn � Q we refer to Qo as a proper representation of Q and to {Qn } as a r'egular decomposition of Q, relative to u .
366
MOSHE MARCUS AND LAURENT VERON
LEMMA 5.4. Let u E U(n) . A q-open set Q C an is a-regular if and only if it has a proper representation relative to u. In particular every pre-regular set has a proper representation. PROOF. The 'if' direction follows immediately from the definition. Now sup pose that Q is a-regular. Then Q = Ul" En where En is q-open and pre-regular, n = 1 , 2, ' " . By Lemma 2.6, each set En can be represented (up to a set of capacity q q zero) as a cOlmtahle union of q-open sets {An,j}j I such that An,j e An,HI C En . We may assume that An,j e En; otherwise we replace it by An,j n En . Put -
If k + j = n then Ak,j
q
C
Qn = Uk +j=n Ak,j '
Ak,j+l C Qn+ l . Hence q - q Qn e Qn+1, Qo : = UQn � Q.
THEOREM 5.5. Let D be a q-open set such that C2/q ,q ' (D) (i) Suppose that (5.3)
liminf f3 O , .....
I (ufl)q(p " Ii �,
.
-
o
> 0,
f3) dx < 00.
Then D is pre-regular, (ii) Suppose that D is a pre-regular set. Then there exists a Borel measure J1 on D q such that, for every q-closed set E e D, (5.4) PROOF. (i) Let {f3n} be a sequence decreasing to zero such that (5.5) By extracting a subsequence if necessary we may assume that {ufn } converges locally uniformly in n to a solution w . Then, by Lemma 4.5, if E is q-closed and q E e D, (5.6) By (5.5) and Fatou's lemma,
Hence, by (5.6), (5.7) Thus D is pre-regular. (ii) By Lemma 5.4, D possesses a regular decomposition {Dj}. Put Wj = [UJ D, .
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PRECISE BOUNDARY TRACE
Then {Wj} is increasing and its limit is a solution Wo < w with w as defined in (5.6). Thus Wo is a moderate solution. If E � D is a q-closed set then, by (4.6) ,
By Lemma 2.5, for every k E N there exists an open set Ok and a natural number jk such that C2jq ,q, (Ok) < 11k and E \ Ok � Djk • By Theorem 3.9 Since [ulo.
-+
0 we conclude that
q [wol E = rul E VE e D : E q-closed.
(5.8)
If Wo is moderate then tr [wolE = {lXEtr wo, which implies (5.4) . We turn to the case where Wo is not moderate. The solution and we denote {lj = tr wj
w)
is moderate
{I = lim J.Lj .
,
By (4.6), Wj = [wj+klDj ' Therefore J.Lj = J.Lj+ kX oj = J.LXD{ Therefore if E is q closed and E � Dj for some j, (5.4) holds with /-t as defined above. If E is q-closed
q and E C D then E � E' := U(E n Dj ) = U(E n Dj). Put Ej := E n Dj . It follows q
-
that
tr rul E; =
J.LXEj
i /-tX E"
Since D is pre-regular, rulE is moderate. Put Ej = E \ Dj and observe that nj'" Ej is a set of capacity zero so that (by Lemma 3.8) UE� ! 0 and hence lim[ul Ej ! O. Since
rul E < rul E, + rul E' and rul E' 1 0 1
we conclude that
1
On the other hand tr rul E > tr rul E;
-+
WX E'
= /-tXE'
This implies (5.4) .
0
COROLLARY 5.6. Let D be a q-open set such that C2/q , q' (D) > O. Suppose that, q for every q-open set Q such that Q e D, -
(5.9)
Then D is pre-regular. PROOF. This is an immediate consequence of Lemma 2.4 and Theorem 5.5.
0
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MOSHE MARCUS AND LAURENT VI
5.2. Behavior of the solution at the boundary. In this subsection we provide a characterization of regular and singular boundary points of a positive solution by the limiting behavior of the solution as it approaches the boundary in a q-open neighborhood of each point. THEOREM 5.7. Let u be a positive solution of (1.1) in n and let Q be a q-open subset of 8n of positive capacity. (i) If � E S tu) then, for every nontrivial q-upen neighborhood Q of �,
(5. 1 0)
00 ,
udS �
as (3
->
O.
Eo (Q) (ii) If � E R(u), there exists a q-open regular set D s'uch that � E D , Further there exists a q-open set Q such that � E Q c Q e D. Consequently
(5. 1 1 )
sup udS < 00, 0
UQ
= l'1m
fJ� O
u"Q "
. t
P-X'tS 05
and uQ is moderate. If J..tQ : = tr [uJ Q then, j01' every q-closed set E C q
Q,
tr [uJ E = J..tQXE•
(5.12)
PROOF. (i) If Q is a q-open set for which (5.10) does not hold, there exists a sequence {{3n } converging to zero such that
r
IE{Jn (Q)
udS � a < 00.
This implies that there exists a constant C such that (5.5) holds (with D replaced by Q). By Theorem 5.5, Q is pre-regular and by Lemma 5.2 every point in Q is a regular point. Therefore, if � E S ( u) (5.10) holds. (ii) By Definition 5 . 1 there exists a q-opell regular neighborhood D of �. By Lemma 2.7 there exists a q-open set Q such that � E Q c Q e D. By Theorem 4.4 and Lemma 4 ..') , ,
-
JU]Dc ({3, ·)dS = O. (l�O Q
u < [ul D + [U]Dc and lim
Therefore
lim sup u ({3, · )dS < lim sup [u]vC{3, · )dS < 00 /3� 0 JQ /3->0 Q so that (5.3) holds. In view of this fact, Theorem 5.5 and the arguments in its proof imply assertion (ii). 0
r
5.3. q-perfect measures.
DEFINITION 5.8. Let J..t be a positive Borel measure, not necessarily bounded,
on an. (i) We say that J..t is essentially absolutely continuous relative to C2/q,q' if the fol lowing condition holds: If Q is a q-open set and A is a Borel set such that C2/q,q' (A) = a then
J..t ( Q \ A) = j..t(Q).
This relation will be denoted by J..t
j-< C2/q,q"
369
PRECISE BOUNDARY TRAUE
(ii) /-I is regular relative to q-topology if, for every Borel set E C an, /-I(E) = inf{tL C D) : E c D c an , D q-open} (5.13) = sup{Ji(K) : K c E, K compact}.
/-I is outer regular relative to q-topology if the first equality in (5. 13) holds. (iii) A positive Borel measure is called q-perfect if it is essentially absolutely con tinuous relative to C2/q,q' and outer regular relative to q-topology, The space of q-perfect Borel measures is denoted by Mq(on). LEMMA 5.9. If /-I C2/q,q' (A) = 0 then
(5.14)
/-I C A)
=
E
Mq (an) and A c an is a non-empty Borel set such that
00
o
if /-I ( Q \ A) = 00 VQ q-open neighborhood of A, otherwise .
If /-10 is an essentially absolutely continuous positive Borel measure on 812 and Q is a q-open set such that J.Lo(Q) < 00 then /-Io l Q is absolutely continuous with respect to C2/q,q' in the strong sense, i. e., if {An} 'is a sequence of Borel subsets of an, C2/q,q ' (An ) 0 :. /-Io (Q n An) O. Let /-10 be an essentially absolutely continuous positive Borel measure on 812. Put -+
-+
/lo(E) := inf{/-Io(D) : E c D c 812, D q-open},
(5.1 5)
for every Borel set E
C 812. Then
Jio < Ji, /-Io (Q) = Ji(Q) VQ q-open (b) /-tI Q = /-LoI Q for every q-open set Q such that /-Lo(Q) < 00
(a) (5 , 16)
.
Finally /-L is q-perfect; thus Ji is the smallest measure in MIq which dominates Ito ·
PROOF. The first assertion follows immediately from the definition of Mq. We
turn to the second assertion. If tLo is an essentially absolutely continuous positive Borel measure on an and Q is a q-open set such that Jio (Q) < 00 then /l,o 'X.Q is a bounded Borel measure which vanishes on sets of C2/q,q,-capacity zero. If {An} is a sequence of Borel sets such that C2/q,q' (An) -+ 0 and /-tn : = /-Io 'X.QnAn then Hence u!'n -+ 0 locally uniformly and /-In --' 0 weakly with respect to C(8n) . /-to (Q n An) -+ O. Thus /-to is absolutely continuous in the strong sense relative to C2/q, q' . Assertion (5.16) (a) follows from (5.15). It is also clear that /-t, as defined by (5. 15), is a measure. Now if Q is a q-open set such that /-Io(Q) < 00 then /-I(Q) < 00 and both /-to l Q and /-II are regular relative to the induced Euclidean topology on Q an. Since they agree on open sets, the regularity implies (5. 16) (b). If A is a Borel set such that C2/q,q' (A) = 0 and Q is a q-open set then Q \ A is q-open and consequently /-I(Q)
J.Lo(Q) = ,lO (Q \ A)
/-I(Q \ A). Thus /-I is eSHentially absolutely continuous. It is obvious by its definition that /-I is 0 outer regular with respect to C2/q,q" Thus /-I E Mq( an). =
=
MOSHE MARCUS AND LAURENT VERON
370
5.4. The boundary trace on the regular set. First we describe some prop erties of moderate solutions. In this connection it is convenient to introduce a related term: a solution is strictly moderate if
lu i < v ,
(5.17)
v harmonic, n
vqp dx <
00 .
A positive solution u is strictly moderate if and only if tr u E W+ 2/ Q,q (80) (see [15]). Notation 5. 1 . Let II : O!3o ---> 80 be the mapping given by II(x) = a(x) (see Notation 1.1) and put II!3 := fI E".
1 : If cP is a function defined on 80 put cP* := ¢ a II. This function is called the normal lifting of ¢ to 0!3o ' Similarly, if ¢ is defined on a set Q c 80, ¢* is the normal lifting of ¢ to O!3o (Q) . 2: If 'P is a function defined on E!3 we define the normal projection of 'P onto
ao by
'P�(O = 'P (II13 1 (0),
Vf ..
E
ao,
If v is a function defined on O!3o then v� denotes the normal projection of v(/3, ') onto 80, for /3 E (0, /30 )' PROPOSITION 5.10. Let u be a moderate solution of (1.1), not necessarily pos
itive. Then:
(i) u E L1 (0) n Lq (O; p) and u possesses a boundary trace tr n given by a bounded Borel measure /1 which is attained in the sense of weak convergence of measures: (5.18)
r n�¢dS = f3-u .Jan
ulji*dS = lim
lim
lji d/1,
13-0 Ell an for every ¢ E C(80) . (ii) A bounded Borel measure /1 is the boundary trace of a solution of (1.1) if and only if it is absolutely continuous relative to C2/ Q, Q" When this is the case, there exists a sequence {/1n} C W-2/q q (80) such that /1n -> JL in total variation norm. If /1 is positive, the sequence can be chosen to be increasing. Note that these facts imply that /1 is a trace if and only if 1 /1 1 is a trace. Q 2 (iii) u is strictly moderate if and only if Itr ul E W- / ,q(80) . In this case the boundary trace is also attained in the sense of weak convergence in W-2/q,q (80) of {u� : f3 E (0, f3o) } as /3 O. In particular (5. 18) holds for every Iji E W 2/ q q (80) U C(80) . (iv) If JL : = t.r u and {JLn} is as in (ii) then u = lim ul'n ' In particular, if u > 0 then u is the limit of an increaSing sequence of strictly moderate solutions. (v) The measure /1 = tr u is regular relative to the q-topology. (vi) If u is positive (not necessarily strictly moderate), (5. 18) is valid for every ' , / q rp E ( W� Q n Loo ) (80) . ,
--->
,
'
Remark. Assertions (i)-{iv) are well known. For proofs see [15] which also contains further relevant citations. Proof of (v). If JL is a trace then 1-1+ and /1- are t.races of solutions of (1.1). Therefore it is enough to prove (v) in the case that /1 is a positive measure.
37 1
PRECISE BOUNDARY TRACE
Every bounded Borel measure on 00 is regular in the usual sense: p,(E) = inf{p,(0) : E c 0, 0 relatively open} =
sup{p,(K) : K c E, K compact}
for every Borel set E c oO. Since J.L(E) < inf{p,(D) : E c D c 00, D q-open} < inf{J.L(O) : E c O c oO, 0 relatively open} it follows that such a measure is also regular with respect to the q-topology. Proof of (vi). By (ii) there exists an increasing sequence of strictly moderate solutions {vn} such that Vn i u. If fJn : = tr Vn then
r vncP*dS = ��O JE� lim
cP dfJn,
an
for every ¢ E W2/q,q' (00). Since {P,n } increases and converges weakly to p, J.L in total variation. Hence it follows that P,n .......
an
an
¢ dp,n .......
=
tr u
¢ dp,
for every bounded ¢ E W2/q , q' (00) . If, in addition, 1> ;;: 0, we obtain lim inf u¢*dS > ¢ dp,. ��o Ep an On the other hand, since u� fJ in the sense of weak convergence of measures, it follows that .......
(5.19)
lim �mp
u�dS < p"
lim inf
i u�dS
;;:
J.L
E for any closed set E c oO, respectively, open set A c oO) . It is easily seen that, in our case, this extends to any q-c1osed set E (resp . q-open set A) . Therefore if A is q-open and then (5.20)
If ¢ E W2/q,q' (00) n LOO (an) and I e 1R is a bounded open interval then, by [1 , Prop. 6.1.2, Prop. 6.4. 10] , A : = ¢-1 ( 1) is quasi open. Without loss of generality we may assume that ¢ < 1 . Given k E N and m = 0, . . . , 2k - 1 choose a number am, k in the interval (m2- k , (rn + 1) 2- k ) such that P,n( r/> - I ( { am,d) = 0. Put k A71l,k = ¢ - l ( (am ,k , am+1,k]) m = 1 , . . . , 2 - 1 , AO,k = ¢- l ( (ao,k' al,k ] ) and 2k _ l !k = L m2- k XAm,k 71l=0 Then fk ....... ¢ uniformly and, by (5 .20), lim fk u�dS /3-0 an This implies assertion (vi).
=
an
fk dfJR'
o
372
MOSIIE MARCUS AND LAURENT VERON
THEOREM 5.1 1 . Let u E U(D) .
The regular set R(u) is a-regular and consequently it has a regular decomposi tion { Q } . (ii) Let
(i)
n
v1'. := sup{ [uJQ : Q q-open and regular } . (5.21) Then there e:cisf.� an increasing sequence of moderate solutions {wn} such that (5.22) SUPPbn 1J)n (Thus V1'. is a-moderate.)
� R(u),
Wn r V"-'
q (iii) Let F be a q-closed set such that F c R(u) . Then, for everll t > 0, there exists a q-open regular set Q€ such that C2/q,q' (F \ Q.) < e . If, in addition, [ull" is
moderate then F is Tegl1.1ar; consequently there exists a q-open regular set Q such that F � Q. (iv) With {Qn} as in (i), denote
(5.23) vn := [ulQn ' J1.n := tr vn , v := lim vn , J1.1'. := lim J1.n' Then, ( 5 24 ) Furthermore, for every q-open Teg'ltlar set Q, (5.25) J1.RX Q = tr [uJQ = tr [vnl Q · .
Finally, J.1.1'. is q-locally finite on R(u) and a-finite on Ro(u) := UQn . (v ) If {wn} is a sequence of moderate solutions satisfying conditions (5 . 2 2 ) then, ( 5 26 )
J1.n. = lim tr Wn (vi) The regularized measure J1.n. given by (5.27) J1.n.(E) : = inf{J1.,,(Q) : E c Q, Q q-open VE c aD, E Bore l} is q-perfcct. u � v". (vii) .
(viii)
n(u)
q
Fm' every q-closed set F c R(u):
[ujp = [V,J F . (5.28) If, in addition, J.1.1'.(F) < 00 then (UJF is moderate and tr ('lLl F = J1.RXF' (5.29) (ix) If F is a q-closed set then J1.R(F) < 00 .: :. [U]F is moderate .: :. F is regular. (5 . 30)
PROOF. (i) By [1 , Sec. 6.5.11] the (a:,p)-finc topology possesses the quasi Lindelof property. This implies that R(u) is a-regular. By Lemma 5.4 R(u) has a regular decomposition { Qn}· Recall that Qn C Qn+l and C2/q ,q' (R(u)\Ru(u)) = O. (ii) This assertion is an immediate consequence of (5.21) and Proposition 3.3. (iii) By definition, every point in R(u) possesses a q-open regular neighborhood. Therefore, the existence of a set Q., as in the first part of this assertion, is an
373
PRIWISbJ BOUNDARY TRACE
immediate consequence of Lemma 2.5. Let O. be an open set containing F \ Q, such that C2/q ,q' (0,) < 2f. Put FE := F \ 0, . Then F, is a q-closed set, F, C F,
q
C2/q,q' (F \ F. ) < 2 € and FE C Q,. q Assertion 1. Let E be a q-closed set, D a q-open regular set and E c D . Then there exists a decreasing sequenee of q-open sets {Gn};:O 1 such that (5.31) and [ul en
(5.32)
->
rulE in U (n, p) .
By Lemma 2.6 and Theorem 4.4, there exists a decreasing sequence of q-open sets {Gn} satisfying (5.31) and, in addition, such that [ul Gn 1 [ul E locally uniformly in n. Since [ulGn :::; [ulD and the latter is a moderate solution we obtain (5.32). Put
En
:=
U�= l Fl /m' Dn := U�= l Q l/m'
Then En is q-closed, Dn is q-open and regular and En /!:. Dn . Therefore, by Assertion 1, it is possible to choose a sequence of q-open regular sets { Vn } such that
(5.3 3) By Theorem 4.4,
[ulF < [Ul En + [ul E'\En and [UJF\En 1 O. Therefore [ulEn i [UJ F. If, in addition, [UJF is moderate then and consequently, by (5.33),
[U,J En i [1tlF in U (0, p)
[ulvn -> [ulF in Lq (n, p). Let {Vn k } be a subsequence such that
(5.3 4) q
Vn n F and that C2Iq,q, (F \ En) 1 0. Therefore C2/q ,q, (F \ v,..) -> O . Consequently F /!:. w := n);" 1 Vnk and, in view of (5.34), [uJw is moderate. q Obviously this implies that W is pre-regular (any q-closed set E C W has the property that [ltlE is moderate) and F is regular. Finally, by Lemma 2.4, every Recall that En
C
q-closed regular set is contained in a q-open regular set. (iv) Let Q be a q-open regular set and put IJ.Q = tr [uJQ. If F is a q-closed set such that F
(5.35)
q
C
Q then, by Theorem 5.5,
[uJE' = IJ.QXr
In particular the compatibility condition holds: if Q, Q' are q-open regular sets then
(5 .3 6 )
- IJ.QnQ' = IJ.QXQnQ - - , = IJ.Q'XQnQ
,'
374
MOSHE MARCUS AND LAURENT VERON
With the notation of (5.23), [vn+kJQk = Vk and hence ftn+kXQ· k = ftk for every k E N. 0 it Let F be an arbitrary q-closed subset of R(u). Since C2/q,q, (F \ Qu) follows that ->
(5.37)
In addition, [VJ F > lim [vn l F
=
[UJ F and v < u lead to,
(5.88)
[UJ F
=
[vl F .
If Q is a q-open regular set, [u)Q = lim[vnlQ < lim v" = : v and so vn < v. On the other hand it is obvious that v < vn. Thus (5.24) holds. By (5.35) and (5.37) , if F is a q-closed subset of R(1L) and [UJ F is moderate, (5 .39)
which implies (5.25). This also shows that ftnXF is independent of the choice of the sequence {ftn} used in its definition. This remains valid for any q-closed set F t. R(u) because C2/q'
(v ) If W is a moderate solution and W ftn.. Indeed [wJ Qn < [vJ Qn
=
::; vn.
vn , [wJ Qn r W
and sUPP�I! W ;.
tr [wJQn r
r
t. R(u) then
< lim tr Vn
=
r :=
tr W <
ftn.·
Now, let {wn} be an increasing sequence of moderate solutions such that F" sUPP�f! Wn t. R(u) and Wn I vR. We must show that, if Zln := tr Wn:
:=
(5 .40)
By the previous argument ZI < ft,,-: The opposite inequality is obtained as follows. Let D be a q-open regular set and let K be a compact subset of D such that C2/q,q, (K) > o . Wn < [wn J IJ + [Wn)Dc
: vn. = limwn < lim[wn)D + UDc .
The sequence { [WnJD } is dominated by the moderate function [VJD . In addition tr [Wn] D = ZlnX]) r ZlX])· Hence, ZlX ]) is a bounded measure and [WnJD r u"x ]) where the function on the right is the moderate solution with trace ZlX ]). Consequently vR = lim wn < uvx . + UDc . D
This in turn implies
( [VJ K - u"x . ) + < inf(UDc , U}( ) the function on the left being a subsolution and the one on the left a supersolution. Therefore ( [vJJ( - u"x J+ < [[U]Dc JK = o. D D
Thus, [VJ K < uvx and hence ftnX K < ZlX JS. Further, if Q is a q-open set such that ]) Q t. D then, in view of the fact that SUP { ftnXK : K E Q, K compact}
= /l-XQ,
375
PRECISE BOUNDARY TRACE
we obtain, (5.41 )
Applying this inequality to the sets Qm, Qm+ l we finally obtain II ,R/\. vQ r-
H�
< vx -
Qm+l -
< vXQ -
m+2
.
Letting m 00 we conclude that /-in < v. This completes the proof of (.'i.40) and of assertion (v). (vi) The measure /-in is e��entially absolutely continuous relative to C2/q,q' (see Definition 5.8). Therefore this assertion follows from Lemma 5.9. (vii) By (4.5) -)
By Theorem 3.9(c) Hence
[u] i:I!1\Qn 1 [U]a!1\Ro '
lim(u - [u] QJ = u
Vn
< [U] an\Ro
so that u e Vn � O. Since Vn < u this is equivalent to the statement U � vn. R Ro (viii) (5.28) was established before, see (5.38). Alternatively, it follows from the previous assertion and Theorem 4.7. If fLn(F) is finite then (5 . 29 ) is a consequence of (i) and (5.35), Indeed, Fn := FnQn 1 Fo q F. Hence, [uJp < [uJPn + [UJF\Fn and C2/q , q' (F \ Fn) 1 O. Hence [UJF = lim [u]Fn and tr [uJ pn = /-i "xFn 1 /-inX.o = /-inXP' Since /-inXp is a bounded measure, [u]p is moderate and (5 29) holds. (ix) If /-in(F) < 00 then, by (viii) , [UJF is moderate and, by (iii), F is regular. 0 Conversely, if F is regular then [u] p is moderate and, by (5.25), fLn(F) < 00. -
�
.
5.5. The precise boundary trace.
5.12. Let qc � q and u E Urn). a: The solution vn defined by (5.21) is called the regular component of u and will be denoted by u,'eg' b: Let {vn} be an increasing sequence of moderate solutions satisfying con dition (5.22) and put /-in = /L,.JU) := lim tr vn . Then, the regularized measure il", defined by (5.27), is called the regular boundary trace of u. It will be denoted by trnu, c: The couple (tr"u, S(u») is called the pTecise boundary trace of u and will be denoted by tr Cu. d: Let v be the Borel measure on an given by
DEFINITION
(5.42)
veE) =
(tr "u) (E) if E c R(u), veE) = 00 if E n S(u) # 0,
for every Borel set E c an. Then v is the measure representation of the precise boundary trace of u, to be denoted by tr u. Note that, by Theorem 5.11 (v), the measure fLn is independent of the choice of the sequence {vn}.
376
MOSHE MARCUS AND LAURENT VERON
THEOREM 5. 13. Assume that u E Urn) is a a-moderate solution, i. e" there exists an increasing sequ ence {un} of positive moderate solutions such that Un u. Let J.Ln : = tr Un, ILo := lim J.Ln and put ->
J.L(E) : = inf {J.Lo (D) : E C D C an, D q-open},
(5.43)
for every Borel set E c an. Then: (i) J.L is the precise boundary trace of U and J.L is q-perfect. In particular independent of the sequence {un } which appears in its definition, (ii) If A is a Borel set such that tI(A) < 00 then tI(A) = J.Lu (A) . (iii) A solution U E Urn) is a-moderate if and only if
j.t
is
U = sup{v E urn) : v moderate v < u},
(5.44)
which is equ ivalent to
u = sup{u". : T E W+ 2/q,Q (an),
(5.45)
T
< tr u},
Thus, if u is a-moderate, there exists an increasing sequence of strictly moderate s olutions converging to u. (iv) If u , w are a-moderate solutions,
tr w < tr u ':
(5.46)
w
:.
< u.
PROOF. (i) Let Q be a q-open set and A a Borel set such that C2/Q,q, (A) = O. Then tIn eA) = 0 so that tIo (A) = O. Therefore tIo (Q \ A) = J.Lo (Q). Thus tIo is essentially absolutely continuous and, by Lemma 5,9, J.L is q-perfect. Let {Dn} be a regular decomposition of R(u). Put D� = R(u) \ Dn and observe that D� 1 E where C2/q,q' (E) = O. Therefore U" n xD' 1 o . R
Since, it follows that
u"
,.- n. x
Now Hence
'R.(u )
u - [uls(u) <W
-
:=
lim u"r-nx D
n
< - u"
,-n x D }
lim u",... n xn(u)
=
n
lim u"
.......
O.
x Dn
< - u r eg.
This implies U e [uls(u) < u c But the definition of ureg( = v,,) implies that ureg < u e [uls ( ) . Therefore lim ul'n X = ureo' Thus the sequence {Ul'nxD) satisfies Dn condition (5,22) and consequently, by Theorem 5.11 (iv) and Definition 5.12, r
,-n
g'
u
(5.47)
If e E Stu) then, for every q-open neighborhood Q of e, In [ul�pdx = 00. This 00. To verify this fact, assume that, on the contrary, there exists implies: J.Ln (Q) a subsequence (still denoted {tIn}) such that SUP tIn(Q) < 00. Denote by vn,Q and wn , Q the solutions with boundary trace XQtr Un and X Qctr Un respectively. Then --)
Un S; Vn,Q + wn,Q
:. u
< VQ + WQ ,
vQ
=
lim Vn,Q ,
WQ
=
lim Wn,Q'
PRECISE BOUNDARY TRACE
377
e E If Then is a q-open set such that D is moderate and w vanishes in Q. D c vQ Q D c Q then [wQl D 0. Therefore �
min(u , UD) < min(v Q' UD) + min(w Q' UD )
:. [Ul D < [Vol D '
which brings us to a contradiction. In conclusion, if � E S(u) then J-Lo (Q) = 00 for every q-open neighborhood of � . Consequently /-i(e) = 00 . This fact and (5.47) imply that J-L is the precise trace of u. (ii) If J-L (A) < 00 then A is contained in a q-open set D such that /-io (D) < 00 and, by Lemma 5.9, /-i(A) = /-io(A) . (iii) Let u E U(o') be O"-moderate and put (5.48)
u·
:=
sup{v E U(o')
: v
moderate v < u}.
By its definition u* < u. On the other hand, since there exists an increasing sequence of moderate solutions {un} converging to u, it follows that u < u* . Thus u = u* . Conversely, if u E U(O) and U = u· then, by Proposition 3.3, there exists an increasing sequence of moderate solutions {un } converging to u. Therefore u is O"-moderate. In view of Proposition 5.10 (iv) , u* < sup{ur : T E W+ 2/Q,Q(an),
T
< tr u} =: ut.
On the other hand, if u is O"-moderate, T E W+ 2/q,Q (ao') and T < tr u then (with /-in and 'Un as in thc statement of the theorem) , tr (ur e Un ) = (T - J-Ln l + 1 0. Hence Ur e Un 1 0, which implies that Ur e u = 0, i.e. Ur < u. Therefore u > ut . Thus (5.44) implies (5.45) and each of them implies that u is O"-moderate. Therefore the two are equivalent. (iv) The assertion :. is a consequence of (5.45) . To verify the assert.ion .; it is sufficient to show that if W is moderate., u is a-moderate and W < tr u. < u then tr W Let {un} be an increasing sequence of positive moderate solutions converging to u. Then Un V W < u and consequently Un < Un V W r u. Therefore tr (un V W ) r j.£' < tr u so that tr w < tr u. D
THEOREM 5.14. Let U E U(o') and put v = tr u. (i) u" eg is a-moderate and tr uceg = tr"Ru. (ii) If v E U(o') (5.49)
v
;· tr v < tr u.
If F is a q-closed set then (5.50)
(iii) A singular point can be characterized in terms of the measure (5.51)
v
as follows:
� E S(u) .; :. v(Q) = 00 VQ : e E Q, Q q-open.
(iv) If Q is a q-open set then: (5.52)
Q pre-regular ·:
(5.53)
Q regular .;
;. :.
v(F n Q) < 00 VF
3
q
C
Q : F q-closed
Borel set A: C2/Q,Q' (A) = 0, v(Q \ A) < 00.
378
(v)
MOSHE MARCUS AND LAURENT VERON
The singular set of Ur,g may not be empty. In fact
(5. 54 )
�
stu) \ bq (S(u» C S(ureg) C stu) n 'R.(u), where bq(S(u» is the set of C2/q,q, - thick points of stu), (see Notation 2. 1). (vi) Put So (u) : = { � E an : v(Q \ S Cu ll = DO VQ : � E Q , Q q - open} . (5.55) Then
(5.56) Remark. This result complements Theorem 5.11 which deals with the regular bound ary trace. PROOF. (i) By Theorem 5. 1 1 (ii) un g is u-moderate. The second part of the statement follows from Definition 5.12 and Theorem 5.13 (i) . (ii) If v < u then n(u) C n(v ) and by definition vreg < ure. By Theorem 5.13 (iv) tr vrog < tr ur g and consequently tr v < tr u (5.50) is an immediate consequence of e
.
(5.49). (iii) If � E n(u) there exists a q-open regular neighborhood Q of e. Hence v(Q) = tr u ( Q) < DO. If e E S( u) it follows immediately from the definition of precise trace that v(Q) = DO for every q-open neighborhood Q of �. 'R
,
(iv) If Q is pre regular then [ulF is moderate for every q-closed set F t Q and Q t n(u) . By Theorem 5.11 (viii) this implies: tr [ul F = (JIn(u»XF and consequently v(F n Q) ( 1-' (u» (F n Q) = (1-'",(u» (F) < DO. Therefore (5.52) holds in the direction ;. . ConverHely, if Q is a q-open set, F a q-closed set, F t Q and v(FnQ) < DO then, q by Definition 5.12, F n Q c n(u) which implies F c n(u) and 1-'7<.(u) (F) < DO. Therefore, by Theorem 5.1 1 (viii) [ulF is moderate. This implies (5.52) in the opposite direction. - q If Q is regular there exists a pre-regular set D such that Q e D. Therefore (5.52) implies (5.53) in the direction :-. On the other hand, =
",
v(O \ A) < DO ===};.
0 t n(u)
and 1-'1<-(0) = I-'n(Q \ A) < 00. Hence, by Theorem 5 . 1 1 (ix), [ul Q is moderate and Q is regular. -
�
(v) Since sUPPb!1 u" g C n(u) and n(u) c n(u,.,J we have S(u.,.)
c
n(u) n S( u) .
Next we show that S(u) \ bq(S(u» c S(u ) If � E S(u) \ bq (S(u» then n(u) u {O is a q-open neighborhood of �. By (i) its trace is q-perfect. UTeg is u-moderate and consequently (by Theorem 5.13 (i » Therefore, if Qo is a q-open neighborhood of � and Q = Qo n ({O u n(u» then r, g
'
( t r ur,.) (Q) = ( tr ureg) (Q \ {OJ = (t r u) (Q \ {OJ· The last equality is valid because Q \ {O c n('ll) . Let D be a q-open set such that � E D c iJ c Q. If tr u e D \ {O J < 00 then, by (iv) , D is regular and � E n(u), contrary to our assumption. Therefore kurD \ {O ) = 00 so that tr u",oCQo \ { O ) =
379
PRECISE BOUNDARY TRACE
for every q-open neighborhood Qo of �, which implies � E S(ure)' This completes the proof of (5.54). (vi) If � rf. bq(S(u)) , there exists a q-open neighborhood D of � such that (D \ {O) n S( u) 0 and cOIHmquently
00
=
(5.57)
(tr ureg) (D \ {O )
=
(tr urc g) ( D \ S (u)) = (tr u)(D \ S(u) ) .
If, in addition, � E So (u) then ( tr u) ( D \ S(u)) = (tr u
re
) (D \ {O) = 00 .
If Q is an arbitrary q-open neighborhood of � then the same holds if D is replaced by Q n D. Therefore (tr ureg) (Q \ {O) = 00 for any such Q. Consequently � E S(ureg), which proves that So eu) \ uq (S(u)) c S(ur,g) ' On the other hand, if � E S(ureg) \ uq (S(u)) t.hen there exists a q-open neigh borhood D such that (5.57) holds and (tr ur ) (D) = 00 . Since u" cy is u-moderate, (t r ureg) is q-perfect so that ( tr ureg) (D) = ( tr ureg)(D \ {O) = 00 . Consequently, by (5.57) , (tT u)(D \ S(u) ) = 00 . If Q is any q-open neighborhood of f. then D can be replaced by D n Q. Consequently (tr u) (Q \ S(u) ) = 00 and we conclude t.hat f. E So (u). This completes the proof of (5.56) . o eq
LEMMA 5.15. Let F
c
812 be a q-closed set. Then S (UF ) = uq (F) .
PROOF. Let f. be a point on 8n such that F is C2/q,q,-thin at f.. Let Q be a - q q-open neighborhood of � such that Q c Fe. Then [Ud Q = UFn Q = O. Therefore � E R(UF)' Conversely, assume that � E F n R(UF)' By Theorem 5.7 there exists a q open neighborhood Q of � such that uQ = lirnp->o u� exists and uQ is a moderate -
q
solution. Let D be a q-open neighborhood of f. such that D C Q. Then Lemma 4.5 implies that [Ul D is moderate so that D c R(u) . In turn this implies that C2/q,q' (Fn D) = 0 and consequently F is q-thin at �. 0
5.6. The boundary value problem. Notation 5.2.
(5.58)
a: Denote by !m+ (8r2) the space of positive Borel measures on 812 (not nec essarily bounded) . b: Denote by Q:q (8r2) the space of couples (T, F) such that F C IX! is qclosed, T E !m+ (8r2) , q-SUpp T C 812 \ F and TX ao\F is q-Iocally finite. c: Denote by 'Jf : Q:q(8n) ...... !m+ (8r2) the mapping given by v = 'Jf(T, F) where 1/ is defined as in (5.42) with S(u), R(u) replaced by F, FC respec tively. 1/ is the measure representation of the couple (T, F) . rl: If (T, F) E Q:q (8r2) the set FT
=
{� E an
:
T
( Q \ F) = 00 VQ q-open neighborhood of f. }
is called the set of explosion points of T .
Remark. Note that FT C F (beca1lse TXaO\F is q-locally finite) and FT C (because T vanishes outside this set) . Thus (5. 59)
-----
ar2 \ F
MOSHE MARCUS AND LAURENT VERON
380
THEOREM 5. 16. Let (i)
II
be a positive Borel measure on an. The boundary value problem 'U > 0 in 0,,
(5.60)
tr ('u)
possesses a solution if and only if II E Mq(an). (ii) Let (T, F) E \tq(on) and put II : = ll'(T, F) . Then (5.61)
T
E Mq(an),
v on on
=
II E
Mq(an) if and only if
F = bq(F) U FT '
(iii) Let v E Mq(on) and denote (5.62)
Ev ; = {E : E q-quasi-closed, veE) < oo } ,
Dv := { D : D q-open, D -
q
�
E for some E E Eq} .
Then a solution of (5.60) is given by u = 'U Ell UF where (5.63)
G := U D,
F := 8n \ G,
'Dv
v : = sUP {U X : E E E } "'
",
E
.
Note that if E E E", then vXE is a bounded Borel measure which does not charge sets of C2/q,q' -capacity zero. Recall that if 11- is a positive measure possessing these properties then uJ1. denotes the moderate Iwl'lltion with boundary trace 11- . (iv) The solution u = v Ell UF is (J"-7Twdemle and it is the unique solution of problem (5.60) in the class of
(5.64)
tr u =
v
By Theorem 5 . 1 1 , ur c.q is (J"-moderate and u R;j U . Therefore R(u) reg
(tr u) x",CU) = (tr ureg) XR.Cur
By Theorem 5.13, 11-R. := tr ur E Mq(on). If v is defined as in (5.63) then eg
(5.65)
v = sup{[U]Q : Q ,,-open regular set}
=
ure9'
where the second equality holds by definition. Indeed, by Theorem 5 . 14, for every q-open set Q, [u]Q is moderate if and only if v(Q \ A) < 00 for some set A of capacity zero. This means that [u]Q is moderate if and only if there exists E E E", such that Q :£ E. When this is the case, -
tr [u] Q = ILR.(U)XCl = JLR(U) xF. = lIXE'
Thus v > urcg' On the other hand, if E E E"" then FJ C R(u) and 11-",(u) (E) = 11-1« u) (E) < 00. Therefore, by Theorem 5.11 (ix), E is regular, i.e., there exists a q q-opcn regular set Q such that E C Q . Hence uVXE � [u]Q and we conclude that v < ureg' This proves (5.65) . In addition, if E n stu) of- 0 then, by Definition 5.12, veE) = 00 . Therefore II is outer regular with respect to q-topology. Next we must show that II is essentially absolutely continuous. Let Q be a q-opcn set and A a non-empty Borel subset of Q such that C2jq,q' (A) = O. Either
- q
-
-
PRECISE BOUNDARY TR.ACE
381
v(Q \ A) = 00 in which case v(Q \ A) = v(Q) or v(Q \ A) < 00. In the second case Q \ A c R(u) and -
Let � E A and let D be a q-open subset of Q such that � C D c D c Q . Let Bn be q n a q-open neighborhood of A n D such that C2/ q,q' (Bn ) < 2- and En C D. Then -
En := D \ B" .
[Ul D ::; [UlEn + [ul Sn '
Since lim [ul B = 0 it. follows that [Ul D = lim[u]En . But n
II [Ul En IILQ(fl,p) < Cv(EII) < CV(Q \ A). By assumption, v(Q \ A) < 00 . Therefore lI [u] n IiL" Cfl,p) < :xl which implies that. D C 'R(u). Since every point of A has a neighborhood D as above we conclude that A C 'R(u) and hence veAl = (trRu)(A) = O. In conclusion v is essentially ablOlutely continuous and v E Mq(80 ) .
Secondly we prove: (B) Suppose that ( T, F) E Q:q(80) satisfies (5.61) and put v := ']['(T, F) . Then the solution u = v (f) UF, with v as in (5.63 ) , satisfies tnt = v. By (5.64), this also implies that v E Mq (80) .
Clearly v is a a-moderate solution. The fad that 7 is q-locally finite in FC and essentially absolutely continuous relative to C2 /q q implies that ,
G := 80 \ F e 'R(v ),
(5.66)
It follows from the definition of v that FT
C
'
(tr v)xr. = 7Xc'
S(v). Hence, by Lemma 5 . 15,
(5.67)
Hence, by (5.6 1 ) , stu) = F, v = 'ILr and 7 = tr u Thus tr u = (7, F) which is equivalent to tr u = v. Next we show: (e) Suppose that ( 7, F) E Q:q(80) and that there exists a solution u such thal tr Cu = ( 7 F) Then, " g'
••
,
C
.
7 = tr u = t.r ureg' F = S(u). If U := Ureg (f) UF then tr U = tr 'IL and u < U. U is the unique a-moderate solution uf (5.60) and (7, F) satisfies condition (5.61 ) . AHsertion (5.68) follows from Theorem 5.11 (i) and Definition 5.12. Since ur is a-moderate, it follows, by Theorem 5.13, that T E Mq(80). Bv ... Theorem 5.11 (vi), u � u . Therefore 1V := u O u reg vanishes on n(u) so R(u) �'f'!} that w < UF. Note that u - Ureg < 1J) and therefore (5.68)
'R.
••
u< - �lre g (f) w < U. By their definitions So (u) = FT and by Theorem 5.14 (vi) and Lemma 5 . 1 5 , (5 .69)
(5.70)
-
stU) = S(ll'reg) U S(UF) = S (ur
U bq (F)
=
FT U bq (F).
On the other hand, 'R(U) :) 'R(uR) = 'R(u) and, as u < u, n(U) c n(u) . Hence R(U) = n(u) and S(U) = S(u) . Therefore, by (5.68) and (5.70), F' =
382
MOSHE MARCUS AND LAURENT VERON
S(U) = FT U bq (F) . Thus (7, F) satisfies (5.61) and tr "U = (7, F). The fact that U is the maximal solution with this trace follows from (5.69). The solution U is O"-moderate because both u'R and UF are O"-moderate solu tions. This fact, with respect to UF, was proved in [16] . The uniqueness of the solution in the class of O"-moderate solutions follows from Theorem 5.13 (iv). Finally we prove: (D) lf v E Mq(80) then the col1ple (T, F) defined by v := sup{1L"xFJ
(5.71)
: E E £,, },
7 := tr v, F := 80 \ R (v)
satisfies (5.61 ) . This is the unique couple in (t(80) satisfying v = '['(7, F) . The solution v is O"-moderate so that 7 E Mq (80) . We claim that u := v EEl UF is a solution with boundary trace tr e u = ( T, F) . Indeed u > v so that R(lI) c R(v). On the other hand, since T is q-Iocally finite on R(v) = 80 \ F, it follows that S(lI) c F. Thus R(v) c R(u) and we conclude that R(v) = R(u) and F = stu). This also implies that v = ureg Finally stu) = S (v) U S(UF) = FT U bq (F), so that F satisfies (5.61 ). The fact that, for v E Mq(80), the couple (7, F) defined by (5.71) is the only one in (t(80) satisfying v = '['(T, F) follows immediately from the definition of these spaces. D Statements A-D imply (i)-(iv). as
Remark. If v E Mq(80) then G and v alternative representation: v := sup { u"xQ :
(5.72)
F" : =
(5.73)
defined in (5.63) have the following
G = U Q = U E,
Q E F,, },
Tv
{Q : Q q-open,
£v
v( Q) < oo}.
To verify this remark we first observe that Lemma 2.6 implies that if A is a q-open set then there exists an increasing sequence of q-quasi closed sets {En} such that A = Uj'" En . In fact, in the notation of (2.22) , we may choose En = Fn \ L where L = A' \ A is a set of capacity zero. Therefore
U D C U Q C U E =: H. -
On the other hand, -if E E £" then p'R( u) ( E) = Pn(u) (E) = v (E ) < 00 and, by Theorem 5.11 (ix), E is regular, i.e., there exists a q-open regular set Q such that q E C Q. Thus H = UD D. If D is a q-open regular set then D = Uj'" En, where {En} is an increasing sequence of q-quasi closed sets. Consequently, v
U"X = lim u"x D
Therefore
En
.
sup{ U"XQ Q E V,, } < sup{ UVXQ Q E Fv} < sup{ :
:
UVXE :
E E ev}.
PRBCISE BOUNDARY TRACE
383
On the other hand, if E E E" then there exists a q-open regular set Q such that E t Q. Consequently we have equality. Note that, in view of this remark, Theorem 1.3 is an immediate consequence of Theorem 5.16.
Acknowledgment. Both authors were partially sponsored by an EC grant through the RTN Program Front-Singularities, HPRN-CT-2002-00274 and by the French Israeli cooperation program through grant No. 3-1:352. The first author (MM) also wishes to acknowledge the support of the Israeli Science Foundation through grant No . 145-05.
References [IJ
Adams
D . R. and Hedberg L. r . ,
Fu nction spac,,_, and
potential
theory, Grundlehren Math.
Wissen. 314, Springer [2J
[3J [4J [5J [6J [7J
[8J
[9J
(1996). Beni/an Ph. and Brezis H., Nonl inear preoblems related to the Thomas-Fermi equation, J. Evolution Eq. 3 , 673-770 (2003). Dynkin E. B. Diffusions, Superdiffusions and Partial Differential Equations, American Math. Soc., Providence, Rhode Island, Colloquium Publications 50, 2002. Dynkin E. B . Superdiffu8ions and Positive Solutions of Nonlmear Partial Differential EqtLa tions, American Math. Soc. , Providence, Rhode Island, Colloquium Publications 34, 2004. Dynkin E . B. and Kuznetsov S . E. Superdiffusions and removable singularities for quasilinear partial differential equations, Comm. Pure App!. Math. 49, 125-176 (1996). 1/.'" dominated by harmonic junctions, Dynkin E. B. and Kuznetsov S. E. Solutions of Lu J . Analyse Math. 68, 15-37 (1996) . Dynkin E. B. and Kuznetsov S. E. Fine topology and fine trace on the boundary as.,ociated with a dass of quasilinear differential equations, Comm. Pure App\. Math. 51, 897-936 ( 1998). Kato T . , Shrodinger operators with Ringular potentials, Israel J. Math. 13, 135-148 (1972). =
Kuznetsov S.E. a-moderate solutions of Lu
Sc. Serie I [lOJ
Labutin D.
326,
(2003) . [l1J [12J [13J [14J [15] [16J [17J
1189-1 194 ( 1998).
=
u'" and fine trace on the boundary, C.R. Acad.
A., Wiener regularity for large solutions of nonlinear equations, Archiv
Legall J. F . , The Brownian snake and solutions of tl.u Fields 102,
393-432 (1995).
=
fiir Math.
u2 in a domain, Probab. Th . ReI.
J. F . , Spatial branching processes, random snakes and partial differ-entiat equations, Birkh1iuser, Basel/Boston/Berlin, 1999. Marcus M. and Veron L., The boundary trace of positive solutions of semilinear elliptic eAJuations: t he subcritical case, Arch. rat. Mech. Anal. 144, 201-231 ( 1998). Marcus M. and Veron L . , The boundary trace of positive solutions of semilinear elliptic equations: the supercritical case, J. Math. Pures Appl. 11, 481-524 (1998). Marcus M . and Veron L . , Removable singularities and boundary trace, J. Math. Pures Appl. 80, 879-900 (2000). Marcus M . and Veron L., Capacitary estimates of positive solutions of semilinear elliptic eAJuations with absorption, J. European Math. Soc. 6, 483-527 (2004). (2001). M selati B . , Classification and probabilistic representation of the positive solutions of a semi linear elliptic equation, Mem. Am. Math. Soc. 168 (2004). Legall
DEPARTMENT OF
MATHEMATICS,
TEcm-aON, HAIFA
32000,
ISRAEL
E-mail address: marcusmlDmath. technion . ac . i1 LABORATOIRE DE MATHEMATIQUES,
FACULTE
TOURS, FRANCE
E-mail address: veronllDlmpt . univ-tours . fr
UES
SCIBNCES, PARC DE GRANDMONT,
37200
Contemporary Mathematics Volume 446, 2007
Blow-up In Nonlinear Heat Equations with Supercritical •
Power Nonlinearity Hiroshi Matano
Dedicated to Prof.
Haim
Brezis on the occasion oJ his sixtieth birthday
ABSTRAGr. In this art icle we study blow-up of solutions of the nonlinear heat equation Ut = �u + luI P - 1u. We focus on the phenomena that are character istic to the supercritical range p > Ps : = N > 3, where N is the space dimension. We assume radial symmetry of solutions. What we present here is for the most part an overview of the author's recent joint work with Frank Merle [301 on supercritical blow-up and the author's recent paper [281 on the new application of the braid group theory to blow-up problems. Among other things we discuss (a) classification of blow-up profiles; (b) nonexistence of a 6-fundion type singularity; (c) continuation beyond the blow-up time; (d) determining type II blow-up rates via the braid group theory.
�+�,
1. Introduction
Blow-up in nonlinear heat equations has been a subject of extensive mathemat ical studies. The motivation comes from various fields of science such as plasma physics, combustion theory and population dynamics, a� well as in connection with some geometrical problems. See, for example, [4) for some of the physical back grounds. In the present article we consider the problem
(Ll)
Ut
=
(x E D, t > 0)
�u + lulp- 1 u
u(x, O)
=
uo(x)
(x E D),
where either 0 = RN or D = BR : = {x E RN I lx l < R}. In the latter case, we impose the Dirichlet boundary condition ( 1 . 2)
u(x, t)
=
0
(x E aD, t > 0).
Primary 35K55, 35B40, 74H35; Secondary 20F36. Key words and phrases. blow-up, nonlinear heat equation, singularity, braid group. This work was partly initiated during the aulhoc'. visit to Cergy-Pontoise UniverS ity and to U niversity of Paris-sud. 2000 Mathematics Subject Classification.
385
386
HIROSHI MATANO
The exponent p is supercritical in the Sobolev sense, that is,
P>P
(1 . 3)
E
..
N+2 ;= N 2 '
N
_
?.
3
and we assume Uu LOO (!1) n C(!1) . Throughout this article we mainly deal with radially symmetric solutions. In other words, u is expressed in the form
u(x, t) = U ( l x l t), ,
where the function U (r, t ) satisfies the equation
( 1.4 )
Ut
=
U'"7" +
N- 1
Ur + 1 U 11'- 1 U.
r Problem ( 1 . 1 ) posseset; a classical solution at least locally in time. We say that a solution u blows up in finite time if, for some T > 0, lim sup II u (·, t ) IIu'" = + 00. t�T Here T is called the blow-up time. The simplest example of a blow-up solution is that of the ordinary differential equation du/dt = l u l ,,-1 U, in which case u(t) = ±K ( T-t) -1/(p- l ) with K ; = (p _ 1)-1 /(P- l) . Another simple example is a self-similar blow-up solution, which is given in the form
( x-a ) u(x, t) = (T _ t) - P-l tjJ 1
(1.5 )
v'T - t '
where a is any point in RN and tjJ(y) is a bounded solution of the equation 1 for y E RN l:l.tjJ - � y . 'VtjJ tjJ + 1 1jJlp- l tjJ = 0 ( 1.6)
2
In both cases we have
p-1
)
II u (- , t) II L� = O (T - t) - P:, .
(
More generally, blow-up in ( 1 . 1) is categorized into the following two types; Type I
:
Type II :
lim sup (T - t ) p ' II u (- , t ) IIL<'" < 00 , t�T
1
lim sup (T - t ) t ..... T
p
1
' IIu ( ·, t) II L = 00 . oo
One ean show that if a blow-up is of type I, then the solution is asymptotically self-similar as t -+ T in a certain sense. It is known that, in the subcritical range 1 < P < Ps ' only type I blow-up can occur; see [ 1 9 , 20, 21, 32] . On the other hand, Herrero and Velazquez [22, 23] found examples of type II blow-up in the range
N> 1 1 and P > PJL
(1. 7)
4 . := 1 + N - 4 - 2 JN - l
Here P ( > Ps ) is the so-called Joseph-Lundgren exponent, which has been known ([24] ) as a critical number for certain spectral properties of the singular stationary solution (see Subsection 2.4). The method of [22, 23] is based on a matched asymptotic analysis, as we explain in Section 5. The range P s < p < PJL was left open, but later Merle and the author [29] showed that no type II blow-up occurs if Ps < P < P JL as far as radially symmetric solutions are concerned. In the critical case P = Ps ' no type II blow-up occurs for positive solutions ( [11, 29] ) , but it may occur for sign-changing solutions according to the formal analysis of [11] . J
L
BLOW-UP IN NONLINEAR HEAT EQUATIONS
387
Apart from the examples of Herrero and Vel;b:quez, the nature of more general type II blow-up solutions in the supercritical range - such as their profiles and exact blow-up rates - remained largely unknown until very recently. This is one of the main focuses of the present article. Another notable difference between the subcritical range 1 < P < Ps and the supercritical range P > Ps is the existence of "incomplete blow-up" . In late 1980's, Baras and Cohen [3] showed that every blow-up is "complete" if 1 < p < Ps ' which roughly meaIls that there is no way to continue the solution beyond the blow-up time. On the other hand, in the supercritical range P > P S ' some solutions can be continued beyond the blow-up time in a certain weak sense. In a joint work with M. Fila and P. Polacik [9] , the author has shown that such weak solutions become classical immediately after the blow-up time. We will discuss further properties of complete and incomplete blow-up. The organization of the article is as follows: In Section 2, we present funda mental estimates for supercritical blow-up. The highlight is the "no-needle lemma" , which shows that a 8-funetion type singularity never occurs if p > Ps ' This is in marked contrast with the critical case p Pa , for which such singularities are ob served in the formal analysis of [11] . This lemma plays a key role in deriving various properties of supercritical blow-up. In Section 3, we classify blow-up profiles and characterize type I and type II blow-ups in Lerms of the profiles. In Section 4, we discuss continuation of solutions beyond the blow-up time. Among other things we discuss the relation between the rat,e of blow-up and the rate of regularization after the blow-up time. In Section 5, we present an intriguing application of the braid group theory to blow-up probems. The goal is to determine all type II blow-up rates by analyzing the topological properties of certain braids. The braid group method for scalar par abolic equations was introduced by Ghrist and Vandervorst [17] as generalization of the zero-number argument. We further develop their theory to make it applicable to our blow-up problem. Sections 2 to 4 are based on the author's joint work with F. Merle [30], while Section 5 is based on the author's paper [28] . Note that Mizoguchi [36] has also obtained a result somewhat similar to [28] via a braid-group method, by partly borrowing the idea of [28] and partly using a different approach. =
2. Fundamental estimates Early studies of blow-up, including the pioneering work of Kaplan [25] and Fujita [13] , were mainly concerned with the question as to what conditions cause blow-up and what conditions guarantee global existence. In the mid 1 980's, Weissler [40] showed an example of single-point blow-up, which lead people to realize that blow-up iii generally a highly localized phenomenon. This triggered a flow of re search on the local and global structure of singularities that appear at the time of blow-up; see for instance [12, 6, 19, 20, 14, 39] for some of the early results. We define the blow-up set of a solution by
B ( uo )
:=
{x E n I :lxn
--4
x , tn / T such that I u(xn , tn ) 1
--4
oo},
where Uo denotes the initial data of solution u and T is the blow-up time. Each element of B(uo ) is called a blow-up point of u.
HIROSHI MATANO
388
2 . 1 . Rescaled equation. In studying the local structure of singularities in ( 1 . 1) , it is useful to introduce the following rescaled coordinates as in [18J . Given an arbitrary point (a, T) E D x (0, (0 ) , we set ;=
(2 . 1 )
=
where
y=
(T - l) v
1
( l)
l U X,
(T - t) v'l u(a + v'T - t v, t)
x-a
s = - log(T - t) .
v'T -t ' REMARK 2.1. The point (a, T) is the center of rescaling. Usually T is chosen
to be the blow-up time of u, but sometimes it is set slightly off the actual blow-up time in order to obtain extra \l�eful e�tirnates Ilear the blow-up time. Examples of such techniques can be found in Subsections 2.2, 2.3 and 4.2. The function Wa ,1' (y, s) satisfies the rescaled equation 1 ow 1 w + IwIP-1w, = �w - - y . \lw (2 . 2) 2 as p- 1
and allY solution of equation (1 .6) is a stationary solution of (2.2). In other words, a solution u(x, t) is self-similar if Wa,T (Y, s) is independent of s. Equation (2.2) is defined on the domain if 0 = { I x l < R } , {y E RN I l y + eS/2al < R es/2 } 0" ;= ,
,
We associate with (2.2) the following energy functional;
1 1 1 E ( w) = ( 2.3 ) -2 1 \lw I2 + 2 (p - l ) I w l2 - p + l IwIP+ ! p(y) dy , JRN where p (y) = (47r) - N/ 2 e- l y I2/4 . Here and in what follows it will be understood that Wa,T is defined for all y E RN even if 0 =1= RN , by setting Wa,T = 0 outside Oa. s . For any solution w (y, s) of (2.2) , we have
r
d
ds E(w(-, s»)
(2.4)
<
-
w; pdy < 0, r JRN
provided that n is convex, hence E(w) is a Lyapunov functional for (2.2) . If the blow-up is of type I, then Wa,T(Y, s) remains bounded as s -> 00 . It then follows from standard parabolic estimates and the existence of the Lyapunov functional E that w( y, s ) approaches a set of stationary solutions as s -> 00 . This roughly implies that any type I blow-up solution is asymptotically self-similar. In the case where 1 < p < Ps ' the boundedness of w follows from the fact that W is defined for all large 8 , as shown in [19, 21J; hence every blow-up is of Type I. Here radial symmetry is not required. To see this, we first observe that I d 2 p-l d -Iw l P+ 1 p dy w p y = - 2E ( w) + 2 ds RN P + 1 RN (2.5) p+ l 2 p- l > for s > So , - - 2 E ( w) +
£
.
p+l .
389
BLOW-UP IN NONLINEAR HEAT EQUATIONS
where So = - log T and W = Wa ,T . Since W is defined for all s > So , the quantity J w2 p dy docs not blow up in finite time. From this and the fact that E(we, s)) is nonincreasing in s, we obtain -2E(w) + ��� Cf w2 P fly) p�' < 0, hence (2. 6)
for s >
(2. 7)
80 ,
for
where G is some constant and Eo(a) p-1
:=
8
>
80,
E(Wa , T ( - . so) ) . From (2.5) one can also get
wwsp dy + 2E( m) �
1
2 P dy w J ) ( J w; p dy) + 2En ( a) ( < (G Eo"i1 (a)) ( � E (w ) ) 1 + 2Eo (a) ,
<
hence
�
,
-
where G' is some constant. Thus Wa, T satisfies linear equation of the form 1 EJ'II'} 1 -- Llw - Y ' \lw 1 w + g( y , s) w , 2 P as where the coefficient 9 : = Iw l p- 1 satisfies =
(2. 9)
jSl+J ( l
-
_
Ptl1 )T
Ig(y, s) l p
dy
ds < Gil
lyl< l with r = 2 . Here Gil is a constant depending on Eo (a) . In view of (2.7) and (2.9), we can derive the following pointwise estimate from a linear parabolic theory (see theorem 8.1 in [27, sec.III] ) , provided 1 < p < (N + 2 + 2r-1 )/ (N - 2 + 2r- 1 ) : '1
I'Wa,T (y, s ) 1 < GEo(a) pi,
for IYI <
�,
SE
h + �, 8 1 + 8J .
Wa,T(Y, S ) = WO,T(y + eS/2a, S ) , the above estimate implies an LOO bound of the form
Since
(2. 10)
IWO,T ( Y, s ) I < G EO"+ 1 ,
where Eo := SUPa Eo (a). Since uo is bounded, w and \lw are hounded for ear-h fixed s > 80 . Thus, by redefining the initial time if necessary, we may assume without loss of generality that Eo < 00. Consequently w remains bounded as 8 -> 00 . The estimate (2. 10) implies that only type I blow-up occurs provided that 1 < p < (N + 2 + 2r- 1 )/(N - 2 + 2r-1) with r = 2. This is sufficient for our later argument, but actually (2. 9) holds for any 2 < r < 00 , hence ( 2 . 10 ) holds in the entire subcritical range 1 < p < Ps ' This can be shown by using the same bootstrap argument as in [38]' the details of which are omitted here (see [21]). The estimate (2. 10) also yields an U'O bound on u, provided that Eo is suffi ciently small. To see this, observe first that (2. 10) implies (2.11) lu(x, t) 1 ::; BK(T - t) - p 1 "
390
HIROSHI MATANO
/'(, : = (p - 1)e = ",-l CEO'+ 1 . where If Eo is small enough so that () < 1, then one gets (2.12) for some constant M > O. This is easily seen by constructing a supersolution for (1.1) of the form ()",(T - t + h(x, t» - p where h is nonnegative and has compact support. Since the proof relies on a local comparison argument, we need to assume (2.11) only in a small ball. Hence we can derive a local bound on u from a local energy estimate. Moreover the argument works for all p > 1 without the restriction of subcriticality nor radial symmetry, thus extending the earlier result in [20, Theorem 2.1] (for 1 < P < Ps and for small 0) with a much simpler proof. See [30, Section 2] for details. 2.2. Loo bounds for the supercritical range. If p > p" , an Loo bound on W does not follow directly from (2.7) and (2.9). However, since we are dealing with radially symmetric solutions, the equation is essentially one-dimensional in the region away from the origin, which means that any power p is subcritical (or even satisfies 1 < p < (N + 2 + 27"- 1 )/(N 2 + 2r- 1 ) with r 2, hence (2.9) follows directly without bootstrap argument). This yields a bound of the form for lyl > 1, 8 � 80 + 15 (2 . 13) I WO,T (Y, 8 ) 1 < CEr' for any small constant 15 > O. Now we apply the same argument to WO,T, , where Tl is a parameter ranging in [15 T, T], and T is the blow-up time of Then we obtain IWO,T, (y, 8) 1 < CEr' for Iyl > 1, 8 > log Tl + 15, ( 2.14 ) where Eo : = sup sup E (Wa,T, (., log T1 » ) . T, E [O T, Tl aER" Since Wa,T, (y , - log Tl) = A6 ' Wa,T( ,jAQy, - log T ) with Ao = TIT1 , it is easily seen that Eo < provided that both uo and V'uo are bounded, Observe also that 1 " T t - =-A( 2.15) WO.T (y, 8) = A p ' WO,T, ( V A y, s + log A) , P
,
'
"
1
"
=
-
a,
,
u.
_
,
-
-
,
-
00
Tl
-
t'
Now we let t vary over [o T, T) and Tl over (t, T] , Then we see that (2.15) holds for any 8 � - log((l - (5)T), A E [1, ) In particular, for each y with 0 < Iyl < 1, we can choose A = 1/1 Y l2 and apply (2.1 4 ), to obtain for 0 < ly l < 1 , 8 > - log (l - (5)1' + 0, I WO,T(Y, 8) 1 < CT I y l - p where CT = o (EO'''' ) . Combining this and (2.13), we get (2. 16 ) IWO.T(Y, 8) 1 < CT ( l + I Y I - :' ) for l y l > O, 8 > - log(1 - 15)T + 15, This and standard parabolic estimates imply that the derivatives of w are bounded in the region Iyl � 1 . Tlms the same rescaling method as (2.15) yields j , / < Iyl (2 ,1 7) 1 + W ( ) (j = 1 , 2, 3). 1 , 8) CT (Y lV' i Q,T Next we derive a global 1/)0 bound from the above estimate. Recall that the singular stationary solution of (2.2) is given by 'P* (y) = �* ( Iyl), where with (C· ) p-l = 2 1 (,"1' - 2 - p -l 2 ). (2.18) (0 .
2
1
,
'P
:l!ll
I:lLOW-UP IN NONLINEAR HEAT EQUATIONS
-
Eo is small so that CT < c* , then (2. 16) implies that WO,T (y, s) stays below CT + /tcp' (y), where It := CT (C* ) - l < 1 . Then one can construct a ' supersolution with initial data CT + ItCP that becomes bounded immediately. This gives an LOO bound for W in the range S 2: - log( l - o)T + 0 + 0 ( � So + 30). Rewriting 30 as 0, we obtain the following proposition: If the initial energy
( [30] ) . Let Ps
< P <
ex:: .
Then for any 0 > 0 there exist positive constants 10 , M depending only on p, N and 0 such that if w(y , s) is u radially symmetric solution of (2.2) defined on some interval So < s < ex:: and if
PROPOSITION
2.2
Eo
sup
:=
aERN sup
1
E
(-\0'':1
W(
< 10,
fi;;' y + a , so) )
then
(2.19 ) The above proposition and (2.12) yield an
I l u (- , to ) IIH�_l n'
is small , where
(2.20)
IIv Il H� _I OC
Ilu(·, to + o ) I I L
2.3. No-needle lemma.
=
=
LOO bound for
sUP aE r!
'U.
More precisely, if
�x -a l< l (lV'v I 2 + v2 )dx, then .
_
M ( l l u (-, to ) I I H�_ I J .+ 1 .
<
In this subsection we present what we call the "no
needle lemma" , which states that convergence outside the origin automatically
im
plies convergence at the origin. Thus a needle-like singularity does not appear in the supercritical range p
> Ps '
Despite its simple proof, this lemma is exceedingly
useful in the study of supercritical blow-up. For example, it easily follows from this
lemma that, in any type II blow-up, the rescaled solution WO,T(y, s) converges to the singular stationary solution cp* (y) or
-cp* (y)
as
s
-t
ex::
(Theorem 3 . 1 ) .
LEMMA 2 . 3 (No-needle lemma for w). Let Ps < p < ex:: and 1 , 2 , 3, · . . ) be a family of radially symmetric classical solutions
defined for y E
RN ,
So
<
s
<
ex::
and satisfy
E (wn ( - , so ) )
SUPn
<
let wn (y, s) (n = of (2.2) that are 00. Suppose that
in C 1 (RN \ {o} ) (n --> oo) for some s' > So and 1jJ E C1 (RIV ) . Then for any sufficiently small exists a constant M1 > 0 such that wn (', s ' ) --> 1jJ
sup
(2.21)
n
PROOF.
IIV'J w,, ( - , s* + o) IIL�
<
for j
M1 = 0,
It suffices to prove ( 2,2 1) for j
and
Iw,, (y , s ' W
< C(l
integrable on RN if P
(
C 1
+ Iyl-
2\;'!'I'» ) ,
since the assertion for j
+ Iyl - P I ) .
> PS '
4
where
Since
p(y)
Lebesgue convergence theorem yields (2 . 22)
E (w,, (. + a,
for some constant
' s )
)
.......
M > O.
is
E (1jJ(. + a))
I w,, (y , s' W+1
where
11
=
1 follows
c ( 1 + I y l - 2i,"!.') ) 4
<
M ( II V'l,b IIE= + 111jJl l i� ) 0<0<
( s) Wll ( vI,>, ( s) y,
>.(s )
<
>
p(y) (1 + I y + a l - p-l + I y + a l - YP+V . - 1 ) is as in (2.3) and a is any point in RN, the
Now, given a constant
iiin (y, s) = >. P
> 0 there
= 0 , 1 , 2 , 3.
from standard parabolic estimates. By (2 . 16) and (2. 1 7) , we have
lV'wn (y , s 'W :$
0
S
1 , we set
+ log >. ( s ) ) ,
( 1 + ( 1 - 0) eS- s" ) - 1 .
as n -+ 00
392
HIROSHI MATANO
Then wn satisfies the saIIle rescaled equation (2.2). Note that wn(Y, s * - log O) = 0/1 wn ( vte y, S O ) ---> OP:l 'I/J ( vte y )
as
n
-
.
00 .
Applying the same estimate as in (2.22) to Wn, we obtain " - 10g B») < M ( B �:':; IIV'l/Jllloo + 0 /1 s lim E (wn C a, (2 . 23) + II'I/J II i= ) . n�oc Choosing 0 small enough, we can make the right-hand side of (2.23) smaller than the constant /0 that appears in Proposition 2.2. This establishes an La;:, bound for wn(Y, s* + 0 - log O) , hence for wn (Y, s * + 0) . 0 REMARK 2.4. It is clear that the above lemma yields a similar no-needle esti mate for solutions Un of ( 1 . 1 ) . There is also another kind of "no-needle" result for u at the blow-up time, which will be stated in Corollary 3.3.
2.4. Structure of stationary solutions. Before ending this section, we sum
marize known results on radially symmetric stationary solutions of (l.1) on RN : �u + lulP-1u = O.
The structure of stationary solutions is deeply related to the blow-up behavior. There are two kinds of positive stationary solutions: the singular one and the regular ones. The singular stationary solution, which we denote by ip"(x) = <.p*(lxl), is given by the following explicit form: (2.24)
l - /1 , * p) * ( C re r = he c = w <.p - (r )
N - 2 - -l 2
2
p-1 pNote that <.p* is also a singular stationary solution of equation (2.2) . The regular solutions, denoted by <.paCr) with a > 0, are defined as the solution of
(2.25)
Urr +
N
- 1 Ur + I Ulp- 1 U = 0 with
•
U(O) = a, U'(O) = O.
r
The solution of (2.2 5) remains positive for all r > 0 if and only if P > Ps ' (The "if" part follows from the Pohozaev identity. ) By the self-similarity, we have
<.pa(r) = a <.Pl (a
(2.26)
2 r) .
p- l
In the critical case P = Ps , the solution is given explicitly by
a <.pa (r) = ( 1 + j.£ a N r2)
(2 . 27)
•
2
N-2 2
,
where J1 = l/N(N - 2). In the supercritical case, the solution can no longer be written explicitly, but its basic properties are well understood ( [24]) . Note that the behavior of <.Pa varies significantly with the range of p, as shown in Table 1 . Incidentally, the non-existence of type II blow-up in the range P s < P < P is proved in [29] by using the fact that Z(O,oo) (<.pa - <.p*) = 00. J
TABLE
P = Ps Ps < P < PJL PJL < P < oo
1 . Asymptotics of <.pa (r)
intersection with <.P2 00
0
limit
as
a
--->
0
<.P-
<.p*
00 (r > 0)
L
l3LOW-UP IN NONLINEAR HEAT EQUATIONS
3. Classification of focused blow-up Since we are dealing with radially symmetric solutions, any blow-up that occurs outside the origin is essentially of one-dimensional nature. In particular, only typc I blow-up occurs; see [29, Theorem 1 . 13] and also [31]. Therefore, real supercritical phenomena can occur only at the origin. \Ve say that a blow-up is focused if the blow-up occurs at the origin. In this section we study such a blow-up, but for the sake of generality, we do not assllme t hat 0 is the only blow-up point.
3.1. Blow-up profiles. Here we summarize the results in [30] concerning I.he
profiles of focused blow-up.
3 . 1 (Characterization of type II blow-up) . Let P s < P < 00 and let Uo E D''' (O) . Suppose that a solution u of (1.1) blows up at t = T. Then the limit THEOREM
w * (y) := slim WO,T ( Y, S ) ( = lim (T - t) v'l u( v'T - t y, t» ) t---+ T -oo e.xists locally uniformly in y E RN \ {O}, and w* is either a bounded radially sym metric solution of ( 1 .6) or the singular stationary solution 'P* (Y) or -'P* (y) . Fur thermore, the following conditions are equivalent: I (a) the blow-up is of type II; (b) lim (T - t) p-l ll u ( · , t) IIL= = Xl ;
(3.1)
(d)
( c) w * ( Y) = 'P* (Y ) or - if * (Y ) ;
t_'l'
x-a II'In
( � <po
u(x
X
)
= 1 or - 1 .
3.2 (Classification of focused blow-up) . Let PB < P < 00 and let u be a sol1ttion of ( 1 . 1 ) that blows up at t T. Then THEOREM
=
Xl
or - 00
finite but =1= ±l, 0
u(x, T) = lim x-a 'P * (x)
1 or - 1
o where
K
:= (p - 1)-
v
1
1
.; .;
:. Type 1 with w· =
or - K
:. Type I with noneonstant w *
.: :. 'IiJPp, .:
K
II
:. no blOW-lip occurs at x = 0,
is the constant solution of (1 .6) .
The following is an immediate corollary of Theorem 3.2. It implies that needle like singularity does not occur at the blow-up time. This result is similar to Lemma 2.8 in itR spirit, but does not follow from it directly.
3.3 (Formation of no needle) . Lp,t PB <
P
< 00 and let u be a solution of ( 1 . 1) defined (at least) for 0 < t < T. Suppose that for each X E O \ {O} lim sup lu(x , t) I < M COROLLARY
for some constant
. t ....T
M > O.
Then u docs not blow up at t = T.
3 . 1 . The main novelty of the theorem is the assertion that the limit (3. 1) exists even if the blow-up is of type II. The no needle lemma (Lemma 2.3) plays a key role in deriving this result. Let us show how we prove (a) => (c). Assume that the blow-up is of type II. Since equation (2.2) has a Lyapunov functional E(w) , and since there is a pointwise bound (2. 16), it is not difficult to show that the w-limit set of 1/JO,T with respect to the topology of C2 (RN \ {O}) is non-empty, compact, connected, and is contained in the set of stationary solutions (regular or sin?;Ular) of (1.6). In order to prove OUTLINE OF PROOF OF THEOREM
HIROSHI MATANO
394
that WO.T converges to 'P* or -'P* , it suffices to show that w (WO.T) does not contain bounded stationary solutions. Suppose the contrary, and let as n -+ oo
in C2 (RN \ {O})
WO.T (y, 5n ) -> 'Ij}{y )
for some bounded solution 'ljJ of (J .6) and some sequence 5 1 < 52 < S3 < . . . 00. Then by Lemma 2.3, Ilwo.T ( - , s" +6) IILoo remains bounded as n -> 00 . On the other hand, since the blow-up is of type II, we have lim sup � Ilwo.T (-, S) IILoo = 00. Renee WO . T (O, 5 ) must oscillate infinitely many times, from which we can derive a contradiction by the zero-number argument . Details are omitted. The assertion (c) � (b) � (a) is obvious. The equivalence (c) <=> (d) is included in Theorem 3.2. 0 ->
s
oo
Theorem 3.2 can be derived from the following two observations: Let w' be as in (3.1) and 'ljJ be a radially symmetric solution of (1.6). Then (3.2)
· u{x, T) I1mO x- 'P* (x )
=
' w* (y) l 1m 'P* (Y) lyl .... 'C lO
1
' lim 'ljJ = 1 .: :. 'ljJ - 'P . Iyl � oo 'P* (y) The identity (3.2) implies that the asymptoties of the local blow-up profile w* (y) are well reflected by the global blow-up profile u ( x , T) . The proof differs between the case w* = ±K and the case w' cf ±K. In the latter case, which we call a highly focused blow-up, (3.2) stems from the following estimate:
(y)
(3.3 )
lut (x, t) 1 < Clx l -
2p
p
for 0 < Ixl <
1
1'0,
to < t < T.
The proof of this estimate is omitted here. The assertion (3.3) can be easily proved by an ODE analysis; see Lemma A.2 of [29) .
3.2. Intersection number for type II blow-up. If a blow-up occurs at the origin, the number of intersections with the singular stationary solution .p* (r) (where 'P * ( y) = cp* ( l y l)) provides useful information. Let u ( x , t) = U(lxl. t) be a solution of ( 1 . 1 ) that blows up at t = T and let 'IlI0, T (y, 8) = Wo.T ( l y l , s ) be the corresponding rescaled solution. In what follows we simply write W(r, s) instead of WU.T (r, s ) . Given a function V(r) defined fOT T > 0, we denote by ZI (V) the number of zeros of V(1') that lie in an interval I C (0, 00). Then ZI (U(·, t) - cp*) denotes the number of intersections between the graphs of U (r, t) and .p* (1' ) in the region r E I. If n = B R , then Z(O. R) (U(-, t) - .p*) is finite for every 0 < t < T and is non-increasing in t. This follows from the result of Chen and Polacik [7] , which is a variation of the earlier result of Angenent [1] . See also [29, Subsection 2.3] . In the case where n = RN , we assume Z(0, 00 ) (U( -, 0) - cp*) < 00 for simplicity. Since Z(o. 00 ) (U(" t) - .p * ) is finite and non-increasing in t , the following limit exists: m(U) := limT ZI(U( · , t) - cp O ) , t...
where I = (0, R) if n = BR and J = (0, 00) if n = RN . In other words, ZI (U(" t) .p*) = m(U) for all t sufficiently close to T. It follows from [7] that the zeros of U(r, t) - .p*(1') are all simple when t is close to T, since otherwise t he number of zeros would drop further. Thus they are expressed by smooth functions (3.4)
(to < t < T) ,
395
BLOW-UP IN NONLINEAR HEAT EQUATIONS
where to is some number sufficiently close to T. One em,ily sees that lim infhT rl (t) = O. In fact, if this were not true, U Cr, t) must stay below
(3. 5)
rj (s) := CT - t ) - � rj (t) = es/2 rj (T - e-S )
for j = 1, 2, ' "
Now we define the number of vanishing intersections by (3.6)
mo t U )
:=
max { j 1 1 < j <
,
m (U ) .
m (U ) lim inf rj (t) = O } . t�T ,
Given an integer m > 1 , we use the term m-intersection blow-up to mean m = motU). The case motU) = 1 is called a single-intersection blow up Figure 1 shows an image of two-intersection blow-up. Clearly any self-similar solution satisfies mo t U ) = m(U), since rj (t) = OhlT - t ) for all j . -
.
'P*
� -
..:::._ ...
_ _
Ixl
FIGURE l . Two-intersection blow-up
THEOREM 3. 4 . Let p > P JL and let u(x, t) = U(lxl, t) be a solution of ( 1 . 1 ) that blows up al t = T. Suppose that the blow-up is of type II. Then moCU) 2:: m* Cp, N), where m*(p, N) ( > 2 ) is the "Morse index" of 'P' to be defined in (3.11) below. Now let us define the number m* (p, N). We linearize ( 1 .6) around 'P' and consider the following eigenvalue problem in the space H� : l I 1 1 p \ I I ( c· )pi:!..'ljJ - -y . "il7jJ 1p = - A1p ( 3.7) , 1p + 2 2 p 1 lyl where c· is as in (2.18), pry) = (47r )-N/�e-IYI2/4 is llli in (2.3) and -
H� = { v I
RN
P ( l "ilvI 2 + v2)dy <
The eigenfullctions are written (3.8) ,cw : = w" +
N-1 l'
1J!1
_
as
00 ,
v(y) is radially symmetric } .
'lj;(y) = w ( l y j) , where w (r ) satisfies
r i W 2
_
P
1
-
1
1J! +
p(C* )p-l 1' 2
W = -,,\1J!
for
l'
>
O.
HIROSHI MATA:-
396
It is shown in [22, 23) that the spectrum of (3.7) consists of countable eigenvalues Aj =
(3. 9) where
� p� +
1 +j
(j = 0, 1 , 2, · · · ) ,
-(N - 2) + l(N - 2)2 - 4pB 0' = ' B = p 2 1 (N - 2 - p:'1 ) ' (3.10) 2 Note Lhat 0' < - p:'l ' We denote by I)!j (r) the eigenfunction corresponding to Aj . It can bc shown that Z(O ,(X») ( lJtj) = j . Now we define: m* (p, N) : = max{j I Aj < O } + 1 ( = min{ j I Aj > O } ) , (3. 11) thaI. is, m*(p, N) is the number of negative eigenvalues of (3.8). Therefore it iH loosely interpreted as the Morse indcx of the Hingular st.ationary solution
_
_
. .
.
-->
s�oo
-->
Combining (3.13) and (3.14), we obt.ain mo (U)
>
mO.
o
397
BLOW-UP IN NONLINEAR HEAT EQUATIONS
4. Continuation beyond blow-up Some solutions of ( 1 . 1) can be continued beyond the blow-up time in a cerlain weak sense. In this section we discuss properties of such continuation.
4.1. Concept of Minimal Continuation.
Let us recall some basic notiollS
of continuation beyond blow-up for an equation of the following general form:
(4.1)
Ut
where
I( u)
=
�u + f(u) (x
n, t > 0) ,
E
u(x, O)
uu(x)
=
E
en
LOO,
is a monotone increasing positive smooth function and
in RN . \Ve begin with the concept of "proper ext.ension" due to
ular!
n is
= 0,
a domain
[3] .
DEFINITION 4.1 (Proper extension (minimal solution) ) . Let u be a solution of (4. 1 ) with Uo > 0 and T be its blow-up time. The proper extension of u beyond t = T is defined as the limit of an increasing seqllenm of approxima.ting solutions
11(.'1:, i)
where each
u", (x, t)
lim
=
m· ----t oo
'lLm (:!:, t) ,
is a classical solution of the approximating equation
u(x, O) = uo(x)
ul = �u + /m (u),
(under the Dirichlet boundary condition if n i= RN) , and 0 < h (u) < h (u) < Is (u) ::; . . . are a sequence of bounded, locally Lipschitz, monotone non-decreasing
functions converging to J ( 'lL ) locaUy uniformly in Since
'lL
>
um (x, t) is min { f (u ) , m}.
is bounded, the function
I", (u)
A typical example is
solution of the approximatiIlg equation, it satisfies
t > O.
'um ( . , t) =
et � ,
eM U.o +
Im(u)
=
(4.2)
we can let
globally defined for x Since each
10t e(t-T)� 1", (1l", ( - , T)) dT
In view of the monotonicity of the sequences the operator
O.
m --+
for 0 <
{um } , { 1m (um ) }
Um
t
is a classical
< 00 .
and the positivity of
00 in the above integral identity, to obtain
t
alue
Q
v
where the integral identity is understood in a generali:.led sense, allowing the + 00 . Moreover,
u
is the smallest element among all functions that satisfy (4.2) in
this generalized sense (see sequence o <
[3] ) . In
partir:l1iar, 'lL is independent of the choice
Next we set (4.3)
Te := sup{ tl
> 0 I u(x, t l ) < 00 for a.e. x
where u denotes the proper extension of 'U. Clearly from the indentity (4.2) , we have
DEFINITION
u(x, t)
T < Te
= 00 for every
4.2 (Complete blow-up) . Let
T = Te
T be
E
xEn
and every
t > Te.
the blow-up time of solution
incomplete if T
Te.
Ps i see
it is known that every blow-up is complete if
As regards 1('lL) =
P u ,
and
for
< 00. As one eilliily sees
complete
if
u
n },
We say that the blow-up is
[3] .
of thc
and is thus uniquely determined by UQ . It is clear that u = Sec [15] for a more general treatment of proper extension.
{1m}
t < T.
E n,
However, in the supercriticai range Ps
< P
<
< 00, there
incomplete blow-up, which is the main focus of this section.
are
u.
< p < cases of
1
HIROSHI MATANO
398
The following result ([26]) holds for a wide class of feu) satisfying a certain growth condition (e.g. feu) = uP and etl) provided that n is bounded and convex: Uo < Uo , Uo =f:- uo ;. T (iLo) > Tc(uo ) , (4.4)
wherc T(uo) and Tc(uo) denote, respectively, the blow-up time and the complete blow-up time (possibly 00) of a solution of (4. 1 ) with initial data uo. There is another approach to define continuation beyond blow-up based on the notion of limit L1 solutions. The following definition is a slightly modified version of what is found in [9]. In what follows n is a bounded domain in RN . DE�'INI'l'ION 4.3 (limit L 1 solution). By a limit Ll _solution on the interval o < t < T* « 00 ) we mean a function u(x, t ) that can be approximated by a sequence of classical solutions un (x, t) of (4. 1) in the following way:
UO,n
( 4.5)
Un( · , t)
(4.6)
f (un)
-->
-->
:=
Un c, 0)
ue , t)
fe u)
-->
as n -->
Uo in C(n)
in Ll (n)
for every
in L1 ( n x (0, t »
tE
00,
[0, T* ) ,
for every t E [O, T· ) .
We call a limit L l -solution a minimal L l _solution if it has an approximating se quence that is monotone increasing in n. By the strong comparison principle, we can show that a minimal L l solution is indeed the minimal element of all limit £1 solutions for a given initial data uo, hence it is independent of the choice of the approximat.ing sequence Un. Since each Un is a classical solution, it is easily seen that a limit £1 solution U satisfies (4.1) in the sense of distributions. Similarly, it is also a mild solution: u(·, t)
(4.7)
=
etA Uo +
t
°
e(t-r)A f(ue, T)) dT for ° < t < T* .
Consequently any limit L1 solution U belongs to C([O, T* ) , £1 (n» . It is clear that U = u for 0 < t < T(uo ) , where u denotes the classical solution of (4. 1 ) and T(uo) its blow-up time. Thus we may call u a limit £1 continuation of u. Note that, if UO ,n is increasing in n, then (4.6) follows automatically from (4.5) by virtue of the Kaplan estimate [25] and the monotone convergence theorem. The minimality of the proper extension u and (4.7) imply u < U for any limit £1 solution U. Hence T* < Tc (uo ) , since u(·, t ) < 00 for a.e x E n, t E [0, T* ) . Conversely, if UO,n < Uo (n = 1, 2, 3" , . ) the comparison principle yields Un < u, hence U < u. Thus the minimal L1 continuation U and the proper extension u coincide on the interval 0 ::; t < T* . Moreover, since (4.4) implies T(uo.,,) > Tc (uo), we can choose T* = T (uo) if U is minimal. If the solutions are radially symmetric, (2. 16) yields an estimaLe of the form ,
c
(4.8)
(
l un (x, t ) I < C lxl - p'l + (T* - t) - /l
)
for X E n, t E [bT* , T* )
for n = 1 , 2 , 3, . . . , where the constant C depends on T* , b but is independent of n . Thus the same estimate holds for a limit £ 1 solution U. Combining this estimate and the integral identity (4.7) , we see that U E C ( (0, T*) ; H1 (n) n U (!"!) for any q < N(p 1 )/2 ; see Proposition 2 .15 of [9]. Here is an example of a minimal L1 solution of ( 1 . 1) . Let n = RH, P > Ps and uA be the classical solution of ( 1 . 1) with initial data Uo = AV, where vex) = V(lxj) > ° is a given smooth function. Then, as is easily seen, there exists A * > 0 -
BLOW-UP IN NONLINEAR HEAT EQl1ATTONS
399
such that uA -> 0 as t -> 00 if A < A', while uA blows up in finite time if A > A· . Choosing a Requence Al < A2 < . -> A' and letting u : = lillln�oo 'uAn , we obtain a minimal Ll solution that is defined for all t 2:: O. We can show that u blows up in finite time and decays to 0 as t -> 00. The same is true for n = RN , provided that v E HI n Loo. See [30] for details. The above idea of constructing an unbounded global weak solution is due to [37] . Initially it was not known whether u blows up in finite time or remains smooth for all time. Later [15] confirmed blow-up for Ps < P < 1 + IV 6 10 , The upper restriction on p was removed in [34, 30] . 4.2. Speed of regularization after blow-up. In a joint work with M. Fila and P. Polacik [9] , the author has proved that any minimal £1 solution of ( 1 . 1) becomes classical immediately after the blow-up time, provided P s < P < P The paper abo deals with the equation Ut = �u + A e u with 3 < ]II < 9. The following theorem shows how fast regularization occurs after blow-up. It also extends the above result of [9] on (1.1) to all P > PB and all (possibly non-minimal) L l solutions under the as!:>umption that the blow-up is of type 1. .
.
JL '
THEOREM 4.4 (Type I regularization [30] ) . Let Ps < P < 00 and let u be a solution of ( 1 . 1 ) that blows up at t = T . Assume that the blow-up is of type I and let u be a limit Ll continuation of u defined on [0, T*) with T* > T . Then lim sup (t - T) /' llu( . , t) IILoc < 00.
(4.9)
t ",.T
OF PROOF. Since the blow-up is of type I, the local blow-up profile w* (y) = W* (lyl) is a bounded solution of ( 1 .6). Put 11 : = limr�oo W* (r)j<.l.>* (r). As mentioned in (3.3), we have 11 f. 1. Now suppose that (4.9) docs not hold. Then there exists a sequence tl > t2 > t 3 > . . . -> T such that O U TLINE
(tn - T) P':l l l u(- , tn ) IIL= -+ OO
Define
as
n -> oo .
1
un (X, t) := A:; ' u( A x, A n t + T), An = tn T. Then Un is a minimal L l solution and satisfies Ilun e 1 ) IILo< -> 00. Since a pointwise bound similar to (4.8) , we have, for some constant C > 0, -
en =
o )..;; - '
(
1
u satisfies
).
Thus, by parabolic estimates, we ean choose a subsequence of {un} converging to some function u outside the origin. This function u is again a minimal £ 1 solution of ( 1 . 1 ) and is defined for all x E RN \ {O}, t E R, since A"fL O. Clearly we have lu(x, t)1 < CixlMoreover it is easily seen that p
(4.10)
,
-+
l .
u(x, t )
=
(-t)- p':'r
w* (�)
for t < 0,
u(x , O )
=
IlCP* (x) .
By Remark 2.4 (no needle lemma for 11. ) , we see that lIu(', 1 8) 11£'", = 00, where 0 < 15 < 1 is arbitrary. Therefore u(x, t) is singular for every 0 < t < 1. Given 0 < Tl « 1, we rescale u as i n (2. 1 ) and denote it by WO,T, (y, s) . Since u is a limit of a sequence of classical solutioIlH in HI n Lq with q > p + 1, we see that the energy E(WO,T1 ( s» is nondecreasing in s, and that Lemma 2.3 (no-needle lemma) holds for WO,T, . Therefore limB�OO WO.Tl = cp' , since Wo,T, is unbounded. On the other hand, (4. 10) implies wo.r, (y , - log Tl ) = IlCP*. Thus we must have E(IlCP*) > E(cp* ), but this is impossible since E(llcp*) = (if ) J p(cp')P+ldy achieves its strict maximum at /.L = 1 . This contradiction proves the theorem. 0 -
-,
;:;:
HTR.OSHI MATANO
400
5. Type II blow-up and the braid group If a blow-up is of type TI, its exact blow-up rate is not easy to determine. This is because type II blow-up solutions, by definition, do not obey the standard scaling law. Herrero and Velazquez [22, 23] constructed examples of type II blow-up for the range p > pJL and computed their blow-up rates explicitly. However, the problem of determining all type II blow-up rates has long remained open. Recently the problem was partially solved by Mizoguchi [35], who proved that any type II rate coincides with one of the Herrero-Velazquez rates, provided p > PL : = 1 + N�1O ( > P.IL ) · The proof of [35J uses a three-step argument : first to show the validity of the eigenfunction expansion away from the origin by using integral estimates, second to give an upper bound for the growth rate at the origin by the zero-number argument, and finally to use a matching argument as in [22, 23J . In thiH section we show how the braid group theory can improve the result of [35J while significantly simplitying the proof at the same time. More precisely we extend the above result to the range p > P J L under a weaker atlsumption. Our proof is bll.'led on a remarkably simple topological argument and is less reliant on heavy technical estimates. This flection is based on the paper [28J . As mentioned in Introduction, a somewhat similar result has been obtained in [36] by partly borrowing the idea of [28J and partly using a different argument. For simplicity, hereafter we deal with only positive solutions.
5.1. Herrero-Velazquez solutions. RecaU that a blow-up is of type II if alld only if the rescaled solution w := 'IliO ,T ( Y , s ) of (2.2) converges to the singular
Htationary solution :p* (y) : = c* lyl -2/(p- 1) as s -> 00 (Theorem 3 . 1 ) . In a naive view, such solutions lie on the stable manifold of :p* , so an eigenfunction expansion around :p* may give a good first-order approximation. However, since :p* does not belong to the space where ( 1 . 1 ) is well-posed (such as Lq(l1) with q > N(p - l)j2), the standard linearization technique does not work. The idea of Herrero and Velazquez [22, 23J is based on a matched asymptotic method. They used an ansatz that the behavior of the rescaled solution w away from the origin is well described by the linearized equation, while its profile near the origin is approximated by a family of stationary solutions of (2.25). Assume P > PJL and let Aj (j = 0, 1, 2" , . ) be the eigenvalues given in (8.9), and 11<j ('I') be the corresponding eigenfunctions. Then it can be shown that (5.1 ) for some positive constant Cj , where a is the constant defined in (3. 10) . Next let <Pa (T) be the stationary solution of (2.25). Then <Pa (T) is increasing in a > 0 and (5.2)
<1'>U (T)
for some constant
= <1'>* ('1')
- k(a) T-In: + o(r- Ial )
'I'
->
00
k(a) > O. Now for each positive integer m, we set 17m
=
( Am la l -
2 p
_
1
) -1
'
It is shown in [22 , 23] that, for each Tn such that (5.3)
as
Am > 0
BLOW-UP IN NONLINEAR HEAT EQUATIONS
401
there existR a classical solution Wmer, s ) of (2.2) that is decreasing in r and possesses the following asymptotics for some positive constants K, <7 with <7 < 1/2:
e7=S
(5.4)
Z(O,oo) (Wm(- , s ) -
(5.5)
for r E [0, Ke - 7Jm S J ,
'1m S , e""s ] , e K r fo r E [ = m.
Here the parameter a is determined by the condition k(a) = em , with k (a) and em being the constants in (5.2) and (5.1) with j = m. This is a matching condition for the inner and outer asymptotics. The growth rate of Wm and the blow-up rate of the corresponding solution (5.6)
Um (r, t) := (T - t ) - P-' 1
arc thus given as follows, where (3 (5 . 7)
=
Wm ( (T - t)- 2 r, - log(T - t » 1
1 and C is some constant: � (Ial ) p 1 P 1 2
The existence of such a solution Wm has been proved rigorously by using a fixed point theorem; see also [33] , which restates the argument of [23] .
5.2. Braid group. The braid-group method for one-dimensional parabolic equations was introduced recently by Ghrist and Vandervorst [17] as generalization of the so-called zero-number argument that has been widely used as a powerful tool for qualitative analysis of various parabolic equations. While the standard zero number argument counts the number of intersections between the graphs of two solutions, the braid-group method of [17) keeps track of the entanglement among an arbitrary number of solutions. Let us briefly illut:)trate this approach. Given a C l function vex) on an interval [0, L], we define a continuous curve in R3 by ,[v]
(5.8)
:=
{ (x, v (x), v'(x» ; ° < x < L}.
Now let VI , 112 , . . . , Vk be C l functions on [0, L] whose graphs are nowhere mutually tangential. Then the curves , [VI ] , ,[112], ' . . , ,[11k] do not intersect with each other, hence they form a "braid" on k strands (see Figure 2). Naively, a braid is understood to be a topological type of mutually disjoint strands whose endpoints are aligned on two parallel lines. A precise defiuition will be given later. Hereafter we will expret:)s a braid by a "braid diagram" shown at the right end of Figure 2 .
o
L
�------�---. x
FIGURE
2 . Graphs and their braid expression (the case k = 2)
In what follows we only consider the case where k = 3. In this case, any braid diagram can be realized by joining the elements <71 . <72 , <711 , <7;1 shown in Figure 3
HIROSHI MATANO
402
in a suitable order. Thus it is expressed as a word of finite length composed of the letters 0'1 , 0'2 , 0' 1 1 , 0'; 1 Such a word is called a braid word. Note that thp- length of a braid word coincides with the total number of crossings that appear in the corresponding braid diagram. ,
x
X 0'1
/
X
-1 0'2
-1 0'1
0'2
FIGURE
X
3. Generators of braid words (the case k = 3)
A product of two braid words is defined by simply joining the two words (see
Figure 4). The set of all braid words form a free group with respect to this multi plication procedure.
J '....
-"'" _
_ _
FIGURE 4. Product of braid words
It is shown by Artin [2J that two braid words on k-strands are topologically equivalent if and only if they can be deformed to each other by using the relations
O'iO'j = O'jO'; ( I i - i l > 2) . O'iO'i+ 1 0'i = 0'i+10'i O'i+ 1 , In the present case where k = 3, the latter relation is meaningless, so they reduce to the following single relation (see Figure 5): (5.9)
By identifying equivalent braid words through the relation (5.9), we obtain a group 83 called the braid group, each of whose elements is called a braid. The unit element of 83 is denoted by 1, which is represented by a braid diagram with no crossing.
FIGURE 5. Artin's relat.ion: 0'1 0'2 0'1
=
0'20'1 0'2
Note that no negative crossings (namely 0'1 1 , 0'; 1 ) appear in the braid expres sion of graphs as in Figure 2. This is because ,,), [vJ in (5.8) is dp-fined in such a way that a curve with a larger derivative always come in front of a curve with a smaller derivative when they cross. A braid word without a negative crossing is called a positive braid word, and a braid that is represented by a positive braid word
BLOW-UP IN NONLINEAR HEAT EQUATIONS
403
is called a positive braid. We denote by Bt the set of all positive elements of B3 . This is a semigroup generated by 0"1 , 0"2 . A posi tive braid A E Bt may not have a unique expression by a positive braid word, but the length of the positive braid word representing A is uniquely determined. This is called the length of A and will be denoted by teA). Garside [16] has shown that if two positive braid words A, B are equivalent, then they are "positively equivalent" (see also Birman [5 , Theorem 2.4] ) . This means that A can be deformed to n through the relation (5.9) without using neg ative elements 0"1) , 0";) during the entire deformation process. Now we introduce the concept of rigidity for the convenience of later arguments. DEFINITION 5.1. A positive braid word is called rigid if it does not contain a segment of the form 0") 0"2 0"1 nor 0"20"]0"2. It is called non-rigid if it contains such a segment.. An element of Bt is called rigid (resp. non-rigirlJ if it is represented by a rigid (resp. non-rigid) positive braid word. 2 0" For example, O"f 0"1 O"f is rigid, but 1 0"2 O"f is non-rigid . It is clear from the above-mentioned result of Garside [16] that a rigid positive braid is expressed by a positive braid word in a unique way. The following lemma follows easily from the identities (0")0"20"1)0") = 0"2 (lT]O"20"1 ) and 0"1 (IT]O"20"d = (IT]O"2 lTJ ) 0"2 : LEMMA
A
5.3.
5.2. Let A E Bt be non-ridid. Then there exist C, D E Bt such that =
lT10"2 0"1 C
( = lT2lT1lT2C) , A = DCTI0"2CT1 ( = DCT20"1 0"2) '
Parabolic reduction of braids. \"Ie now explain how the braid group
theory comes into parabolic equations. We first follow the idea of [17] , then later explain our own development of the theory. I Given C functions VI (x), V2 (X) , V3(X) on [0, L], we denote by ,6[ Vb V2 , V3 ] the positive braid defined by the strands ,[VI] , , [V2] , ,[V3] as shown in Figure 2. Need less to say, ,a[ VI , V2, V3 ] is well-defined only if the graphs of v] (x) , vAx) , V3(X) are not tangential to each other. Let VI (x, t), V2 (x, t) , V3 (x, t) be solutions of a scalar parabolic equation Ut =
a(x)uxx + b(x)ux + f(x, u),
for x E (O, L) , t > 0 ,
where a(x), b(x), I(x, u) are smooth functions and a(x) > O. Assume that the boundary values Vl (0, t) , V2 (0, t), V3 (0, t) remain distinct for every 0 < t < tl , and that the same is true of VI (L, t) , 1)2 (L, t), 1)3 (L, t ) . Then it follows from the results on zero-number properties ([1, 7] ) that the number of intersections between the graph of Vi and that of Vj (i 1= j) is finite for every t E (0, ttl and that there is a discrete subset 8ij of (0, t1] (possibly empty) sueh that
(1) the intersections between Vi (', t) and Vj (" t) are all transverse if t E (0, t1 ] \ 8ij , hence they are locally constant; (2) the number of intersections between Vi and Vj strictly drops at every t E 8ij . From the property (1) above, thc braid ,6[ V1 (" t), V2 (" t), V3 (', t) ] is well-defined for every t E (0, l)] \ S, where 8 ;= 812 U S23 U 831 , Moreover, since the length of thc braid f3[ VI ( " t ) , V2 (', t), V3 (-, t) ] coincides with the total number of intersec tions among VI , V2 , V3 , the property (2) above implies that the length of the braid ,6[ Vl C , t), V2 ( - , t ) , V3 (', t) ] drops strictly at every t E S.
404
HIROSHI MATANO
Furthermore, a careful examination of the arguments in [1 , 7] shows that the dropping of the intersection number at each t E S occurs by deleting the term af or a{ from the braid word, as illustrated in Figure 6, or as a finite sequence of such procedures. Indeed even if many intersections cancel simultaneously, we can decompose such an event into a sequence of simple cancellations as shown in Figure 6. This observation leads to the following notion of "parabolic reduction" : ,
-
---
-
--
.... ' )0 , ... ••• •••• Cu2D I
FIGURE
<'-.. L.J
... >0 0< , ... . -
- - - -
_
...
... ..
. -
CD
-
-
_
-
- . -
--- - - _ ...
----
---
-
6 . Simple parabolic reduction (the dotted areas remain unchanged )
5.3 (Parabolic reduction) . We say that B E B:1" is a simple par abolic reduction of A E Bt if there exist C, D E Bt and i E {1, 2} such that A = Cal D, B = CD. We denote this relation by DEFINITION
A I>, B . We say that B E Bt is a parabolic reduction of A E Bt if there exist AI , . . . , Aj E Bt such that A 1>, AI l>, . . . 1>, Aj = B. We denote this relation by A I> B. The above reductioIl process itself is also called parabolic reduction. We use the notation A I> B to mean either A I> B or A = B . Note that A 1>, B implies e(A) -f(B) = 2 . Therefore, if A I> B, then £(A.) -feB) is a positive even integer. Now we apply the above idea to equation ( 1 04) in the (r, t) coorditanes:
N-1 -l p Ur + lUI U for r > 0, t > to. Ut = Urr + ( 5. 10) r Given a solution triple U1 , U2 , U3 , we consider a braid f3[ U1 ( - , t ) , U2 (·, t) , U:l (-, t) 1 on some interval fro, r1l with 0 < ro < rj . Then the following proposition holck
Let U1 , lh, U3 be solutions of (5. 10) defined for to < t < t1 and suppose that the values Uj(ro, t) (j = 1, 2, 3) remain mutually distinct for every t E [to, ttl and that the same is true of the values Uj (rl , t) (j = 1, 2, 3) . Then P ROPOSITION SA .
(5 . 1 1)
f3 [ Ul e-. to) , U2 ( -, to ) , [h ( - , to ) 1
I> f3[ UI ( · , t t), U2 (-, t t l , U3 ( - , t I ) J .
REMARK 5.5. By (5. 1 1 ) , the length £(f3[ Ud· , t), U2 ( - , t), U3 ( - , t) J ) is non-increas ing in t, hence it is a Lyapunov function for (5. 10) . However, (5. 1 1 ) contains far richer information on the dynamics of solutions than what e(f3) alone can tell us.
The following proposition provides a useful tool to check parabolic reducibility of braids. An outline of its proof will be given in Subsection 5.5.
PROPOSITION 5.6 ([28) ) . Let A, B E Bt . Then for any P conditions are equivalent:
(a)
A I> B
(b)
PA
I>
PB
(c )
AP
E
I>
Bt , the following
BP.
405
BLOW-UP IN NONLINEAR HEAT EQUATIONS
5.4. Main theorem and its proof. Let u(x, t) = U ( l x l , t) be a positive �olution of ( l . 1 ) that blows up at t = T. In the case where n = RiV , we as,mme: (5.12) Clearly the Herrero-Velazquez solutions (5.6) ( "HV-soluLions" for Hhort) satisfy this condition. IL can be shown that if p > P JL then (5.12) implies
T'j (t)
(5. 13)
-->
as
0
t
-->
T
(j = 1 , 2, · " , mo (U» ,
where rj is as in (3.4) and mo (U) is the number of vanishing intersections defined in (3.6) . See [28] for a proof based on a )lera-number argument . The following is the main theorem of this section. It states that the blow-up rate of a type II blow-up solution is uniquely determined by mu (U):
THBOREM 5.7. Let P JL < P < 00 and let U(lxl, t), U ( l x l , t) be positive solutions of ( 1 . 1 ) that blow up at t = T with a type II rate. If n = RiV, assume also (5. 12). Denote f,y W(lyl, s), W(lyl, s) the corresponding rescaled solutions of (2.2 ) . Then -
�
-
(i) if mo ( U) < mo(U) , there exist T > 0 and So > - log T such that
II W e , s) I I £O'" < I I W ( " s + T ) IIL =
(5. 14)
fo r s E [so , 00 ) ;
(ii) if mo(U) = mo ( U) there e.rist T > 0 and So > - log T such that -
,
(5. 15)
I I W e , s - T ) IIL= :S II W( · , s ) IIL= < II W ( · , s + T ) I I L= --
�
for s E [So, 00) .
Recall that if U ( r, t ) is a type II blow-up solution, then mo (U) 2: m* (p, N) by Theorem 3.4, hence Amo ( U ) > O. Therefore, if >"mo(U) '" 0, we see from (5.3) and (5.5) that there exists an HV-solution Urn (and its IeHcaled solution Wm ) with � mo (Um) = mo(U) . Putting W = Wm in (5.15), and considering that the growth rate of WTn is exponential, we see that II W ( s) II L� / I I WTn ( . , s) II L= remains bounded boLh from ahove and below. Thus we obtain the foHowing corollary: .,
COROLLARY 5.8. Assume that 0 is not an eigenvalue of (3.8) . Let u (x, t) : = U(lxl, t) be a positive solution of ( 1 . 1 ) that blows up at t = T with a type II blow-71p rate. If 0. = RN , ass'ume also (5. 12) . Then I IW(·, s)II L= and II U( ·, t) IILoo satisfy the same estimates as in (5.7) with m = mo (U). PROOF OF THEOREM 5 . 7 . Since the assertion (ii) follows from (i) by exchanging the roles of W, W, it suffices to prove (i) . Now for p,ach A > 0 , we define �
tP'- (r, t) := >.. /1 fj ( .,,0. r, T - ).. (T - t» . Then fj>. i� again a solution of ( 1 . 4) and it blows up at t = T with a type II rate. Note that
(5 . 16)
where y, s are as in (2 . 1 ) with a = 0. Therefore, proving (5. 14) is equivalent to showing the following estimate for some t o E [0, T) and a sufficiently small >.. > 0: for t
E
[to , T) .
By Lemma 5.10 below, we have II U(-' t) IIL� = U(O, t) for t sufficiently close to T, therefore what we have to show is the following estimate for some to E [0, T) :
(5.17)
U(O, t) < fj>.(O, t)
for t E [to, T) and 0 < >.. «
1.
HIROSHI MATANO
406
Choose to sufficiently close to T so that the number of intersections between U(r, t) and <1>* (r) remains constant in to ::; t < T as in (3.4). Then the intersections between U(r, t) and <1>* (r) are all transverse for every t E [to T) . As in (3.4) , we denote the zeros of U(r, t) - <1>*(r) by r1 (t) < r2(t) < . . . < rm(U) (t) . Similarly, the zeros of U(r, t) - <1>* (r) will be denoted by i\ (t) < T2 (t) < . . . < 1'm(U) (t) . By virtue of (5.13) , there exist ro, c > ° such that ,
(5. 18)
for t sufficiently close to T. (Here we understand that rmo ( U) -H (t) = 00 if mo (U) = m(U» . By redefining to if necessary, we may assume that (5.18) holds for all t > to · Then a simple comarison argument yields that, for some constant t5 > 0, (5. 19)
IU(ro, t)/<1>* (ro) - 1 1 > t5
for to < t < T. I
Since <1>* (r) is invariant under the transformation VCr) zeros of [f:" (r, t) - <1>* (T) are given by
'ft (t) < r� (t) < . . < _
f�(U) (t),
�
,\ P
"
V ( J>: r') , the
fJ (t) := v'>..- lrj (T - '\(T - t» .
Therefore, if ,\ is chosen sufficiently small, we have (5.20)
fJ(t) >
for mo(U) + 1 < j < m(U), to < t < T. -
TO
-
By Theorem 3.1, W(lyl, s ) -> <1>*(lyl) as S 00, hence the right side of (5.16) converges to (T - t)-/ 1 <1>* « T - t)-1/ 2 Ixl) = <1>* (Ixl) as '\ 0, Consequently -
-4
-4
(5.21)
fY' (r, to)
�
tfJ* (r)
in C1 «0, ro])
for 0 < ,\ «
1.
Since U(r, to) and <1>* (r) intersect mo (U) times transversely in the interval (O, ro] , we see from (5.21) that, for ,\ is sufficiently small, (5.22 )
Z(O,Toi (U( - , to) - iP'(- , to » ) = mo CU) .
One can also show that, if ,\ is chosen sufficiently small, then - A (T , t)/4.>* Cro ) - 1 1 < t5 for to < t < T. (5.23 ) IU O
This follows from [30, Lemma 4.6] , which is used t.o prove (3.2). Hereafter we fix to , ro , '\ such that (5. 1 8) to (5.23) all hold. Figure 7 gives a schematic description of this situation, where
407
BLOW-UP IN NONLINEAR HEAT EQUATIONS
r=O I
I 00
!
./
I I
,
I I
1\
V
V
f\
1\
\
r
"
- TO
I I
!
:I
�
FIGURE 7. A schematic graph at t
I I
,
=
u(-, to)
to (for mo (U) = mo (U'),")
=
6)
Consequently there is some t2 E (to, tl) such that U(O, t 2 ) = UA(O, t 2 ) ' This means that the graphs of U and UA become tangential at r = 0, hence they lose at least one intersection at this moment, by the result of [7]. 'Without loss of generality, we may assume that the intersections between U and UA are transverse at t = tl, since degeneracy occurs for at most finitely many values of t. Now we choose a sufficiently small e > ° such that the two braids O [ U( " t l ) , fjA ( . , tl), <1>*] and O [ U(" tl), fjA(., tl +e), <1>* ] are topologically equivalent. Since UA ( . , t + c ) blows up at t = T e , there is t3 E (tl ' T) such that U(O, t3) = UA (O, t3 + c ) . This means that an intersection is again lost at T = O. As t approaches T e , the mo (UA ) inten>ections between UA ( - , t + c) and <1>* are all swept away to the left end (i.e. r = 0). Thus, for an appropriate t* E (t3, T c), they all lie on the left side of the intersections between U and <1>*, as illustrated in Figure 8. , U(r, to) , U(T, to) , U (T, tl) -
-
-
--+)
r
r=U I I I
- • - - - • -
.
I ' ",
\ "
=
I I I
ro
-.�_I___+___,I_+--I----"\_-_+_....J,._I__4_1___4-_I_
. . . . .. .\ . . . ,' �
fjA (r, t* + e) .
I I
�
'---4----+---;- if). (
�------------�
<1>'
-,
t' + c)
U(" t')
I
,
FIGURE 8 . A schematic graph at t
=
t* (with trapped crossings)
As mentioned earlier, we lose at least one intersection between U and UA at r = ° as t passes t2 and at least one between U and U A (', t + c ) as t passes t3 . Since both losses occur at T = 0, we can interprct these losses as being "trapped" behind r = ° rather than being completely lost. This creates "trapped crossings" between the two strands, as indicated by the broken line at the left end of Figure 8. More intersections may be lost at r = 0, but the number is even since fjA(O, to) > U(O, to) and fjA (O, t* + e) > U(O, t*) . Therefore, we can reduce the number to two by an artificial parabolic reduction of the braid. Note also that U(r, to) and fjA (r, t* + c) may have other intersections in the right region of Figure 8, but we can again delete them by an artificial parabolic reduction. Thus a combination of the natural PDE based parabolic reduction and an additional artificial parabolic reduction convert the graph in Figure 7 to that in Figure 8.
HIROSHI MATANO
408
The braid diagrams corresponding to the graphs in Figures 7 and 8 are shown in Figure 9 ( for the case mo (U) = mo (U) = 6). They represent the braids -
A : = f3[ U (·, to), fPc-, to ) , 1>0 ] ,
B : = f3[ u(-, to), UA(-, t o + E) , 1>* ] .
To clarify the dependence of A , B on m = mo (U) , 'iT! mo (UA ) , we write them as A m,in , Em , in ' Their exrpessions differ depending on whether m, m are even or odd: 2 a) k oT, 2n (aT2 ay, 2n T (aT T = = A2k,2n A2k+l,2n B2k+l ,2n = �2�2n��2k+l ( 5 . 25 ) 2 k 2n+l 2n+ l (iT iT 2 k ( a) T T T aT, a) T A2k,2n+l A2k+l,2n+l , l 2k l 2 + 2n 2 + n k 2 tl 2 = T T, a l a a = a r B2k,2n+l B 2 k + 1,2n+ , =
v
=
.
v .
,
=
Here and in what follows the symbols a, r stand for 171 , 172 for notational simplicity. Conditions ( 5 . 1 9 ) (5.23) guarantee that no crossing between U and UA enten; nor exits from the right end as t varies in [to , to]. Thus, by Proposition 5 .4, we have A m,in t> Bm,in. However, this contradicts Lemma 5.11 below. This contradiction 0 disproves the hypothesis (5 . 24), establishing (5. 17) . The theorem is proved. ,
A B
FIGURE 9 . Corresponding braid diagrams (for t = to and t
=
to)
REMARK 5.9. The above idea of replacing UA (T, t l ) by U>' (r, tl + c) has its prototype in [35] ' where a result similar to but weaker than our Corollary 5.8 is proved by using the zero-munber arglUllent aided by more technical estimates. Our braid-group method gives a much more transparent argument. 5.lD. Let U be as in Theorem 5. 7. Then there exist ro > 0 and to E [0, T) such that Ur (r, t) < 0 for every 0 < r < ro, to < t < T. LEMMA
OUTLINE OF THE PROOF. This can be shown by counting the number of intersections between U(r, t) and the solutions of (1 .4) with initial data I\;(T ± E)0 where E is a small parameter. See [30, Remark 3. 15J for details. P
1
"
5 . 1 1 (Main lemma) . Let Am,in, Em,in be as in (5.25) with m < m. Then A m,in if Bm ,in ' LEMMA
PROOF. We first note that the following identities hold:
A2k-I,2n- l TaT = A2k_2,2n_l ar2 CTT, = B2k-l,2n-lrar B2k,2n B2k_2,2n _ l ar2 CTT. They can be derived by using the identities (ar2iT)T = T(ar2 a) , aj (Ta) = (rCT )Tj . A2k,2n
= A2k- l,2n Ta = B2k- l , 2nra
=
=
409
BLOW-UP IN NONLINEAR HEAT EQUATIONS
Thi� and Proposition 5.6 imply .: )- A2k-1 2n-1
I> B2k-l 2n- l
': ;- A2k-2 ,2n-l
I>
B2k-2 , 2n- l ·
Therefore, it suffice� to prove the lemma when m and m are both even integers. Moreover, since A2k,2nT(l 2 T = A2 k+ 2,2n and B2k, 2n T(l2 T I> B2k+2,2n , we have !
,
A2k,2n I> n2k,2n
)- A2k+2,2n
I>
HZk '· 2,2n .
Thus it suffices to consider the case where m m = 2n. Observe that T2 and (lT2 rr arc commutative, and that the braid T2 (1T2 (1 = T(lT(lTa commutes with every element of Bt . Consequently A2n,2n can be written as A2n,2n = (T2 aT2 a)" and it commutes with every element of Bt . Hence =
A2n,2nO"2 = (72 A2n�2n ':::::::' (J2 T2n (J"_2 (a2T 2 ) "-1 (1. j
Using tills observation, one can show that
A2n,2n(l = (I2 T2n aT2 A ' , B2n,2nO' = a2 TZn aT2 n' , A' (a2 T2 )n-1 , where B ' = T 2( n-1 l a2 By Proposition 5.6, A2n,2n I> B2n ,2n if and only if A' I> B' , so it, suffices to show that the latter does not hold. Observe that A' is written as a product of (12 and T 2 . This means that A' is rigid, and that its simple parabolic reduction has the same property. Consequently any parabolic reduction of A' i s achieved by �imply deleting the term (12 or T2 one after another without changing the order of letters. =
However, the number of the letter T on the left side of (I in 8' is 2(n 1), while no letter a in A' has that many T'S on its left side. This shows that B' is not a parabolic reduction of A' . This contradiction proves the lemma. 0 -
5.5. Proof of Proposition 5.6. We only give a brief outline. We start with the following lemma:
LEMMA 5.12. Let A, B E Bt and suppose that (lA 1>, aB. Then A PROOF. By
the
1>,
B.
a�sumption" there exist C, D E Bt such that
(lA = Cp2 D, aB = CD
where p = (I or p = T.
Case 1: Suppose that C is written as C = aCI for some C1 E Bt . Then A = C1P2 D, B = C1D, so the conclusion holds. Case 2: Suppose that C is non-rigid. Then by Lemma 5.2, the situation reduces to Case 1 , so the conclusion holds.
Case 3: Suppose that D is non-rigid. Then by Lemma 5.2, D is written as D = TaTDI for some Dl E Bt . Note that, by (5.9), we have p2 (TaT) = (TO'T)p2 , where p T if P = a and f; = a if p = T. Consequently, =
where
aA = C/(T(lT)Dl = C(TO'T)p2 D1 = C1 P2 D l ,
aB = GlD1 ,
C1 : = C(TaT). Since C1 is non-rigid, the situation reduces to Case 2.
Case 4: Suppose C, D are rigid, and that C = TG\ for some C1 E Bt . Then O'A = TC1p2 D,
aB = TCID.
Therefore, both a A, aB are non-rigid. Since TCI , D are bolh rigid, the non-rigidity of aA, O'B holds if and only if Cl = C2 pp, D = pD1 or C1 = C2 p, D = PpDl for
1 10
HIROSHI MATANO
some C2 , V 1 E Bt . In either case we can argue as in Case 3 above, and reduce the 0 situation to Case 2. This proves the lemma. Ot:TLINE OF THE PROOF OF PROPOSITION 5 . 6 . Since the assertions (a) � (b) and (a) � (c) are trivial, we only need to prove (b) � (a) and (c) � (a) . Let us show (b) � (a) ; the other assertion is essentially the same. It suffices to consider the case where P = (7 or P = T , since the general case follows by induction. \Vithout loss of generality we may assume P = (7 . Thus, what. we have to show is
(7A [> (7B �:. A [> B. We argue by induction on j := (f(A) - f(B))/2. (5.26)
[Step 1] If j
1 , the assertion is true by virtue of Lemma 5.12 above. [Step 2] Let the assertion be true for j = 1, . . . , m and set j = m + 1 . Then there exist A 1 , " . , Am E Bt such that =
(5.27)
Case 1 : Suppose that some Ai is written as Ai = (7Ai for some Ai E Bt. Then (7 A [> (7 Ai , (7 Ai [> (7B. The assumption of the induction argument then implies A [> Ai , Ai [> B, hence A [> B. Cillie 2: Suppose that no Ai can be written as A i = aA i . In view of Lemma 5.2, thi� means that every Ai is rigid and is written as Ai = TAi , i = 1, · . . , m. \Ve 0 can derive A I> B by induction on feB) . See [28] for details. -
-
-
-
-
-
6. Open problems A needle-like singularity in the critical case: As we have stated in Corollary 3.3, a needle-like singluarity does not appear at the time of blow-up if P > Ps ' The same is true in the suhcritieal range 1 < p < Vs ' However, the formal analysis of [11] indicates that a needle-like singularity may appear for sign-changing solutions if V = Vs ' Thus the critical case P = Ps seeIllS to be highly exceptional. So far, there is no rigorous study which fully explains this phenomenon. Uniqueness of limit L1 continuation: A limit L 1 continuation u of a solution u of (1.1) is defined as the limit of a sequence of approximating classical solutions Un �ati�fying (4. 5), (4.6). Limit L1 continuations form a subclass of "L 1 continuations" . The latter consist of all continuations of u beyond the blow-up time as an U solution. It is known that an L 1 continuation of a. given solution is not necessarily unique if Ps < P < P ( [8, 10] ) . However the question still remains open as to whether a limit L1 continuation of a given solution is unique or not. JL
Reversed blow-up profile: As shown in Theorem 4.4, an incomplete type I blow-up yields a type I regularization. One may then wonder if such a solution is asymptotically self-similar as t ',. T. This amounts to asking if the limit w· (y) := lim (t _ T) .�' u( Jt - T y, t) t'"T
always exists. The question is still open. The difficulty comes from the fact that the energy for the forward rescaled equation Ws = 6w + � y . 'Vw + P�1 W + l·iVIP-1W, where y = x/ Jt T, s = log(t T), is finite only for a. narrow class of funcitions. -
-
411
BLOW-UP IN NONLINEAR HEAT EQUATIONS
Non-radial problems: Because of lack of adequate a priori estemates, the nature of blow-up for equation ( 1 . 1) is largely unknown in the supercritical range without the assumption of radial symmetry. For example, it is known ([29, 30]) that no type II blow-up occurs for radially symmetric solutions in the range Ps < P < P J L ' but we do not know if the same is true for non-symmetric solutions. Even in the subcritical range 1 < P < P the situation is not so simple if the domain n is not convex. Thi� iH because the energy E(we, s ) ) may no longer be monotonically decreasing in s due to the contribution from the boundary of n. s '
Exponential nonlinearity: The equation (6. 1 )
is another important model equation for the study of blow-up. Intriguingly, the range 3 < N < 9 for (6. 1 ) looks much like the range Ps < P < P JL (the lower supercritical range) for (l.1), as far as the behavior of stationary solutions is con cerned. In fact, any radially symmetric stationary solution of (6. 1 ) on RN inter sects with the singular stationary solution log 2(�r22) infinitely many times, just as a (r ) does with * (r) (see Table 1 in Subsection 2.4) . In view of this, we are lead to speculate that every (radially symmetric) blow-up for (6.1) in the range 3 < N < 9 is of type 1. Here, a type I blow-up for equation (6.1) means that lim SuPt �T (u(x, t) + 10g(T t) < 00 . This question is still open. -
References
[I]
S. Angencnt,
The zero 8et of a solution of a parabo lic equatlOn, J. reine
angew. Math., 390
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Garside, The braid group and other groups, Quart. J. Math. Oxford Ser., 20, No. 78 ( 1 969) , 235-254. [17J R. Ghrist and R. Vandervorst, Scalar parabolic PDE's and braids, preprint [18] Y. Giga and R. V. Kohn , A.symptotically sdf-.•imilar blowup of semilinear heat equations, Comm. Pure Appl. Math. , 38 (1985), 297-319. [ 1 9] Y . Gig" and R.V. Kohn, Chamcterizing blowup using similarity variables, Indiana Univ. Math. J . , 36 (1987), 1-40. [20] Y . Giga and RV. Kohn, Nondp.generacy of blow up for semilinear heat eq'uulions, Comm. Pure Appl. Math . , 42 (1989), 845-884. [21J Y. Giga , S . Matsui and S. Sasayama, Blow up rate for semi/inear heat equation with sub critical nonlinearity, Indiana Univ. Math. J., 53 (2004) , 483-5 14. [22] M . A . Herrero and J . .J.L. Velazquez, Explosion de solutions des equations paraboliques umilineaires supercritique" C.R. Acad. Sci. Paris, t . 319 ( 1 994) , 141-145. [23] M.A. Herrero and J.J.L. Velazquez, .4. blow up result for scmilinear heat equations in the supercritical case, (1994) unpuhlished. [24] D.O. Joeeph and T.S. Lundgren, Quasi/inear' DiTichlet problems driven by positive source_., Arch. Rat. Mech. Anal. 49 (1973), 241-269. [25] S. Kaplan, On the growth of solutions of qua8ilinear parabolic equations, Comm. Pure Appl. Math. 16 ( 1963), 327-330. [26] A.A. Lacey and D. Tzanetis, Complete blow-up for a semilinear diffU8ion equatinn with a sufficienlly large initial cond,tion, IMA J. Appl. M at h . , 41 (1988), 207-215. [27] O.A. Ladyzhenskaya, V.A. Solonnikov, and N.N. Ural'ceva, Linear' and Qu asilinear Equation of Parabolic Type, Translations of M at hemati cal Monographs 23, AMS, 1968. [28J H . MaLano, Determining type II blow-up rates for nonlinear hmt equations via the braid Y',vup theo,'y, preprint. [29] H. Matano and F. Merle, On nonexistence of type II blow up for a supercritica/ nonlinear heat equation, Comm. Pure Appl. Math . , 57 (2004), 1494-1541. [30] H. MaLano and F. Merle , Classification of Type I and 7'iJpe II behaviors for a super'critical nonlinear heat equatio n, preprint. [31J J . Matos, Unfocused blow up solution, oj semilinear pambolic equations, Discrete and Con tinuous Dynamical Systems, 5 (1999), 905-928. [32] F. Merle and II. Zaag, Optimal estimates for blow-up rate and behavior for nonhnear heat equations, Comm. P ure Appl. Math., 5 1 (1998), 139-196. [33J N. Mizoguchi, Type II blowup for a semilinear heat equation, Adv. Different.ial Equat ions 9 (2004) , 1279-1316. [34] N , M izoguchi, Boundedness of global solutions for a supern,licaL heat equation and its application, Indiana U niv. Math. J . , 54 (200ri), 1047-1059. [35] N. Mizoguchi, Rale of type II blowup for a semilinear hea.t equation, preprint. [36] N. Mizoguchi, Blowup rate of type II and the braid group theory, preprint. [37] W.-M. Ni, P.E. Sacks and J. Tavantzis, On th e u'Y1nplotic behavior of solutions of certain quasilinear parabolic equati o ns, J. Differential Equations, 54 ( 1984), 97-120. [38] P. Quittner, A. priori bounds for global solutions of a semilincar parabolic problem, Acta Math. Univ. Comenian"", 68 (1999), 195-203. [39J J.J.L. Velazquez, Estimates on the (n - l) -dimensional lIausdorff mea.mre of the blow-up sct for a semilinear heat equati on, Indiana Univ. Math . .1., 42 (1993), 445-476. [40] F. Weissler, Single point blow-up for a semilinear initial value problem, J. Differential Equa tions, 55 (1984), 204-224. [16]
F.A.
'
GRADL.'<;I'E SCHOOL
OF
MATHEMATICAL SCIENCES, UNIVERSITY OF TOKYO, TOKYO
E-mail address: matano(Qms . u-tokyo . ac . jp
JAPAK
15�-89 14,
ContempOl-ary Math1:!IIlatic!:i Volume 446, 2007
Sobolev maps on manifolds: degree, approximation, lifting Petru Mironescu
To Haiin Brezis, wtth deep esteem and affection A ASTRACT.
space X
=
WS,P(}v!: N),
N
In this paper, we review some basic topologic al properties of the where AI and
are compact Riemannian manifold
without boundary. More specifically, we discuss the following questions: can one define a degree for maps in X? are smooth or not-far-from-being-smooth maps dense in X? can one lift § l -valued maps?
1 . Introduction Sobolev maps hetween manifolds appear naturally in different contexts: har monic maps, liquid crystals and the Ginzburg-Landau equation are some of them. In this survey, we describe what is known about three natural questions concerning maps u : 1'v1 --+ N in a Sobolev spac:e X = W',P(M; N); here, M and N are com pact Riemannian manifold without boundary. Question 1. Does u has a degree? Q uest ion 2. Can one approximate u with smooth N-valued maps? Question 3. If N = § l , can one write u = e"P , with 'P as smooth as u? Though these questions are motivated by applications, this paper is rather a quick introduction to research problems than applications oriented. Degree is discussed in Section 2. Sections 2.6 and 2.7 may be viewed as an introduction to the problem of describing the connected components of X : this is beyond the scope of this paper, and we send the reader to the beautiful papers [BL] and [HaL2] . Section 3 deals with lifting. Section 4 is about approximation with smooth maps. This matter is the source of additional research directions; some of them w:ill be addressed in Sections 4.3-4.4. We sometimes go beyond Sobolev spaces and investigate properties of VMO or BV maps. The aim of this paper is to give a quick overview of the existent literature. Therefore, proofs are rather sketchy; often, " hint" would have been more appro priate than " proof' . However, we hope that the main ideas are sufficiently clearly exposed. On the other hand, the bibliography is reduced to the strict minimum; it contains only the papers that were quoted in the text. 1991 Mathematics Subject Classification. Primary 46E35, 47Hll, 55M25. Key words and phrases. Sobolev spaces, compact manifolds, degree, lifting,
approximation.
©2007 American Mathematical Society
113
414
PI£TRU MIRONESCU
Part of the open problems is not new. Some of them are directly inspired by the surveys [BI], [B2J . Others raisp.d from useful discussions with Halm Brezis, to whom the author wants (,0 express his gratitude. 1.1. Notations. For 0 < S < 00 and 1 < p < 00, we equip the Sobolev space WB,p with the norm u 1-4 lIuliLP + lulw'.p . If s is an integer, lulw •.p = II Dsu llu' . If not, write s = m + (J" , with m integer and 0 < (J" < 1 . Then, in IRn or on an n-dimensional manifold, l u ltv
•.v
=
In IRn , the BMO norm is I l u l l B MO = l Iu l l£! + luIBMO , where
r> O .fB (x,r)
lulB MO = �up sup
lu(y) - u(z) ldy dz.
S(x,r) When IRn is replaced by an embedded manifold 1vI, averages are computed on B(x, r ) n M. In BV, we consider the norm lI ull BV = lI u ll£! + lulBV, where 111.IBV is the total variation of the measure Du. If N is a compact manifold embedded into �l and M. is any smooth manifold, then WS,P (M; N) = {11. : M -t �l ; 1J.(x) E N a. e. and lu l w',p < oo}. One defines similarly BV(M; N) and BMO(M; N) . §n is the unit sphere in IRn+ 1 . Bn is the unit ball in � n . On a compact n-dimensional manifold M embedded in an Euclidean space, a mollifier is a Coo-map (0, 00) x M x M 3 (E, X, y) Pe (x, y) such that supp Po (x, y)dy 1 and liD] p� 1I < CE-n-j . With somp. abuse p., (x, ·) c B(x, E) n M, x
1-4
=
of notations, we will denote
J 11.(y)pg (x, y) dy by u
*
Pe (x) .
2. Maps from §n into §n
2.1. Degree. In this part , X = ws ,p (§n ; §n) . The basic question we address
is: can one define a degree for maps in X? If we have in mind the way the topological degree is defined for continuous mapti, the natural strategy would be: Step 1. Prove that §n-valued smooth maps are dense in X; Step 2. Prove that the degree, init.ially defined for smooth maps, extends by density to the whole X . It turns out that Step 1 works. THEOREM 2. 1 . coo(§n; §n) is dense in X for each s and p.
PROOF. When sp > n or s = n and 11 = 1, X is embedded in Co , by the Sobolev embeddings. In t.his case, the proof is trivial: approximate 11. E X hy a smooth family 11.;; = 11. * Po' Then lug l 1 uniformly. For sufficiently small 10, take v£ = 11.£/111.£ 1 . It is easy to see that v£ 'U in X. When sp < n, this result is due to Escobedo [E] when s is not an integer; however, his proof adapts to the case where s is an integer. Here it is how it works: fix any point x in §n and consider, fur small E: > 0 , a smooth map 1r = 1rx ,e: : §n §n \ B(x, E) projecting the cap B(x, E) n §n onto its complement in §n. This can be done such that 1r =id outside B(x , 210) n §n and I I Dk 1r 1 l < Cc k . If 11. E X , then v = v"" , = 1r 0 11. is still in X and has the additional property that it can be appro ximated by §"-valued smooth maps. Indeed, v takes values into §n \ B(:r., E) , which -?
-t
-->
SOBOLEV MAPS ON MANIFOLDS: DEGREE, APPROXIMATION, LIFTING
415
is topoLogically like a ball. By composing v with an appropriate diffeomorphism, we are in the situation where the map is JRn-valued, and then approximation is standard. It remains to prove that, if we pick appropriately c � 0 and x = x(c) , then the corre,sponding v's converge to u. Here it is where the hypothesis sp < n comes into the picture: it implies that
r !lVX,E - ullf-v" p dx isn
--
0 as
f: �
O. This
leads immediately to the desired conclusion. When sp = n, the idea of the proof goes back to Schoen and Uhlenbeck [SUJ; their proof works for w 1 ,n (§n ; §n ) maps. The general case is due to Boutet de Monvel and Gabber (Appendix to [BGP]). The starting point is that a map u in X belongs to VMO (=vanishing mean oscillation=the closure of smooth maps in BMO). AnaxESn
lytically, this means that, with h (u) = sup we have
f
f
B(x,Ii)'lsn .
lim Io (u) = O.
(2 . 1 )
lu(y) - u(z) l dy dz,
B(x,5) 'lsn
O�O
(For a proof of the Sobolev embedding w s,p(lR!n) � VMO when sp = n , see, e. g., [BNI] , Section 1.2). Next, the key ingredient is that, for VMo(§n; §n) maps, we have (2 .2)
lu. 1
-->
1 uniformly as E
->
O.
Indeed, if y E B (x, c:) n §n , then lu(y) - 'u. (x) 1 < c 1- l u. ( x ) 1 =
B(x . )n§n ,
( l u (y ) l - I u£ (x ) l) dy <
cf'
l u (y) - u(z) l dz. Thus B(x,£) n§n
B(x,£)nSn
f
lu(y) -u(z) ldy dz,
B(x,.)n§n
and the last integral tends to 0 uniformly in x as c: O. For further use, we note that the above argument implies that, if u EVMo(§n; F), where F is any set, then (2.3)
dist (u. (x), F)
�
0
Once (2.2) is proved, one may proceed
->
, uniformly in x as
c: �
O.
as in the case of continuous maps.
0
Troubles come from Step 2. The following result essentially due to Brezis and Nirenberg [BNI] .
THEOREM 2.2. There is a degree (equivalently; the degree oJ smooth maps is continuous with respect to the ws,p -norm) in X if and only if sp > n .
PROOF . When sp > n or s = n and p = 1, we deal with continuous maps, so that we are done. When sp < n, we may construct a sequence of smooth maps of degree 1 converging in X to a constant; thi� implies that the degree of smooth maps does not pas� to the limits. The construction is the following: fix a smooth map v : JR." � §n, of degree one and equal to a constant P at infinity. Since sp < n, the scaled maps O. By stereographic projection, we v£(x) = v(x/c) tend to P in ws,p as c may transport these maps on §n and obtain a sequence of maps of degree 1 and converging to P in X. It remains to study the limiting case sp = n. Here, VMO comes again into the picture: maps in VMO(§" ; §n) do have a degree, i. e. the degree of smooth maps is continuous for BMO convergence. This is Theorem 1 in [BNIj. The argument there ->
416
PETRU MIRONESCU
applies to the case where §n is replaced by arbitrary compact oriented manifolds. We will present in the next section a different approach, tailored for §n-valued 0 m�.
maps. In this part, maps u are in VMo(§n; §n) . If u is smooth, then there are two ways of computing its de!!;ree:
2.2. Formulae for the degree of VMO
deg u
( 2.4 )
=
and
(2.5)
f'
§n
deg 7J, =
det(Du, u)
f
.
1l,,+1
Jac
v.
In the first formula, u is considered as an JRn+1-valued map, Du stands for the tangentiaJ jacubian matrix and the determinant is (n + 1 ) x (n + 1 ) . In the second one, v : Bn + 1 -4 Rn+ l is any Lipschitz extension of u, and Jac stands for the jacobian determinant. It tlJrns out that the second formula is "the right one" , in the sense that we may use it to define the degree even if u is merely VMO. In this case, one has to take a special extension 1J. The idea is borrowed from [BBM3] . Let u be the harmonic
u/l u l, if lui > 1/2 2u, if lui < 1/2
extension of u and set v =
(2.6)
lu(x) I
. When u EVMO, we have
1 uniformly as I x l
-4
-4
1.
This is proved in [BNII] , Appendix 3. Though the rigorous proof is delicate, the result is intuitively clear: if Pr is the Poisson kernel (so that u = u * Pr), then in some sense Pro is close to a mollifier P l -r, so that morally this result is similar to (2.2). This implies that the right-hand side of (2.5) is well-defined. Indeed, near §n , V is §ll-valued, and thus its jacobian determinant vanishes. On the other hand, far away from §n, v is Lipschitz. \Ve may now try to define deg u as the right-hand side of (2.5). Note that, when u is smooth, v is Lipschitz, so that we fall back to the classical degree. The following result completes the proof of Theorcm 2.2 and incidentally gives, for §n-valued maps, an alternative proof of Theorem 1 in [BNI] .
PROPOSITION 2 . 1 . The map u I->
f
Bn +l
Jac
v
is continuous in VMO(13n; 13" ) .
PROOF. Consider a. sequence {ud converging in VMO to some u. Then there is a fixed r < 1 such that the corresponding harmonic extensions satisfy IUk(x)1 > 1/2 and l u (x) 1 > 1/2 whenever Ixl > r . To prove thh;, it. suffices to check that the .
.
argument leading to (2.0) yields uniform estimates when applied to a convergent sequcnce. Thus, with obvious notations, we have
B(O,r)
since
Uk
-4
U in C1 (B(O, 'f'» .
J ac vk -->
1 ,.----,-,:--; IBn +l l
B (O,r)
Jac v = deg u,
o
SOBOLEV
MAPS ON MANIFOLDS: DEGREE, APPROXlMATION, LIFTING
417
2.3. Estimates for the degree. Here, X = ws,p (§n ; §n) and sp > n (so that the degree exiHts) . A natural question is whether it is possible to estimate this degree in terms of the WS'P-norm. Before giving the answer, let us consider the more familiar situation where maps are continuous or better. If 1.1. is merely continuous, then there is no possible estimate, since the sup norm of 1.1. is always 1, while its degree can be any integer; the salIle argument shows that there is no estimate for the degree of VMO maps. However, if 1.1. is slightly better, then there is a control; for example, if 1.1. is Holder continuous, then its degree is controlled by its Holder semi-norm ( this can be shown as in the proof of Theorem 2.3 below ) . In view of the Sobolev embeddings, we would thus expect the following: if sp > n , there is a control, while, if sp = n , there isn't. Surprisingly, the answer is [BBM3] THEOREM 2.3. If sp =
n,
then
I deg 1.1. 1 < C i u l fv" ,p · Consequently, if sp > n there is an estimate of the degree in terms of l u lw'.p •
(2.7)
Second assertion follows simply from the first one and Sobolev.
PROOF. The case s may assume p > 1 -
=
n and p
=
1 is easy to treat ( using
(2.4)), so that we
We start with a simple remark: if we know how to prove this result when s is small, then we know how to prove it for all s. This follows from the Gagliardo-Nirenberg type embedding l-V" P Loo '-> wr, q if sp = rq and 0 < r < s,
(2.8)
n
valid except when s is an integer, p = 1 and r > s 1 is not an integer. The case p = 1 being settled, it thus suffices to treat the case s < 1 . We rely on (2.5). Let, for x E §n , r = rx E (0, 1) be �he smallest p such that lu(tx ) 1 > 1/2 for t E (p, l). Thus the set where the jacobian of v does not vanish is contained in U = {px ; x E §n , p < rx } . Since u is the harmonic extension of a map of modulus 1 , we have 1 Du(y) 1 < C(l - l y l ) - l and thus l Jac v(y) 1 < C(l - I y l ) -(n+l). -
Integration of this inequality over U yields I deg 1.1.1 < Ix denoting the 1 /2 Thus
<
segment
lu(x) - u(rxx)1
G
connecting x to rxx, we have either rx
<
§n
=
0 or
( 1 - rx) n/P l ulcn/p (Ix) < G(l - rxt/P I U I W (n+l)/p,P (I.) .
(2.9)
Integrating this inequality yields, with the help of trace theory, that I deg 1.1. 1 < Ci ulfv,.p + Gf • Finally it is easy to get rid of Gf by noting that, when the semi norm of 1.1. is small, 1t is close in VMO to a constant, and thus its degree vanishes. 0 ,
There is a challenging question concerning estimate (2.7) . We state it for §1 , though it also makes sense for §n . OPEN P ROBLEM L Let 1 < p < 00 . Set u(x ) - u (y) IP . l P 1/P' P(51 ) ; 1 2 dx dy = lu lw G = mf{ 1 = 5' §1 1X - Y Is C attained? Which are the optimal functions?
ll
4 18
PETRU MIRONESCl:
Note that C > O. The answer is easy to give when p = 2 (and unknown when p =1= 2). Indeed, when p = 2 and 'u = L anetn9, an easy computation yields
I = 41l"2 L Inl l an l 2 , while deg u = L nlan l 2 (see Section 2.4). Thus I > 41l"2. Equality requires an = 0 when n < O. Thus u is the boundary value of a holomor phic map. It is easy to see that the holomorphic maps in the unit disk that are of modulus 1 and degree 1 on the unit circle are precisely the Moebius transforms. Thus C = 41l"2 and optimal maps are Moebius transforms. Note that this problem lacks of compactness, which explains why it may be dif ficult to handle: if {ud is a minimizing sequence, then up to some subsequence {U k } converges weakly in W 1/p,p to some u. However, weak convergence does not e 'kO , if 0 < () < 11k , guarantees that the degree of u is 1 (take, e. g., u k (e,9) = otherwise 1, which has degree 1 and weakly converges to the constant 1 ) . Thus it is unclear whether the infimum is at,tained.
2.4. Maps from §l into §1 : an explicit degree formula. Throughout this n § § section, u : 1 1 . If u is smooth and u = L anet 9, then (2.5) reads --->
(2.10)
deg u =
L nlan l 2 ;
this was first noted by Brezis. On the other hand, as we saw right after Open Problem 1 , lu l�'/2 L Inl l an l 2 . Using this remark and the continuity of the '"
degree in H1/2 (Theorem 2.2), we find
PROPOSITION 2.2. Degree formula (2, 1 0) is valid when u E H 1 /2 (§ I ; § l ) . Consequently, (2. 1 0) holds if u E w1/p,p for 1 < p < 2.
Last statement is simply a consequence of the Gagliardo-Nirenberg inequality. In a survey paper [BI], Brczis asked several challenging questions about formula (2. 10). Question 1 : since degree makes sense when u is merely continuous (or even VMO ) , can one give a meaning to the right-hand side of (2.10) in order to recover deg u from lanl? Soon after, this answer revealed to be (presumably) negative (Korevaar [Ko]). This suggested some more "modest" questions. Question 2: if u = L a em9 , v = L b",emo are continuous maps such that l anl = I bnl , is it true that deg u = deg v? Answer: no ( Bourgain and Kozma) . Question 3: same as Question 2 if u, v E W1/p,p and p > 2. This is partially open. Back to Question 1 : the absolute convergence of the series L nlanl2 is equivalent to n
u E H 1/2 Nevertheless, one may still hope give a meaning to its sum. Commonly k used summation procedures consist in taking either S = lim '""' nlanl2 or T = k---1' OCl � k .
-
lim '""' rlnlnlanl2. Korevaar's result concerns these two procedures.
r-+l- �
THEOREM 2.4. If U E CO (§1 ; § 1 ) then S even not exist. ,
or
T could be any real number, or
The proof is explicit: given a E JR, Korevaar exhibits a map u such that S = a (or T a). Of course, one may imagine some other summation procedure, but Korevaar's r:Dnstruction will probably take care of it. =
SOBOLEV MAPS ON MANIFOLDS: DEGREE, APPROXIMATION, LIFTING
419
Concerning Question 3, the first answer was obtained for Holder maps in an un published work of Kahane. In the setting of Sobolev spaces, Brezis [B2] proved the following variant of Kahane's result.
THEOREM 2.5. If U E W1/3,3, then deg u = lim
(2.11)
£-->0
L
sin2 nE . la l2 n nE2
n,oO Consequently, the answer to Question 3 is positive when p < 3.
PROOF. \Ve assume u continuous; the general ease requires some more subtle consideration on lifting, developed in Section 3. Write u = ZdelW, where d =deg u and 1f; is continuous. It is easy to see that t/J E W1/3,3. The starting point is the
identity 2� u(e'(O+h) u(e ,o)d B = 21T L l an l2 sin nh 1m
1. o
2rr =
sin(dh+1f;(e, (/!+h) -1)!(e ,O» dB.
0
A second order Taylor expansion of sin(dh + 1f;(e'(8+h) - 1f; (e,e» yields
L l an l2 si
nn
h - dh
::;
Cl hl 2 + C
o
� 2
11f;(e'(8+h) - 1f;(e'/!)13 dB .
Integrating this inequality over h E (0, 210) and dividing the result by 2102 lead to sin2 nE C l an l 2 - d < CE + L nE2 102 n,oO
12 12rr 11)!(e'(8+h) £
0
0
'
- t/J(e,O) 1 3dO dh.
Using the fact that 1f; E W1/S,s , it is easy to see that the right-hand side of the 0 above inequality tends to 0 with E. In some sense, the above result is optimal: Kahane [Ka] proved that, if u E WI/p,p for some p > 3, then the limit in (2. 1 1 ) may be any real number. However, this still leaves the following
OPEN PROBLEM 2. For p > 3, can one compute the degree of u W I /p,p in terms of I an I ?
=
L aneme E
2.5. Another degree. As we saw in Sections 2. 1-2.2, one may prove existence of a degree first by establishing density of smooth maps, next by using (2.5) . Yet there is another natural way to do it: assume that the integrals in (2.4) or (2.5)
make sense, take this as the definition of the degree, and then prove that the result is an integer. This approach was taken by Esteban and MUller [EM] . In what follows, it is convenient to consider u not as a map from sn into lRn+l, but rather as an Sn-valued map. With tills in mind, the jacobian matrix of u is n x n, and (2.4) rewrites (2. 12)
deg u
=
Jac u.
THEOREM 2.6. Assume that u E wl,n-l (sn; §n ) is such that all the (n - 1 ) x (n - 1) minors of its jacobian matrix are in Ln/(n - 1) . Then the right-hand side of (2. 12) is an integer.
420
PETRU MIRONESCU
Note that the hypotheses imply that Jac u E LI . The argument relies essen tially on the area formula of Federer: if Jac u E Ll , then there is an integer-valued f 0 u(x) Jac u(x)dx = L l -function d on §n such that f (y)d(y)dy whenever
f : §n
�
J
� is smooth. The theorem amounts then to proving that d is constant.
The VMO degree and the degree defined in the above theorem are not related: if u E VMO, we need not have u E w 1 ,n- J . Conversely, a map that satisfies the assump tions ofthe theorem need not belong to VMO: pick a map 'l/' E W l, 1 (§2 ; ( - 1 /2, 1/2» which does not belong to VMO, and set u = (1/J, /1 _ 1/J2 , 0), which is §2-valued. Then the first order minors of u are in I} , since 1/J E W l l , while its jacobian de terminant vanishes, since u is §l-valued. Clearly, u does not belong to VMO. This leaves us with the following very vague question. ,
OPEN PROBLEM 3. Is there a "unified" degree theory?
The above theorem was generalized by Giaquinta, Modica and Soucek [GMSl]. Vie do not quote here their result, which needs notions of cartesian currents to be stated.
2.6. Degree beyond VMO. Let u be a map from §n x (0, 1) k into §n If u is
continuous, then one may define a degree of u as follows: fix any A E (0, l)k and set deg u =deg u(·, A). By homotopical invariance of the degree, this definition does not depend on A and yields a degree which is continuous for the sup norm. The same can be done if u E ws,P, with sp > n + k. Indeed, by Gagliardo-Nirenberg we may assume s < 1. By trace theory, >. u(·, A) is continuous from (0, 1/ into ws- k /p,p, and thu� into VMO. The degree being continuous for the BMO norm, we derive that deg u ( · , >. ) does not depend on A. Thus we may define a degree in WS,P if ws,p embeds into VMO (which is t.he same as sp > n + k). It turns out that the condition sp > n + k can be relaxed. In special cases, the following result was obtained by White [W] ; see also Rubinstein and Sternberg [RSJ . The general case is taken from [BLMNJ. ......
2.7. A ssume that sp > n + 1 . Let u E ws,p (§n x (0, 1) k ; §n ) . Then there is an integer d such that deg u(-, A) = d for a. e. A E (0, l)k. Thus, we may define the degree of u as this integer. In addition, the condition sp > n + 1 is optimal and the degree is continuous for the W·,p-norm. THEOREM
=
1 is settled by the discussion at the beginning at this section. Assume thus k > 2. Note that, for a. e. >., u ( · , >.) E WS,P cVMO, so that the map 1/J given by 'rjJ(A) =deg u(·, A) is defined a. e. With some work, olle may prove that 'r/J is measurable. For a. e. t l , . . . , ti- l , t i+ 1 , ' " tk E (0, 1), the map u( . , t l , " " ti - 1 , ' , tHI , · . t k ) is in W" p . If this is the case, then 1/J(tI , . . . , ti-1 , ' , tHJ, ' " t k ) is constant a. e. (cf the case k = 1). Existence of the degrcc follows from the following elementary
PROOF. The case k
.
1/J : (0, l)k � lR be a measurable function such that for a. e. t k ) is constant a. tl , . . . , ti- l , ti +l > ' " tk E (0, 1 ) , the map 1/J(t1 , . . . , ti- I , ', ti + l , e. Then 1/J is constant a. e. LEMMA 2.1 .
1,et
.
.
·
To prove continuity of degree, let Uk � u in W s ,p . Possibly after passing to a u ( - ' >') in W8,p for a. e. A. We conclude subsequence, we then have Uk (', A) -+
SOBOLEV MAPS ON MANIFOLDS: DEGREE, APPROXIMATION, LIFTING
421
using continuity of the degree in VMO. Incidentally, this proves continuity for the Wr,q-norm as soon as rq > n. To prove optimality, let k = 1 , e E §n and u(x, A) = (x - 2Ae)/lx - 2Ae l . Then 1 , if oX < 1/2 . u E ws,p if sp < n + 1, while deg u ( , A) = D 0, if oX > 1 /2 •
2,7. General manifolds. One may define the Brouwer degree for continuous maps from M into N, provided these manifolds have same dimension, are compact, oriented and without boundary. This degree may be extended to VMO maps as follows: density of smooth maps in VMO(M; N) follows from (2.3). Existence of degree was proved by Brezis and Nirenberg [BNI] . THEOREM 2.8. The map GOO (M; N)
norm.
:1 7L
>->
deg
n
is ClJntinnous for the BMO
Their proof does not use integral formulae for the degree; yet, this can probably be done this way. The above result allows to state, e. g., Theorem 2.7 for maps from M x (0, I)k into N. It is also plausible that Theorems 2.3 and 2.6 are still valid for maps from M into N. For a map u from §n into itself one of the interests of the degree is that it describes the homotopy class of n : HopE's theorem asserts that if two continuous maps have the same degree, then they are humot.opic; t h is holds also for VMO [BNI] . In general, it is more natural to replace the degree with the homotopy class. This can be done indeed: one can associate to V MO maps a homotopy class, which is continuous with respect to BMO convergence ( this is obtained by copying the proof of Theorem 1 in [BNI]) . By mimicking the proof of Theorem 2 .7, one may prove the following result, essentially dne to White [W] ( see also [HaL 2]) : ,
THEOREM 2.9. Let
x
(0, 1)k; N), with M, N compact (but not necessarily of the same dimension) and sp >dim M + 1 . Let, for A E (0, I)k, d(A) be the homotopy class of u(· , A) in G(M; N) (this is well-defined when n ( · , A) E W',P, thus a. e.). Then d is constant a. e. We may thus define in this way [u] , the homotopy class of u . In addition, u >-> [u] is continuous for the WS'P-convergence. U
E
WS,P(M
3. §l-valued maps: lifting The question we address here is: given an §l -valued map u, can one find a lifting of u as smooth as u? In thc well-known case of continuous maps on domains in ]Rn , the answer is positive locally (i. e. on balls) , while globally the answer may be negative, due to the topology of the domain. In the context of Sobolev spaces, the local problem is already interesting. We aHHume throughout this section that u : C §l , where G is the unit cube in ]Rn . ---+
3.1. Lifting of Sobolev maps. Here u E W',P, and we look for a real function 'P E WS,p such that u = et'P. We start with the question of the uniqueness: if 'Pi , 'P2 E W" p lift u, is it true that 'Pi - 'P2 is a constant multiple of 27r7 This amounts to proving that, if 'P : G Z is in �Vs ,P , then 'P is constant . ,
---+
422
PETRU MIRONESCU PROPOSITION 3 . 1 . The only functions in WS,P( C; Z) are constants if and only
if sp > 1 .
In special cases, this was proved by Hardt, Kinderlehrer and Lin [HKL2] and Bethuel and Demengel [BD] . The general case is from [BLMN].
PROOF. Let Q <;; B be a cube. If sp < 1, then the characteristic function of Q belongs to WS,P ( C; Z); this proves the necessity of the condition sp > 1 . Suppose now sp > 1 . Fix some i E { I , . . . , n} . For almost every E (0, 1), j of i, the map v given by vet) t, belongs to WS,P«O, 1 ) ; Z). If we mollify v , implies that the maps VE are close to Z; since these maps are smooth, they have to be close to a fixed integer when e: is small. Passing to the limits, we find that is constant a.e. Lemma 2.1 implies that 'U is constant a. e. D
Xj
= u(xt, . . . ,Xi-!, Xi+b . . . , xn)
(2.3) v
We next give an example of map with no lifting. Let : B2 §I , which belongs to WI,I. We claim that TL has no lifting in Argue by with r.p E For a. e. l' E (0, 1), restricted to the contradiction: circle C(O, 1' ) has a continuous representative, still denoted r.p, such that everywhere on C(O, 1') . This is impossible, since on any such circle the winding number of is 1 , while the one of e'CP is 0. In general, the question of existence of r.p was settled in [BBM1].
u
z/Izl,
u = e'CP,
WI,I.
u(z) =
-->
WI,I.
r.p
u = e'CP
u
a
3 . 1 . There is 2 and sp E [1, 2)
a) n ::::: or b) n > 2, s < 1 and sp E [2 , n). THEOREM
lifting r.p E WS,P for each
'u
E WS,P except when:
PROOF. Several cases are to be considered:
(i) When sp > n, is continuous. We then take r.p to be any continuous lifting of u; it is easy to see that r.p E W" p . (ii) When s > 1 and sp > 2, the idea of the proof goes back to a paper of Carbou [C] . Assume that r.p exists. Since u = e''P , we find that so that here it is where the hypothesis s > 1 plays a role. with The idea is then to solve the equation Dr.p and to prove that the solution is essentially the one needed. One may prove that E ws-I,,,; tills relics on multiplication properties of Sobolev spaces. On the other hand, is a closed then vector field, i. e., Formally, this is clear: if = = + Therefore (at least if is smooth) =
u
Dr.p = F,
Du = wDr.p,
F = -zuDu;
=
F,
BFi/Bxj BFj/ox;. F u I Du2 - U2 DuI. u (3. 1) of;jBxj - OFj/OXi = 2(out/OXjOU2/0Xi
F
F u U I W2 ,
-
out /8x/Jud8xj ) .
The rigorous justification of this equality is obtained by approximating u with smooth maps and requires sp > Next the right-hand side of (3. 1 ) vanishes. Indeed, is §l -valued, so that DU l and are collinear. A variant of Poincare's lemma allows then to write for some 'lj; E Finally, the map ue-'.p is constant (it is easy to check that its gradip.nt vanishes) , so that + C, with C appropriut,e constant, is a lifting of (iii) The case sp = n. In order to keep the presentation simple, we consider the special case of H 1 /2 maps on an interval; the general case follows the same lines. set v(x, e:) We regularize By trace theory, v E Hl ( (O, 1)2). Since EVMO, v has modulus close to 1 for small c . Thus w = is HI and of modulus 1 in (0, 1 ) x (0, 8) for small 8. In view of case (ii) , we may write
2.
DU2
u
W8,p.
u.
r.p = t/J
u:
U
F = D'lj;
= u * Pe:(x).
v/lvl
w = e'.p,
SOROLEV MAPS ON MANIFOLDS: DEGREE, APPROXIMATION, LIFTING
with 1jJ E HI . By taking traces, we have U = e''P , with cp =tr 1jJ E H l /2 . ( iv) Cases a) or b) . Explicit examples show non existence. (v) The case sp < 1. It is the delicate one, and we refer to [BBMI] for details.
423 D
3.2. Estimates for the lifting. Once existence of lifting is established, the The proof natural question is whether we may estimate Icpl w in terms of I nlw of the above theorem is constructive, and yields estimates except when we are in the critical case sp = n. Actually, in this case there is no estimate. Here is an example. Let n = 1 ( so that C = (0, 1)) and let 1 < p < 00. Let cpdx) = if 0 < x < 1 /2 0, 2k1r(x - 1/2), if 1/2 < x < 1/2 + l/k and set Uk = e''I'k . It is easy to see that otherwise 211" , {Uk} is bounded in W1 /p,p . Since CPk helongs to W l /p ,p, any lifting of Uk in W I /p,p is CPk + C, by Proposition 3.1. It is easy to see that { lcpklw1 /p,p } is not bounded. Thus, in this limiting case there is no control of 'P in terms of u . However, the above CPk'S are bounded in W1, 1 . This suggests that there is a control of the phase, if not in Wl /p,p, then in a space containing WI /p,p and WI,I. For some values of p, this was proved by Bourgain and Brezis [BB] . • .p .
• .p
as
THEOREM 3.2. Let 1 < p < 2. Then each u E W I /p,P ( (O, 1); §1) may be wr'itten
(3 2 ) U = e C'P l + 'P 2 )
where I CP1 1w1,1 < C I'IL I �! 1 I1'" , and I cpz l}Vl/P,P < C lulw1Ip,p · More generally, one may replace (0, 1) by (0, l) n , but then U has to be in the clo,g7J,re .
'
,
of §l-valued smooth maps.
Their result is stated only for p = 2, but the proof works also when 1 < p < 2.
P ROOF . The construction in [BB] is explicit, but not elementary; it relies on a Littlewood-Paley decomposition of u . In one dimension, it is easy to establish a weaker form of (3.2), namely U = e''P, with I cp l w1,1+W1 /P,p S C ( l ul w / p p + l ul�r1;p,p ) · For simplicity, we work only with p = 2 and u E Coo, though this is not relevant. Let cP be any smooth lifting of u . It suffices to prove that I cp'IL '+ H -l / " < C ( lulHl / 2 + lul�'/ 2 )' By duality, this amounts to proving
(3.3)
(3. 4)
1
1 1 cp'(
Using the identity
11
cp'(
S
' cP
,
C (luIHl/2 + lul�'/2) ( lI ( I I L'>O + 1(IHl/2 ),
=
=
'
-lUU ,
E CO' (O, 1 ) .
integration by parts shows that,
1 u' (u() 1
'<:/ (
<
lu'IH -l/2 Iu(I H1/ 2
=
luIH 1/2 Iu(I Hl/ 2 ,
and (3.4) follows from the elementary inequality lu(I Hl/2 < I n I Hl/2 11(II L� + 1(1 H'/2. Actually, one can carry out this computation in any dimension, and find an estimate for Dcp in Ll + H - 1/2 However, in n > 2 dimensions, it is unclear whether D Dcp E Ll + H-1 /2 implies cP E W 1 , 1 + H l/2! OPEN PROBLEM 4. Is it true that Theorem 3. 2 is still valid for p > 2 ?
This is not known even in one dimension. An estimate weaker then (3,2) was established in [BBM3] , Theorem 0 . 1 .
424
PETRU MlRONESCU 3.3. Lifting of VMO maps. This was settled in [BNI] .
THEOREM 3.3. Each u EVMO(C; §l) has a lifting
constants.
'P EVMO, unique modulo
PROOF. Uniqueness comes from the fact that integer-valued VMO maps are constant, cf proof of Proposition 3 . 1 . Concerning existence, the idea is to regularize u. l:
that
'P(x) = lim 'Pe(x) exists a. e. and satisfies u = e''I'. It remains to check that .-->0 'P EVMO, and the idea is to prove that the family {'Pe } is relatively compact in VMO. To explain how this is obtained, we replace for simplicity C by §n . With the notations in (2.1), set Mo (u) = sup Io (u) . The necessary and sufficient condition
)..<0
for relatively compactness in VMO is lim sup Mo ('Pe)
Now Taylor's formula yields
o-o e <'o
=
0; see [BNI] .
Thus
M6 ('PE) < Mo (ve) + 1j2 sup sup ),,< 8 xE§n
and the John-Nirenberg inequality implies that the last quantity in the above for mula is less than CMJ ('Pe) ' This leads to
MO ('PE) < MO (vE) + C Ml ('PE) ' This implies easily that, for small 6, we have M8 ('PE) < CM8 (Ve). Compactness follows then from the fact that the last supremum tends to 0 with 6, uniformly in c, since {v,} is compact in VMO . 0 A similar argllment gives a proof (different from the original one) of the follow ing result of Coifrnan and Meyer [eM] .
THEOREM 3.4. There is some 6 > 0 such that, if u EBMO( C; §1 ) and l u lmvlO <
6, then
u = e''I', with I 'PIBMO < C iUII3MO .
Note that a BMO map always has a BMO lifting: for example, its main ar gument is in BMO. The point in this theorem is that, if u is close (in BMO) to a constant, then the phase 'P is controlled in BMO by u.
3.4. Lifting in BV. When u EBV(!1; §l ) , u has a lifting in BV (Giaquinta, Modica and Soucek [GMS2]); here, n could be any smooth domain or manifold.
This result was sharpened by Davila and Ignat [DI] .
THEOREM 3.5. For u EBV(!1; §1) , there is some 'P EBV such that u
(3.5) The constant 2 is optimal.
=
e'
42 5
SOBOLEV MAPS O N MANIFOLDS: DEGREE, APPROXIMATION, LIFTING
The constant 2 cannot be improved. Indeed, let lJ, be the identity on §l and let 'P be a BV lifting of u. Then 'P has to jump somewhere, say at 1 . Since u = e"P a. e., this implies that 1'P(1 + ) - 'P ( l-) I > 2'iT. On the other hand, the variation of 'P on § 1 \ {I} is at least r I Dlll = 2'iT. Consequently, the total variation PROOF.
ls'
of 'P is at least 4'iT, while the one of u i8 2'iT. Concerning (3.5) , we give a proof when u is smooth; this case is of interest only when n is not simply connected, for otherwise we may write u = e''P with 'P smooth and ID'PI = IDul . Let arg be the principal argument (which hat> a jump at 7r) and set, for a E [0, 27r) , Bo (z) = arg( ze'(1f -a ) , which has a jump at a . If e'o is a regular value of u, then 'Pa = (Ja (u) jumps by 27r on the smooth level set La = u- l (e,a) . Outside this set, 'P()t is smooth and ID'Pal = I Dul. Thus, (3.6)
where the measure of La is the (n - 1 )-dimensional Hausdorff measure. The idea is that, for some n, 'Pa will satisfy (3.5). To see this, we integrate (3.6) over a E [0, 27r) and find, with the help of the co-area formula, that 21i 2 71' I La l da + 27rI UI BV = 47rl u IBV . l 'PalBVda = 27r o
0
This gives immediately t.he exi8tence of an appropriate
a.
The heart of the proof in [DI] is to establish the inequality
r 21f l 'Pa lBVda < 47rl u 1 BV 10
when u is merely BV; the argument relies heavily on t,he structure of BV maps.
3.5. Lifting when there is no lifting. Throughout this
n
> 2.
0
If u E WI, I (C; §l ) , then u may not have a lifting in wl , I , by Theorem 3. 1 . However, by Theorem 3.5 u has a lifting in BV, which is a slightly larger space. Another example in the same vein is [BBM2] section ,
THEOREM 3.6. Each U E H1/ 2 ( (0, 1) 2 ; §1 ) has a lifting in HI/2 +BV.
Using Theorem 3.2 and results form Alberti, Baldo and Orlandi [ABO], one could more generally prove that maps in W1/p,P ((0, l)n; § l ) have a lifting in W 1 /p'P+BV when 1 < p < 2. However, this is not known when p > 2.
OPEN PROBLEM 5. Is it true that, for eac/t s , p a.nd n, a. map in W8,P( (0, l)n; §l)
ha.s a lifting in W8'P+BV ?
u
3.6. Lifting in covering spaces. One could envision, more generally, maps from C into some smooth compact manifold N. If Z is a universal cover of N
and 7r is the projection of Z onto N, a natural question is whether we may write U = 7r 0 'P, with 'P as smooth as u. In VMO, this was answered by Brezis and Nirenberg [BNI] . 'P
THEOREM 3.7. Ea.ch U EVMO(C; N)
ma.y
be written as
EVMO(C; Z) .
U = 11'
0
'P,
with
The proof is identical to the one of Theorem 3.3. OPEN PROBLEM 6. Find the analog of Theorem 3. 1 when §! and lR are replaced by N and Z.
426
PETRU MIRONESCU
The case sp > n can be treated as in the proof of Theorem 3.1. Probably, the same holds for sp < 1 . It is unclear how to deal with the remaining case 1 < sp < n. Some partial results on this question were obtained by Chiron in an unpublished work.
4. Approximation
In this section, lYf and N are compact Riemannian manifolds without boundary, respectively n and k-dimensional. In order to simplify the presentation, we will sometimes replace M by Rn or a domain in Rn . Set X = WS,P (M; N) . A natural question is whether Y Coo (M; N) is dense in X. VMO arguments imply that the answer is yes if sp > n. Therefore, in the remaining part of this section we assume that sp < n . Under this assumption, in general the answer is no [SU] : let U : B2 §1, u ( z ) = z / i z i . Then U E W 1 ,1 , but u cannot be approximated in W 1 , 1 by smooth §l-valued maps. Argue by contradiction: let {ud C COO (B2 ; § 1 ) converge to u in W 1 , 1 . Up to a subsequence, {ud will converge to u in W1,1 (thus uniformly) on almost each circle C(O, r) with 0 < r < 1 . For any such r, the degrees of U k on C(O, r) (which are zero) tend to the degree of u on C(O, r ) (which is one)! Far from closing the subject, this cxamplc opcns Direction 1. Characterize S , p, jvI and N for which Y is dense. Direction 2. Find " almost smooth" maps that are dense in X. Direction 3. Characterize the closure of Y in X. =
->
4.1. Approximation methods. We explore here Direction 2. In many situ ations, it is known that maps in X can be approximated with maps that are smooth except a " small" set A : A could be a finite number of points, or curves, . . . , up to a finite union of smooth (n - I)-dimensional submanifolds of 1\11 . We describe below the main approximation methods and the results that can be obtained via them. 4 . 1 . 1 . Projection method. It was first devised, in a different context , by Federer and Fleming [FF]. For approximation purposes, it was first used by Hardt, Kinder lehrer and Lin [HKL1]. This method requires N = §k (or homeomorphic to §k) . It works as follows: let u E X and Us = u * p" . The idea is to project u" (which is Rk+1-valued) onto §k. For this purpose, let a E Rk+1 with l a l < 1/2 and let 7fa be the projection of Rk \ { a} onto §k : thus 7fa (.T ) = a + ),(x - a), where ), > 0 satisfies l a + ),(x - a) 1 = 1 . Let u",a = 7fa u" . (Note that, if l u,, 1 happens to be close to 1 , then uE,a is close to u" and thus to u.) Almost each a is a regular value for U" . For such a, u E ,a is smooth outside the level set u;l ( a) , which is (n - k - I)-dimensional if n > k, empty if n < k, Hopefully, if we choose well a = aCc:), then v" = us, a C") will converge to u in ws,P, providing thus an approximation of u with maps that are smooth outside (n - k - I)+-dimensional submanifolds of M, This is true if n < k [E] ; the proof is the same as the one of Theorem 2.1. When n > k, partial answers are known when k < sp < k + 1. THEOREM 4 . 1 . Assume that n > k and k < sp < k + 1 , and set I = n - k - 1 . Then the class n = {u E COO (M \ A; §k ) ; A = i-dimensional submanifold of M} 0
'is dense in X when: a) [BZ] s is an integer; b) [BBM2] s < 1 ; c) [Bo] 1 < s < 2 .
SOBOLEV MAPS ON MANIFOLDS: DEGREE, APPROXIMATION, LIFTING
427
Actually, [BZ] deals only with s = 1 , but their method works for integer s. On the other hand, [BBM2] treats only the case s = 1/2, p = 2, k = 1 but the proof applies to each s < 1 and k. It iH very plausible that Bousquet 's method [Bo] gives that the above theorem holds for each s. The most delicate ease is sp = k; for non integer 8 , the first result in this direction was obtained by Riviere [R].
PROOF. Wc only treat the case s = 1 (and thus k < p < k + 1); the case where s is not an integer is more tricky. It is easy to see that I D1l'a (x) I < C/lx - al uniformly in x and a; therefore, (4.1 )
Sincc u" is smooth, this implies that, for a regular value of u., we have I Du" ,a (x ) 1 :::; C ,a /dist(x , A) , where A = A" ,a = u; l (a). Since A is smooth and I-dimensional, we obtain u.,a E W 1 ,p (and even better) . Let B = B (0, 3/4) C ]Rk +! and let K = K",a = u" ,�(B). Outside B, we are far away form a, and tlms 1l'a is Lipschitz uniformly in a. Since U = 1l'a 0 u, we have ue,a - U = 1l'a 0 U" 1l'a 0 u; from this, 0 with c:, and this holds uniformly in a. we find easily that lIu.,a - ullwl,p(M\ K ) The choice of a comes into the picture when we want to prove convergence in K. If we integrate (4. 1) over Ke, a, then in a, we find .
-
-t
,
(4.2)
K
I Due ,a lPdx da < C
K
IDue I P/ lue -aIPdx da < C
the last inequality holdH since p < k + 1 , and thus Using t h e fact that I Ke,al
( la - x l -Pda JBk+l
< C <
00 .
0 with c: , it follows that the last term in (4.2) tends to O. Thus, we may pick regular values a such that II U" ,a U II W (K) 0. 0 -t
-
l,p
-+
It is natural to try to remove the assumption sp < k + 1 in Theorem 4.1. However, the natural conjecture is that the singular set should be not ( n - k - 1 ) , but rather (n - [sp] - I)-dimensional.
OPEN PROBLEM 7, Assume that n > k + 1 and k + 1 < sp < n, and let
l = n - [sp]
-
1 . Is it true that the class
n = {u E COO (M \ A; §k)
;
A
=
finite union of I-dimensional submanifolds of M}
is dense in X ? Same q1Lest'iu1L when 13k is replaced by an arbitrary N.
In view of the projection method, it is a bit surprising that k has to be replaced by [sp] . We will see why in Sections 4 . 1 . 2 and 4 . 1 .3. The answer to Open Prohlem 7 is yes when k = 1 [BBM4]. Actually, in this case COO(M; §1 ) is dense in X. 4.1 .2. Good and bad cubes method. It is due to Bethuel [Be2] and works for s = 1 (and thus p < n). For simplicity, we let M = ]Rn . We treat here the case n - 1 < p < n. We will show that maps in W1,P(]R"; N) can be approximated with maps smooth except at a finite number of points. Fix c: > 0 and y E (O, c:t , The cubes C = Cm,o,y = Y + c:m + (0, c:)n , with m E zn , cover ]Rn ; let F = Fe,y be the collection of these cubes. The (n - 1)-dimensional skeleton Cn-1 of F is the lll1ion of t he faces of the cubes. The (n 2)-skeleton Cn2 is the union of the boundaries of these faces, and so on. The O-skeleton Co is formed simply by the vertexes of the cubes. -
PETRU MIRONESCU
428
By Fubini, one may find y = y (e) such that lul�ll , p(C�_ l) < C/e. For such y, u is in W l ,p (thus Holder continuous) on en-I . The idea is to approximate u on each cube C without modifying u on ac, and to glue these approximations, The way the approximation is performed on C depends on how much u oscillates on C. With 0 > 0 small, a cube C is " bad" (=u oscillates a lot on C) if lul�!1'p ( c > ) oen-p+ 1 or lul �!l 'P(8C) > 8en- p- l ; the choice of y implies that the union A = A,; of bad cubes is small (its measure tends to 0 with e). The remaining cubes are "good" , On a bad cube, the main care is to construct a map with few singularities and not too large norm; since there are few bad cubes, this will suffice. E. g., one may
consider a solution
v
r IDw l P ;
of min
le
w
;
C
-;
N, w
less energy than u. By a deep regularity result [HLJ ,
v
= u
on 8C
;
thus
is continuous in
finitely many points in C. It is easy to see that l l DU - Dv l P -; 0,
C
v
has
except
If C is good and 0 sufficiently smail, then the image of 8C is contained in some small ball B, by the Sobolev embeddings; the center of B will depend on C, but not its radius r . If B is the ball concentric to B and twice larger, it is possible to project N on B n N through a map <1>, Lipschitz uniformly in B (flatten locally N, next take the neare:;t point projection; this is where smallness of balls is needed) . On C, we approximate u with VI = 0 u, which agrees with u on 8C. The lIlap VI is B-valued, Morally, this means that VI is IRk -valued; a standard technique allows then to approximate VI with a continuous map v agreeing with u on 8C. (A similar argument appeared in the proof of Theorem 2.1.) One has to check next that VI is close to u. These two maps differ only on the set D where (u(x» # u(x). If X E D and if C is the good cube containing x, then -
-
(4 .3 )
I U I Wl,p(E) _ ue >
<
n-p - l
,
where E = {z E
C ; Zn
= xn } .
Argue by contradiction; if not, u(E) is contained in a ball of size T (by Sobolev embeddings), and this ball intersects B , since 1L(BE) C B. Thus u(E) c B! The O. choice of y and (4.3) imply that IDI -+ 0, so that lu- vl l wl,P(D) < Ciulwl,P (D) To summarize, u has been approximated with maps V that are continuous except on a finite set. By further mollifying V, one may find an approximation with maps which are cx; except a finite set. The remaining case 1 < P < n - 1 requires additional ingredients. For a relat.ively short proof in this case, see Hang and Lin [HaL2J, Section 6. The result obtained through this method is ->
THEOREM 4.2. Let 1 < P < n =dim M and I = n - [P]
-
1 . Then
R = {u E COO (M \ A; N) ; A = finite union of I-dimensional submanifolds of M}
is dense in WI,P(M; N ).
Gluing is a key ingredient in the proof. In WI ,P, this works since two W1,p maps defined on neighbor cubes which have the same trace on their common face are in W l ,p of the union of cuhes. This leaves the hope of adapting this argument in ws,p for s < 1 + lip; when s > 1 + lip, higher order traces appear, and the method needs to be supplemented with entirely new ideas.
SOBOLEV MAPS ON MANIFOLDS: DEGREE , APPROXIMATION, LIFTING
429
4.1.3. Homogeneous extensions method. This works only for s < 1 and is taken from [BBM4] . We explain it when M = �n and sp is not an integer. Let m = [sp] . For a. e. y, the restriction of U to the m-skeletoll Cm of re,y is in W >,P , thus uy,£ as follows: on Cm , uy = u. continuous. For any such y, we define uy Assuming uy defined on Cj , j < n, we proceed to define uy on an arbitrary face F of Cj+ l . If z is the center of F and t E of, we take uy uy(t) on the whole segment from z to t. (This is the " homogeneous extension" technique.) Vie extend in the same way uy from Cn- l to �n, and we end with a map defined in �n . If, for example, m = n - I , then uy is continuous except at the centers of the cubes C. In general, ?Ly is continuous except a countable union of (n - m - I)-planes. Next, the key ingredient is the estimate =
_
f(O,e)n
Iuy,£ - u l � dy -> 0 • .p
as c
->
0,
valid when sp < n and s < 1. It implies that we may find y = y(c) such that Uy("),,, -+ n in ws,p Further mollification allows to replace the uy s by maps that are CDC outside a countable union of (n m - 1 ) planes, and thus prove the following '
-
THEOREM 4.3. Let sp < n =dim M, s < 1 and l = n - [sp] - 1 . Then
n {u E COO (M \ A; N) is dense in W>,P(M; N) . =
;
A
=
finite union of I-dimensional submanifolds of M}
Since gluing is part of the method, one may hope to use it for s < 1 + 1/p. This will not work, even for s = 1: for u E CIf \ {O}, we have ID(uy - U)ILP > C > O.
4.2. Density of smooth maps. We step forward Direction I: density of COO ( M ; N) in X WS,P (M; N). The main known result concerns Wl ,p . It Y was obtained in [Be2] , but both the statement and the proof were incomplete. The corrected result is due to Hang and Lin [HaL2] . It is simple to state only when n - 1 < P < n. =
X
=
THEOREM 4.4. Let n - 1 < P < n =dim M. Then Y = COO (M; N) is dense in
=
WI,P(M; N) if and only if the homotopy group 7rn-1 (N) is trivial.
PROOF. Assume that 7rn- l (N) I- {O}; we will construct a map u which cannot be approximated with smooth maps. There is a map v E CX?(Bn - l ; N) , identically equal to a constant C near §n-2, and such that v is not homotopic to a constant. After locally flattening M, we may assume that M contains Bn Let D (respectively E) be the cone with vertex the North Pole (respectively South Pole) of §n - l and base Bn-l , and set F = D U E. On each horizontal slice S of F, we may transport v by translation and dilation. If u is the map obtained in this way, then, for - 1 < t < 1, the restriction of u to F n {xn t} is smooth and not homotopic to a constant. We extend u to M with the value C outside F; then u E l-V1,p. Assume now by contradiction that Y is dense in X. Possibly after passing to a subsequence, we find {Uk} C COO (M; N ) such that, on a. e. S, Uk u in W1,p , thus in VMO. The homotopy class being stable with re�pect to VMO convergence (cf Section 2.7) , we find that, for large k, Uk IS is not homotopic to a constant, which is absurd. Conversely, assume that 7rn -l (N) = {O}. It suffices to prove that maps in R can be approximated by smooth maps. We explain the method when u has only one singularity; in the general case, we apply this procedure near each singular point. We may assume that M contains Bn, and that the singular point is the =
-+
430
PETRU MIRONESCU
origin. For a < l' < 1 , we extend U from S(O, r) to B(O, r) by homogeneous extension (thus the extension Vr is constant on rays). Let Ur be the map that equals U outside B(O, 1') and Vr inside B(O, 1'). By a Fubini type argument, there is U in lV I ,p . It suffices thus to consider maps U a sequence rn � a such that Urn which are smooth in M \ B (O, '1') and homogeneous in B(O, 'r ) ; assume, e. g., that r = 1 . The map v = 'U lsn -l is smooth and, as a continuous map, homotopic to a constant C. By regularization, there is a homotopy H E COO ( [0, 1] x §n -\ N) such that H(l, , ) = v and H(t, ') = C for t < 1 /2 . For 0 < £ < 1 , we define outside B(O, £) u(x) . It is easy to see that Ue -> 1L in W l ,p. uE (x) = H( lxl /£ , x/lxl ) in B(O, E ) Since the ue ' s are continuous, they can be approximated with smooth maps. D .......
When I < P < n - 1 , the condition 7l"[pl ( N) = {O} is necessary, but not sufficient for density; we send to [HaL2J , Section 6 for details.
OPEN PROBLEM 8. Find, for arbilmry s, p, M and N such that sp < n =dim
M, a necessary and sufficient condition for the density of Y in X .
The answer is known when s < 1 [BBM4J, and the density condition depends on the value of sp. For example, the analog of Theorem 4.4 is
THEOREM 4.5. Assume that n - 1 if 1rn- 1 (N) = O.
::;
The answer is also known when N
=
sp < n . Then Y is dense in X if and only
§ l [BBM4] :
THEOREM 4.6. Assume that N = §l . Then Y is dense in X except when n > 2 and I < sp < 2.
We emphasize the fact that even for the space W2,p the answcr is not known. It is quite likely that understanding this case will unblock the general situation.
4.3. The singular set of a map. Assume that Y is not dense in X , but that we are able to approximate a map U E X with maps 7J.k in the class R. Question:
can one " pass to the limits" the singular sets of the uk 's7 If so, one has a natural notion of singular set of u, and can even dream of proving that u is in the closure of Y if and only its singular set is cmpty. (Thus this question is related to Direction 3 . ) Most of the work in this direction has been done when N = §k. Except at the very end of this section, we let N = §k. To start with, we take n = k + 1 and M = §k+ l ; however, M could be any (k + I)-dimensional manifold. The maps we consider are W l ,P, with k ::; p < k + 1. (When p is not in this range, Y is dense in X.) Actually, it suffices to know how to pass to the limits the singular set when k u E Ufl, ; we take thus p = k. In this case, R consists of maps u smooth outside Home finite set A = A(u) . To each a E A, we may associate a degree, defined as the degree of u on a small geodesic sphere around a on §n , positively oriented with respect to the outward normal at a; this integer is independent of the small sphere and will be denoted deg (u, a ) . Brezis, Coron and Lieb [BeL] discovered the fact that the singular set A can be obtained from u via all analyl.ic formula. deg (u, a) 6a and Ck = l /l §k l , then the action of More specifically, if T = Tu = the distribut.ion
(4.4)
T(()
T is given by
=
-Ck
L
aEA
( OJ ( det(olu, . . L J5k+1
.
, OJ- l U, u, OJ+lU, . . . , 8k+ I 1L) ·
SOBOLEV MAPS ON MANIFOLDS: DEGREE, APPROXIMATION, LIFTING
431
Here, the derivatives are computed in an orthogonal positively oriented frame. Note that the right-hand side of (4.4) makes sense for u E Wl,k If we endow WI,oo (§k+I ; lR.) with the semi-norm ( I--> IID(lI v'o , then T given by (4.4) lies in (Wl ,=) * , and depends continuously on u. Since each u may be approximated by maps in n, one may intuitively think of T as an infinite sum of Dirac masses. This is indeed correct.
PROPOSITION 4 . 1 . For u E WI,P(§k+\ §k ) , one may write Tu for two sequences {P;}, {Nd
PROOF. Assume u E
n.
c
§k+ 1 such that I Pi - Ni I < 00.
L
Then
aEA
deg (u, a)
=
=
L(OP, - ON. ) ,
O. This may be seen either
from topological considerations, or by noting that T(l) = O. Thus the points in A counted with the multiplicity of their degree (a point with degree 2 appears twice as a " positive point" , a point wit,h degree - 1 appears once as a " negative point" , a point of degree 0 does not appear at all) form a list PI , . . . Pm , NI , . . . , Nm of positive and negative points, the positive points being as many as the negative ones. With the points in A listed in this way, we have T«()
m
=
L «((Pi) - ( N; » . i=1
The key ingredient is the following sup-inf inequality devised in [BCL];
( 4.5)
sup
m
L «((P ) - ( Ni» i=1
i
;
ID(I < 1
here, d is the geodesic distance on §k+1 and Sm is the mth symmetric group. Formula (4.4) gives ITu«()1 < CIiDu ll tk IID(II Loo . In view of (4.5), T may be written as T«() = «((Pi) - ( Ni» with 'L d(Pi, Ni ) < CIiDulltk ' A Cauchy sequences argument, combined with the fact that the geodesic distance is equivalent to the 0 Euclidean one, allows to conclude.
L
is tempting t.o consider the set { Pi} U {Ni } as the singular set of u. This is not realistic, since there is a high degree of non uniqueness in the choice of these points; see Ponce [PI] for a thorough discussion on the infinite sums of Dirac masses. In a somehow non intuitive way, one has to identify the singular set of u with the distribution Tu ; when u E n, Tu can further be identified with a set of points. Proposition 4 . 1 has a converse [ABO]: given sequences { Pi } , {lVi} C §k+1 such that I Pi - Ni l < 00, there is a map u E W1,k(§k+\ §k ) such that Tu = L(Op, -ON.). The map u is explicitly constructed using the " dipole construction" in [BCL] . All the above results can be summed in the following IL
L
PROPOSITION 4.2. The map u (WI,OO)', and its mnge is
Tu is continuous from Wl,k (§k+1 ; §k) into _,.------, ( W',oo ) ' I-->
L (Op, - ON.)
•
finite
It turns out that one may define Tu for u E ws,p when k < sp < k + l [BBM3] . By Sobolev and Gagliardo-Nirenberg, it suffices to consider the case sp = k, s < 1 .
432
PETRU MIRONESCU
4.7. Assume that sp k and s < 1 . Then the map u Tu , initially E 'Il, extends by density as a continuous map from W8,p(§k+l ; §k )
THEOREM
defined for
U
=
into ( WI,DO Y , and its range is
-,----____ ( w"
L (OP. - ONJ
finite
f-t
=)-
Formula (4. 4) does not make sense for u E W·,p. The idea is to find another formula for Ttl; this is very much in the spirit of Section 2.2. We take v / l ul, if lui > 1/2 u , with u the harmonic extension of u. Let as there, i. e., v = 2u, if l ui < 1/2 also � be any smooth extension of ( to Bk+2. For u in W I - 1/ (k+1),k+1 n Wl,k, u smooth outside a finite set (call such a 'U a good map), we have, with dk (k + 1 )Ck ' PROOF.
=
this can be easily checked for smooth u, next by approximation. Since good maps are dense in WS'P(§k+l; §k ) , it suffices to prove that the right-hand side of (4.6) depends continuously on u and �; this is done as in the proof of Theorem 2.3. Finally, the range of u f-t Tv- is determined by adapting the dipole construction. 0 While the above result allows to define Tu in ws,p if k < sp < k + 1, it says nothing about the set of all the T" 'so
T k
Assume that k < sp < k + 1 . Charar:te1"ize the distributions which are of the fo rm T = T" for' some u E W ',P (§k + \ §k) . OPEN P ROBLEM 9 .
The answer is not known even when s 1 . Parti al results were obtained, for 1 , by Bousquet [Bo] . E. g., we may find u E Wl,p (§2 ; §1 ) such that T Tu if =
=
�,(W',p/(p- l ) r
-,-
L (op, - ON, )
=
_ _ _ _ _
and only if T E
finite
range of 1L ...... Tu is
. Presumably, in W1 ,p(§k+l; §k) the
L (Op, - ON. )
finite
The following result, due to Bethuel [Bel]' suggests that Tu really describes the singular set of u. THEOREM
4 .8. Let u E X
=
Wl.k (§k+l ; §k ) . Then u E Y if and only ifT"
=
O.
When u E C''''', we have Ttl 0; by continuity, T" 0 if u E Y. The key ingredient in the proof of the converse is the following result, whose proof relies on an explicit construction similar to the dipole one. PROOF.
=
=
LEMMA 4.1 . If A is the singular set of u E 'Il, then there is a map v E W1,k (§k+\ §k ) , locally Lipschitz outside A, such that deg (v, a) 0, a E A, and IID(u - v) 1I1. < C I i Tu ll(Wl.OO)" =
Restricted to a small geodesic sphere S around some a E A, the above v has degree 0; thus VIS is homotopic to a constant. The proof of Theorem 4.4 shows that v E Y . Lemma 4.1 implies that dist (u, Y) < CIiTu ll ��,,= ) , (here, the distance is computed with respect to the W1,k-semi-norm) . By continuity, this inequality holds for each u E W1,k. This completes the proof. 0
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433
The proof of Lemma 4.1 works only for p = k. It can be adapted in ws,P, but only if sp = k (see, e. g., [BBM2]) . This leaves us with the following . OPEN PROBLEM 10. Assume that k < sp < k + I and let 1L E ws,p (§k+ l ; §k ) Is it true that u E Y if and only if T" = 0 7 The answer is yes when s < I Ponce [P2] or when k = 1 [Bo] . We next consider maps in w l ,k (§nj §k ) , with n > k + 1 . In this case, the singular set of a map u E R is an (n - k - I )-dimensional manifold A . We may still associate to u an object Tu, but this time it acts on (n k - I ) forms ( =sections of An- k -1 (T* (§n» , not on functions. For simplicity, we explain how this is done when n = k + 2. (In higher dimensions, the ideas are the same [JS] , [HaLl], [ABO] .) The singular set A of 1L is a finite union of compact flimple curves. To start with, assume that A consists of only one curve, say r. We choose an orientation on r and let T be the tangent unit vector positively oriented on r. \Ve may define the degree of 1L around r as follows: we take an ( n - 1 )-submallifold P of §n , transversal to r at some point x, and oriented positively with respect to r (i. e., the orientation on Tx (r) x Tx (P) is the positive one on §n) . This orientation induces a positive orientation on small geodesic spheres S on P around x . We define degen, r) =deg u l s ; this integer does not depend on x or S. Then the object associated to u is T = Ttl =deg(u, r)Tor, with Or the Dirac mass on f (=the I-dimensional Hausdorff measure restricted to r). When A = U f;, we let T = deg(u, f;)n)r, . This object acts on I-forms, i . e. on smooth sections W of the -
L
cotangent bundle to §n through the formula T(w) = (In case of p-dimensional manifolds, p >
formula T(w) =
L deg(u, I';)
r
Jri
Ldeg(u, fi) l. < w, T > ds .
2, one defines similarly
(4.7)
T (w ) = -Ck L t<1
If U
.fs.. w;j det(011J.) . ) 8;-1 .
u, 1L, Oi+1 u,
W
=
L widxi and we set
. . , OJ _ I U, OJ + 1 U, . . , Gn U ) .
is a good map, then we also have an analog of (4.6): with smooth extension of w, and ai = dk ( - I )k+i, (4.8)
T(w) =
L a; .
. < , 1
T through the
< W , T > ; here, T is a unit p-vector positively
oriented, and W is a p-form.) If wc write, in an orthogonal positively oriented frame, Wij = OiWj OjW;, then the analog of (4.4) is -
-
v
as in (4.6), !1 a
(Bn+l !1ij det (ol v , . . , 8i-1V, Oi+1 V , . . , Gj - 1 V, Oj+1V, . . , On+1V) . J
Recall that, in case of point singularities, positive and negative points in A are in equal number. This may be translated as A = oe, where C is a union of curves in §n, each one with starting point a negative point and endpoint a positive point. Here, the boundary has to be understood in the distributions sense: if we orientate each curve from the negative to the positive point, then it defines a current (still denoted C) as above, and the equality A = oC means Tu (() = C(d(), 'if ( . The count.erpart of these properties in the case of curves is that that the fi'S are closed, which in turn implies that we may write Tv. = as, where this time S iD the current associated to a finite union of surfaces. The analog of (4.4) is
(4.9)
sup{Tu (w) ;
I dwl <
I}
= inf{ I SI
;
oS = T} ;
434
PETHU MIHONESCU
here, l S I stands for the mass (=surface) of S. Once the parallel between the two situations is established, the counterpart of Proposition 4.2 is [ABO]
THEORE� 4.9. The map U f-> Tv. is continuous from WI ,k(§k+2; §k ) into the dual of Lipschitz I -forms, and its range is {8S ; S is a rectifiable surface} .
The delicate part of the proof is the construction of u when S is given; this is done by adapting carefully the dipole construction, A similar statement holds in higher dimensions. A straightforward adaptation of the proof of Theorem 4.7 gives the existence of Tv. when u E yVS,P and sp k. Thus one may consider Open Problems 9 and 10 when §k+l is replaced by §", with n > k + 2, We end this section by considering the case where §k is replaced by a general k-dimensional manifold N. If n = k + 1 and M is n-dimensional, then maps u with point singularitiPB are dense in Wl.k (M; N), If A is the singular set of u, it is natural to associate to a E A a homotopy class [u] (a), namely the class of u restricted on a small geodesic sphere around a . Thc proof of Theorem 4,4 shows that u E Y if and only if [u] (a) is trivial for each a. It is not known whether one can associate to u a distribution that " hears" the singularities of u. =
OPEN PROBLEM 1 1 . Is there a way to associate to a map u with point singu larities a distribution Tv. supported in the sing1Jlar set of u, depending continuously on the W1,k -norm and such that Tu = 0 if and only if u E Y ?
4.4. Relaxed energy. Though the questions raised in this section make sense for general manifolds M and N, we shall consider only M = §k+ 1 and N = §k;
even this special case is not well-understood, If k < sp < k + 1 , then smooth maps are not dense in X = �vs,p(§k+l; §k), However, one may hope weak density of smooth maps, i. e., that given U E X, there is a sequence {uieJ of smooth maps, bounded in X, and such that Uk U a. e, There is no weak density if sp > k. Indeed, let U ( X' , Xk+2 ) x'/ lx' l , which is singular at the poles of §k+l Argue by contradiction and assume that there is a sequence {ud as above. Then, up to some subsequence, on a. e. geodesic sphere S around the North Pole of §k+1, we have Uk U in ws,k/S (S) , and thus in VMO. This leads to a contradiction, since the degree of u on S is 1 , while the one of the Uk 'S is O. Tn view of this example, from now on sp = k. The relaxed energy introduced by Bethuel, Brezis and Coron [BBe] is " the least energy required to approximate u" : ......
=
......
(4.10)
Ere1 (u) = inf{lim inf l u k l ev
"
p
; {uieJ
C
COC (§k+ 1 ; §k), Uk
�
u
a,e,}.
Clearly, Ere1 (u) > lulfvs,p , If R is dense, then the relaxed energy is always finite; this relies on a dipole construction. It is very likely that R is always dense (cf the discussion aft.er Theorem 4,1), The exact formula of the relaxed energy is known only when 8 = 1 , p = k, This formula is related to the singular set of u and establishes a bridge between Directions 2 and 3. If u E R, let L(u) be the right-hand side of (4.5), In a suggestive way, L (u) is called the minimal connection between the negative points Ni and the positive points Pi [BeL] . For a general u, L(u) is defined as II Tu l l (Wl,�) • .
THEOREM 4,10. In
(4. 1 1 )
W 1 , k (§k + 1 ; §k ) , we have Ere1( u )
=
/ k §k Du k II lltp + l l 2 L ( u ) .
sonOLEV MAPS 01\ MANIFOLDS: DEGREE, APPROXIMATION, LIFTING
435
For k > 2 , this result is from [BBC] . We take k = 2 ; when k > 2, the argument is similar. Inequality < in (4. 1 1) is established, for U E 'R, via the explicit construction of a sequence {ud such that Uk U a. e. and I I Duk 117,2 IIDulli2 + 87r2 L(u). By density, such a sequence exists also for a general u. For >, the key argument is that, for fixed ( with ID(I < 1 , the map u >-+ IIDulliO +87r2Tu «() is lower semi-continuous on the convex set of Hl-maps of modulus < 1 . By taking the supremum over (, this implies thaL u >-+ I I Dulli2 + 87r2L(u) is lower semi continuous. Thus, for any sequence {Uk} such that UTe --'> u, we have (4. 12) II Dulli2 + 87r 2 L(11) < lim inf( llDuk IIi2 + 87r2 L (Uk» = lim inf IIDuk ll i2 . PROOF.
-+
-+
This argument does not apply when k = 1 ; in this case, this result was proved using a different method in [GMS2j; for an elementary proof, we refer to [BMPj. 0 Nothing is known when
s # 1.
.-
We end with the following challenging
OPEN PROBLEM 1 2 . Assume that
u E w',P (sk+1; §k ) , s and k.
we have
Ere1(u)
=
0 <
lulf-v
s
•. p
<
1 and sp
+ CL(u) ,
=
where
k.
C
Prove that,
for
depends only on
The only hint towards this question is the fact that, when k Ere1(u) 1111f-v.,p � £(11) [BBM2] .
=
1, we have
-
References B aldo G. Orlandi, Functions with preSCTibed singularities, J. Eur. Math . Soc. 5 (2003), 275-311. [Bell F. Bethuel, A characterization of maps in Hi (B3 , §2) which can be approximated by smooth maps, Ann. lnst. H. Poincare Anal. Non Lineaire 7 (1990), 269-286. [Be2] F. Bet huel, The approximation problem for Sobolev maps between two manifolds, Acta Math. 167 (1991), 1 53-206. [BBC] F. B ethuel , H. Brezis, J.-M. Coron, Relaxed energies for harmonic maps, in Variational M eth ods (H. Berestycki, J.-M. Coron and l. Ekeland eds. ), Birkhiiuser, 1990, 37-52. [BD] F . Bethuel, F. Demengel, Extensions for Sobolev maps between manifolds, Calc. Var. Partial Differential Equat ions 3 (1995), 475-491 . [BZI F . Bethuel, X. Zheng, Density of Smooth Functions between Two manifolds in Sobolev Spaces, J . Funct. Anal. 80 (1988), 60--7 5. [BB] J. Flourgain, H. Brezis, On the equation div Y = f and application to control of phases, J. Amer. Math. Soc. 16 (2003), 393-426. [Bo] P. Bousquet, Topological singularities in w··p(§n , sl ) , to appear. [BBMl] J. Bourgain H . Brezis, P. Mironescu, Lifting in Sobolev spaces, J. Anal. Math. 80 (2000), 37-86. [BBM2] J. B ourgain H . Brezis, P. Miron escu , Hi/2 maps with values into the circle: minimal connections, lifting, and the Ginzburg�landau equation, Pub!. Math. Inst. Hautes Etudes Sci. 99 (2004), 1-115. [BBM3] J. Bourgain, H . Brczis, P. Mironescu, Lifting, Degree, and Distributional Jacobian Re� visited, Comm. Pure Appl. Math. 58 (2005), 529-551. [BBM4] J. Bourgain, H. Brezis, P. Mironescu, Density in W··P, in preparation. [BBNJ J. Bourgain, H. Brezis, H.-M. Nguyen, A new estimate for the topological degree, C. R. Math. Acad. Sci. Paris 340 (2005), 787-79l. [BGP) A. Boutet de Monvel-Berthier, V. Georgescu, R. P urice Boundary Value Problem Related to the Ginzburg-Landau Model, Comm. Math. Phys. 142 (1991), 1-23. [E1] H. Brezis, Degree theory: old and new, in Topological lS'onlinear Analysis II: Degree, Singu larit ies and Variations (M. Matzeu and A. Vignoli eds . ) , Birkhiiuser, 1997, 87-108. [B2J H. Brez is New questions rdated to the topological degree, in Proceedings of the Conference Celebrating the 90th B irthday of l. M. Gelfand , to appt>.ar . [BCL] H. Brezis, J . � M . Coron, E. Lieb, Harmonic maps with defects, Comm. Math. Phys. 107 (1986), 649-705. [ABOJ G .
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Homotopy classe in Sobolev spaces and the existence of energy minimizing maps,
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INSTITUT CAMILLE JORDAN, UNIVERSITE LYON
69622 VILLEURBANNE CEDEX FRANCE E-mail address: mironescu�math . univ-lyon1 . fr
1,
43 BOULEVARD
nIl
1 1 NOVEMBRE 1918,
Cuntemporary Mathematics
Volume 446, 2007
Partial and full symmetry of solutions of quasilinear elliptic equations, via t he Comparison Principle. .
'
Patrizia Pucci, Berardino Sciunzi, and James Serrin To Professor
Hai'm Brezis for his Sixtieth Birthday
Using a comparison principle for degenerate elliptic equations, we establish radial symmetry results for solutions in balls and annuli of equations of the form div{a(r)A(IDuI )Du} + B (r, u) = 0, when B (r, z) E LE','c (fl x JR) is non-increasing in z, ABSTRACT,
1. Introduction In this paper (1.1)
we
consider the equation
div{a(x)A( IDuIlDu} + flex, u) = 0
in n,
where n is a bounded domain in jRN and the solution is understood in the weak distribution sense. We make the following structural assumptions on the operator (HI ) a(x) [a;j (x)] (i, j = 1 , . . . , N) is a locally bounded real positive definite symmetric matrix, i. e , (H2) A is positive and differentiable in jR+ . Also
In some cases we shall not assume ( H2 ) , but only the condition (H3) A is positive and
0(1) as
2000 Mathematics Subject Classification. P rimary 35J70; Secondary 35B05, Key words and phm.. es. Degenerate elliptic operators, geomet ric and qualitative properties of the solutions, comparison principle. Patrizia Pucci and Berardino Sciunzi have been supported by the MIUR national project " Variational Methods and Nonlinear Differential Equations" , ©2007
43 7
American Mathematical Society
438
PATRIZIA PUCCI, BERARDINO SCIUNZI, AND JAMES SERRIN
REMARK 1 . 1 . Condition (H3 ) is weaker than (H2 ) , Indeed, if (H2) holds, then by ( 1 .2), we have cp' (t) tA'(t) + 1 > 0, A(t) - A(t) _
so that is strictly increasing in JR+ .
Assumptions (H2), (H3) include the case A(t) = tp - 2 , p > 1, which givE'B the well known Laplacian when p = 2 and a = IL The operator in (1.1) was first studied in [1o), see also [2J and [11J . For definiteness in the interpretation of (1.1) we put A(O = A ( IWe for � i 0 and A (O) = O. Thus A is continuous on JRN because of (H2 ) , and (1.1) can also be written in the form div{a(x)A(Du)} + B(x, u) = O.
Our starting point is to give sufficient conditioIlH to guarantee that the operator is elliptic. As shown in [2], see also [11], to this end it is important to know when the product of two positive definite matrices i::; positive definite. In particular, using the results in [8, 13, 2] , it was proved in [11] that the operator a(x)A(IDul)Du is
elliptic if
,1, ( 'I'
+ c + 2y'1 + c 2 C) = ' l ei
where Cl and C2 are given by (H2); see [11, Lemma 2 .3.3] and the following Lemma 2 . 1 .
We use the ellipticity of the operator to get a Comparison Principle, H€€ Propo sition 2.1, from which symmetry results for solutioIls of (1.1) follow. In particular, we consider the case when B (x, z) E Lk;'c(fl x JR) is non-increasing in z , and show that if the domain is symmetric in one direction, say e l , then the Holution is sym metric in the same direction, provided that the matrix a and the nonlinearity B are similarly ::;ymmetric. No assumption is needed on the sign of the solution nor need the domain be simply connected. Our main result is Thcorem 3.1. HerA we point out an interesting corollary.
JRN be a ball or an annulus. Let u E Wl�'� (fl) n C(fl) be a solution of the following Dirichlet boundary value problem for (1.1), written in distribution form: for all :p E CO" (fl) THEOREM 1 .2. Let fl
�
C
(a(lx l )A( IDul)Du , D
u(x)
=
g(lxl)
on
ofl,
r B(lxl, u)
Jo
withl B(lxl, z ) E Lk;'c (fl x JR) non-increasing in z . If the assumptions (H1 ) , (II2 ) , (H4) hold, then u is unique and radial (i. e. u = u(lxl) . If finally �(t) < const. t,,- l , ]I > 1 , the same result holds even for solutions u E Wl�': (fl) n C(n) . Theorem 1 .2 will be proved in Section :3. lNote that the
0. is
condition at the boundary in Theorem
1.2
means that
a ban or u assumes two different values on 80. if n is an annulus.
u
is constant
on
an
if
PARTIAL AND FULL SYMMETRY
439
In case a (x) = p(lxl)ll it is enough in Theorem 1 . 1 to assume only that condition (H3) holds and that p is positive and locally bounded in BR \ {o} . See the remark in Section 3 after the proof of Theorem 1 . 1
For the case of the p-Laplace operator, we refcr the reader to [4, 5J and the references therein. In particular in [4] the case of locally Lipschitz continuous nonlinearities with p < 2 is considered, while [5] deals with the case of positive locally Lipschitz continuous nonlinearities for p > 2. When A(t) = tp-2 and a is not the identity matrix (0 < oXl(12 < aij (x)�i�j < AleI2), the ellipticity condition (H4) becomes REMARK 1 . 3 .
p + 2..jp - l < 2;oX '--:lp --'-'--:1 with no condition if p = 2, see [2] and [11, Section 2 .3] .
A
In spite of the elegance of the results, thc reader should observe that nontrivial solutions of the Dirichlet problem in Theorem 1.2 may not exist for arbitrary boundary data. In particular, consider the zero Dirichlet boundary value problem for ( 1 . 1 ) , with B independent of x and non" increasing in z, and B(O) = 0. Let u be a solution, and note that it can be used as test fundion in ( 1 . 1 ) , yielding REMARK 1 . 4 .
0<
(a(x).4 ( I Du I)Du , Du) dx =
B( u)u dx < O.
n n It follows that Du = 0, showing that there are no non-trivial solutions! Nevertheless, there are other cases with different Dirichlet boundary conditions, or with B (O) =I- 0, where this difficulty does not arise and nontrivial solutions exist. In these cases we obtain symmetry results which hold fur broad classes of operator;; and domains. In fact we only need to assume that the domains in question are symmetric in some direction to prove that the solution is symmetric (in that direction) . We note particularly that the domains need not be convex or even simply connected.
If we consider the semilinear non--degenerate case (A (t) = 1 and a = ll), there are in the literature many symmetry and monotonicity results obtained exploit ing the well known Alexandrov-Serrin [12] moving plane method. We mention here the celebrated papers [1, 7] where symmetry and monotonicity results are ob tained for positive solutions with zero Dirichlet boundary conditions, under general assumptions on the nonlinearity. There are cases when the moving plane technique can not be exploited. As an example if the domain is not convex (e.g. an annulus) or if we consider operators that depend on the position (as in our case in view of the matrix a ) . Nevertheless one could expect that if the domain is symmetric then the solution inherits symme try properties. If the domain is a ball or an annulus and we consider the semilinear non--degenerate case, axial symmetry of the solutions is proved in [9] assuming that Morse index information concerning the solution is known and assuming that the nonlinearity is convex. The idea behind [9] , which we shall a.lso exploit here in a different (and possibly degenerate) context and with different techniques, is to consider the solution 1L and its reflection (say 'ii.) directly in the entire domain.
440
PATRIZIA pucer, BERARDINO SCIUNZI, AND JAMES SERRIN
2. Preliminaries The following two lemmas can be found in [11] . For the reader's convenience we recall the proofs.
LEMMA 2 . 1 . (Lemma 2.3.3 of [11] ) . Assume that conditions (H1 ) , (H2 ) , (H4 ) are fulfilled. Then the operator a(x)A(I Dul ) Du is elliptic in n x ]R N , i. e. the Jacobian matrix 8d a(x)A(IWO is positive definite in n x ]RN . Moreover, for any x E n and �, 'T) E ]RN, with � i:- 'T), we have
( a (x)A(I �I)� - a(x)A(I'T)I)'T) , � - 'T)) > O.
(2. 1 )
PR.OOF. By direct calculations we get
that is
1�IA'(IW � 0� } 8d a (x)A(IW�) = a(x)A(IW 1I + A(IW 1 � 1 2 ' 8da (x)A(IWO = a(x)A(IWb(e),
with
( 1 W �IA ' I c( l� l ) = A(l�l) .
By linear algebra the eigenvalues of the matrix b are 1 with multiplicity N - 1 and 1 + c. Note that 1 + c > 0 by (H3)' Therefore b is symmetric and positive definite and by [8, 2, 131, see Theorem 2. 1 of [2], the (symmetric) product ab is positive definite, provided A �-1
that is
A ,-1 A
( ";1 + e - 1 )
<2
if
1 1 1+c
<2
if - 1 < c < 0,
e
>0
JA/A < ep(e) .
Let us first prove (2. 1 ) assuming that ° ¢ [� , 'T)) (that is, ° is not on the segment from � to TJ). We have for some ( E [e, 'T))
(a(x)A(IW� - a (x)A(I'T)I ) TJ, � - TJ) = ( 8da (x)A(I(I)(} (� - .,,) , � - .,,) = A(I(i)(a (x) b (O(� - TJ), � - 'T)) > 0 ,
since we already proved that ab is positive definite. When ° E [�, TJ] ' we can exploit the same arguments in ['T) , O) and [o, �] , using the fact that A is continuous. 0 If a(x) = 1I we have
LEMMA 2.2. (Lemma 2.3.2 of [11)) . A ssume (H3 ) . Then for all � and ,,/ in ]RN , with � =1= TJ,
PARTIAL AND FULL SYMMETRY
441
PROOF. If either of the vectors is 0 the assertion is trivial since A(O) - 0 . Otherwise, since A(t) > 0 for t > 0 and (� , 71) < I� I . 17]1 , we have (A( IW� - A( lrJl)7] , � 7]) -
A( I W I�12 + A(I 7]1 ) 17]12 - A(I W (� , 7]) - A( lrJl ) (� , 7]) > ( IW I � I + ( 17]1 ) 1 171 - ( IW lrll - ( I7] I ) I � 1
=
= { ( I W - ( irJl ) } ( I� I - I7] I ) and the conclusion now comes from the strict monotonicity of .,
o
Exploiting now Lemma 2.1 and Lemma 2.2, we prove the following Comparison Principle, see Theorem 3.3 . 3 of [11] .
PROPOSITION 2.l. Let 0 C l�N be a bounded domain and let 1.1 , v E Wl�';' (0) n C(O) be such that o
(2.2)
( a(x)A(IDuI)Du , D'P)dx <
!'!
0
flex, u)'P dx
(a(x)A( ID11I)Dv, D'P)dx
- r B(x, v) 'P dx. ..r0
for any 'P E CO" (O) . Assume that flex, z) E L�c(O x lit) and is non-incTea8ing in z . Let conditions (Hd, (H ) , (H4) be fulfilled and suppose that 1.1 < v on a�. Then 2 1.1 < v in O.
The same result holds assuming only 1.1 , 11 E Wl�'� (O)nC(O) when (t) < const. tV- I , p > 1. Finally, in the case a(x) = p(lxl)lI with p positive and locally bo'unded, the TeSU/t follows 1Jrith the assumptions (Hd , ( H2 ) , (H4) replaced by the weakeT condition (H3) . PROOF. For c > 0, define
f
=
f"
'P = 'Pc
=
=
(1.1 - V - c)+ . and
{x E O : u(x)
-
v ex) > c } .
Then since 1.1 < v on ao we have supp 'P C f and r c c 0 so rp E W5'OO(0). Therefore, recalling that the matrix a is locally bounded, by density arguments we can use 'P as test-function in (2.2) and get r
(a(x) A ( I Du I Du - a (x)A( IDv l ) Dv , Du - Dv)dx <
[ IB (X , u) - B(x , v)] 'P dx < O.
where [B(x, 1.1 ) B(x, v)]'P is non-positive in I' since 'P ;:: 0 and B(x, z ) is non increasing in z . By Lemma 2 . 1 it now follows easily that Drp = Du - Dv == 0 in the (open) set f. Also D'P = 0 a.e. in 0 \ f. Thus 'P = constant in 0, and in turn since 'P = 0 in 0 \ f we get 'P = 0 in O. That is 1.1 < 11 in O. Next assume that 1.1, 11 E Wl�'; (O) nC(O) and (t) < const. tV- I • Define f = fe , 'P = 'Po as above. In this case 'P E W5'P(0). Also A(Du) < const . IDuI P -1 so that A(Du) E Lfoc (O). Consequently, by density arguments, 'P can be used as test function in (2.2) and the result follows as above. -
,
PATRIZIA PUCCI, BERARDINO SCIUNZI, AND JAMES SERRIN
442
When a(x) = p(lxl)ll the result can bc proved in cxactly the same way using Lemma 2.2 instead of Lemma 2.1. 0
3. Symmetry (and monotonicity) results We exploit the above preliminaries to get symmetry and monotonicity results. Introduce the notation X' = (X 2 , . . . , XN ) , THEOREM 3.1 . Let 0 C ]RN be a bounded domain, symmetric with respect to
the el -direction (that is x E O ¢} x E O) . Let u E WI;;' (H) n C(O) be a solution of the following Dirichlet bonnda'f"'lj value problem faT" (1.1), written in distribution form: faT" all
j� (a(x)A( I Dul ) Du, D
(3.1 )
u=9
on
a�,
with B(x, ) E L�c(O x ]R) non-increasing zn Assume that conditions (Hl ) , (H2), (H4) are fulfilled and suppose z
(3.2)
z.
a(x)
=
a (x )
,
B(x, z) = B(x, z ) ,
g(x)
=
g(i).
Then u is the only solution of (3.1) in Wl�';' (O) n C(O), and is symmetric with respect to the el -direction. The same result holds for solutions u only of class WI�:'(O) n C(O) if
ii(x)
for any x
u(i)
E O. By the change of variables x -> x it follows that for all
n
(a(i)A(IDuIl Dii, D
By the assumption (3.2) it follows that the assertion is proved.
=
( B(x, ii)
on
a�.
u is a solution of (3.1) so that u
u and 0
Pmof of Theorem 1.2. By rotation of coordinates, we can use Theorem 3.1 to obtain
reflection symmetry with respect to all directions in ]RN . Radial symmetry is now apparent ; an explicit proof for this is given in [6] ' Lemma 1.8. This completes the proof of Theorem 3.1. The case when a(x) = p(lxllll in Theorem 1.1 (and condition (H3) holds) is proved in the same way, using however the final part of Proposition 2. 1 .
The ideas of Theorem 3.1 can also be used to prove monotonicity of the solution in the el -direction, provided 0 is partially convex and the conditions on the matrix a and the function n are slightly strengthened.
PARTIAL
AND
FULL
443
SYMMETRY
Let n c l�N be a bounded domain, partially convex in the sense that the intersection of n with any line x' constant is connected, and also symmetric with respect to the el -direction. Let u E YVj�';, (n) n C(n) be such that for all 'P E CQ (n) THEOREM 3 . 2 .
=
n
=
(a(x)A(JDuI )Du, D'P) dx in on
u>0 u=0
n
1 B (x, U)'P dx n
an,
with B(x, z) E L�c(l1 x R) non-increasing in z . A ssume that conditions (HI ) ' (H2 ) , (H4) are fulfilled and suppose B (x, z ) B(x' , z ) . a(x) a(x' ) , (3 .3 ) =
=
Then u is non-dec1'eas'ing in the C l -direction in n- = {x E n J Xl < a}. The same result holds for solution u only of class WI�''; (l1) n C(n) if (t) const. tp- l , p > 1 . PROOF.
For A < 0 and
x
E
x the reflection of the point By convexity if
By (3.3)
x E l1>.
we see that
x
l1 - , define
(X l + 2( A - X l ) ' x' ),
=
across the plane Xl
then X E
n,
specifically
=
z
be two points in
and so
define
l1>.
=
{x E n :
u(y) < v(y)
=
=
n>..
with y
YI + Zl
u(y)
in nA
in
n-
2 =
Xl < A} .
u(:r,) .
2 . 1 applied
A Then Y
=
can
A . Let
nA , as also of course is u. u > 0 in n , one has u < v on an>. n an and u
z
we
=
is a solution of ( 1 . 1 ) in
Since 'I.t = 0 on an and anA \ an. Hence by Proposition Now let y,
and
v(x)
v
<
'
<
=
we find that
1L = V
on
z' and Y I < Zl < O . Choose
O.
o
u(z) .
References [11
Berestydri, L. Nirenberg, On the method of mO'uing planes and the sliding methods, Bol. Soc. B rasil Mat. 22 (1991), pp. 1-37. C . H . Conley, P. Pucci, J. Sen'in, Elliptic equations and products of positive definite matrices, Math . Nachrichten 278 (2005 ) , pp. 1490--1508. L. D»mascelli, Comparison theorems for Bom.e quasilinear degenerate elliptic operators and applications to symmet"y and mono tonicity results, Ann. Inst. H . Poincare. Analyse non lineaire 1 5 (1998), pp. 493-516. L. Damascelli, F . Pacella, Monotonir.ity and symmetry of solutions of p-Laplace equations, via the moving plane method, Ann. Seuola Norm. Sup. Pisa Cl . Sci. 26 (1 998) pp. 689-707. L. Damascelli, B . Sciunzi, Regularity monotonic-ity and symmetry of positive solution., of m-Laplace equations, J. Diff. Equations 206 (2004), pp. 483-5 15. L.E. Fraenkel, An int.,.oduction to Maximum Principles and Symm.etry in Elliptic Problems, Camhridge Tracts in Mathematics 128, Cambridge Univ. Press, 2000 . B. Gidas, W.-M. Ni, L. Nirenberg, Symmetry and related properties via the mU:J;imum prin ciple, Comm. Math. Phys. 68 (1979), pp. 209-243. H.
.
[21 [3J
[4J
,
[5J [6J [7)
PATRIZIA PUCCI, R8RARDINO SCIUNZI, AND JAMES SERRIN
444
[8] D.W. Nicholson, Eigenvalue bounds for AB + SA wit.h A, IR positive definite matrices, Linear Algebra Appl. 24 (1979), pp. 173-183. [9] F. Pacella, Symmetry results for solutions of semilinear elliptic equations with convex non linearities, J. Funct. Anal. 192 (2002), pp. 271-282. [10] P. Pucci, J . Serrin, The strong maximum principle revised, J. Ditr. Equations 1 9 6 (2004), 1--66; Erratum, J. Ditr. Equation" 207 (2004), 226-227. [11] P. Pucci, J. Berrin, The Maximum Principle, book in preparation, 173 pages. [ 1 2] J. Serrin, A symme!7'Y problem in potential theory, Arch. Rational Mech. Anal. 43 (1971) , pp. 304-318. [13] W.C. St rang , Eigenvalues of Jordan products, Arner. Math. Monthly 63 (1962), pp. 37-40. DTPARTIMEN'l'O
D! MATEM ATICA E INFORMATICA,
U N IVERSITA
DEGLI STUD! D! PERUGIA, VIA
VANVITELLI 1 , PERCGIA, ITALY
E-mail
address: pucci0dipmat . unipg . it
DIPARTIMENTO D! MATEMATICA, UNIVERSITA UI ROMA " LA SAPIENZA" , P .LE
A. MORO 2 ,
nOMA, ITALY
E-mail
address: scinDzi0axp . mat . uniroma2 . it
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MINNESOTA, MINNEAPOLIS, MN,
E-mail address:
serrinQwath . 11mn . edu
U.S.A
Contemporary Mathema.tics Volume 446, 2007
Single and Multi-transition Solutions of a Family of PDEs Paul H. Rabinowitz Dedicated to Harm Br'ezis for his 60th birthday
This paper surveys existence results [or the family of partial dif ferential equations described in (PDE) and (Fl l-(F2 ) below. It is shown using elementary minimization arguments that (PDE) possesses a large nllmber of spatial heteroclinic and homoclinic solutions. This repre8ents on the one hand a step towards a PDE version of A ubry-Mather theory and on the other hand a simple Illodel case for possible spatial transit ion behavior. A BSTRACT.
1.
Introduction
Tills paper describes and surveys results and states some open questions for the family of PDEs: (PDE)
where F satisfies
-tlu + F (x u ) = 0 u
,
F(x, z) is 1-periodic in Xl , "
,
"
x
E Rn
xn and in z.
Note that (PDE) encompasses prohlems ranging from the forced pendulum to Laplace's equation and more. In [1] , Moser initiated the study of a more general class of equations a::; a step towards developing a co dimension 1 or PDE version of Aubry-Mather theory [2]-[3] . On the other hand, phase transition models of Allen-Cahn type lead naturally to (PDE) To illustrate this latter connection, consider the equation (1)
-tlu + Wu (x, u ) = 0, x E R2
where W is a douhle well p otential. E.g. for definiteness, let W (x, z) a(x)z2 (1 z) 2 willch arises in modelling phase transitiuns for binary metallic alloys. Here a is positive and 1-periodic in X l , X2 . Now (1) has the "pure state" solutions u 0 and u 1 . It is of interest to find "mixed states" , i.e. solutions u of (1) with 0 < u < 1 and which are asymptotic to the mixed states. Set F(x, -z) = F(x, z) = W(x, z) =
_
This research was sponsored by the National Science Foundation under grant #MCS-8110556. @2007 445
A mer ic an Mathematical So ciety
PAUL H.
446
RABINOWITZ
for Z E [0 , 1 ] and make F 2-periodic in z. This leads to a problem of the form of (PDE) under (Fd-(F2 ) for which solutions between 0 and 1 are solutions of (1). Formally (PDE) arises as the Euler equation corresponding to the Lagrangian
L (u)
=
1 1V'u1 2 + F(x, u). 2
Moser was interm;ted in the existence and properties of solutions of (2)
a - I: FPi (x, u, Du) + Fz (x, u, Du) a Xi
=
°
which comes from the Lagrangian associated with F(x, z, p) . In particular, Moser studied solutions of (2) which were (i) "minimal" and (ii) "without self intersection" . For each "rotation vector" a E Rn, he found ordered classes No< of solutions of this type. The terms in quotes will be made precise in §2. Bangert [4] carried Moser's work furthcr. Let a E Qn and assume that Nex does not foliate Rn+1 , i.e. there is a region in Rn+ I bounded by the graphs of a pair, v < w, of members of No with no other members of No< between v and w. Under this gap condition, Bangert found solutions of (PDE) of type (i)- (ii) that were spatially heteroclinic from v to w and from w to v . In the setting of (1), v = 0, w = 1 , and the heteroclinics represent mixed states or single transition solution of ( 1 ) . More generally, spatial phase traiisitioIlS can be viewed as heteroclinic or homoclinic solutions of the corresponding model equations. Alessio, Jeanjean and Montecchiari [5]-[6] and Rabinowitz and Stredulinsky [7]-[8] found single and multi-transition solutions for ( 1 ) . By multi-transition solutions, we mean solutions that shadow or are near single transition solutions on certain regions. E.g. for (1), there are 2-transition solutions that are homo clinic to ° in Xl and 1 -periodic in X2. These solutions are near a single transition solution heteroclinic from 0 to 1 in Xl and I-periodic in X 2 for Xl < ° and near a single transition solution heteroclinic from 1 to ° in Xl and I-periodic in X2 for Xl > 0. The papers [5]-[8] use minimization arguments to get these solutions. Bangert [4] used rather different nonvariational methods to get his single transition hete roclinic. More recently it has been shown in [9]-[11] that (PDE) under (Ft}-(F2) together with certain gap conditions possesses a remarkable number of single and multi-transition solutions. These gap conditions are both necessary and sufficient. A mixture of ideas from the calculus of variations, partial differential equations, dynamical systems, and geometry is used to obtain these solutions. Thus the class of equations treated here can be viewed as a proving ground for tools, methods, and results which one might hope for in other phase transition problems. In the remainder of this paper, the various kinds of results that have been ob tained for (PDE) will be surveyed. In §2, Moser ' s results will be discussed more fully and Bangert's work on single transition solutions will be described. A vari ational approach using minimization arguments to find single transition solutions will be presented in §3 to treat the simplest case of Bangert. It requires the use of a renormalized functional. \Ne also discuss solutions of mountain pass type. In §4, "doubly" heteroclinic solutions which require a second renormalization are treated. The final two sections handle multi-transition solutions. There are two basic types of such solutions. Those that lie in a gap are di�cllssed in §5 while those that are monotone and cross gaps are studied in §6.
SINGLE AND MULTI·TRANSITION SOLUTIONS
OF
A FAMILY
OF
447
PDES
2. The work of Moser and Bangert As was mentioned in §l, Moser studied (2) which arises formally as the Euler equat.ion of _
Rn
F(x, u, Du)dx.
The function F(x, z, p) E C2 (Rn x R x Rn , R) and is assumed to be convex and coercive in p, and periodic in XI . ' . . , Xn and z. For simplicit,y take these periods to be one. Furthermore F and itH derivative satisfy conditions implying that weak solutions of (2) are classical solutions. In the spirit of Aubry-Mather Theory, Moser was concerned with solutions, u, of (2) that are (i) minimal and (ii) wit.hout self intersections. Here (i) means for all r.p E WI�'; (Rn , R) with compact support,
(3 ) i.e. for any
f (F(x , u + r.p, D(u + r.p» JR"
- F(x, u, Du» dx > O,
n c Rn with a smooth boundary, u minimizes
l F(x, u, Du)dx
over the class of functions which equal u on 80.. Such minimizers were studied by Giaquinta and Giusti [12J and, as we have learned from L. Nirenberg, even earlier by Morrey. Expressed analytically, property (ii) means for any j E zn and jn + 1 E Z, either u(x + j) - j,,+1 > u(x), == u(x) , or < u(x) for all x E Rn. Assuming the structural conditions for F, among other things, Moser proved THEOREM 1 . If u is a solution of (2) of type (i)- (ii) , there is an M > 0 and unique a E Rn such that l u (x) Q x l :::; M for all x E Rn.
The unique a = a(u) E Rn given by Theorem 1 is called the rotation vector of the solution. Now one can ask whether there are solutions of the above type. As a first step in this direction, Moser proved this is the case for a E Qn. Since it is relevant for the material that follows, the proof will be sketched. However for simplicity assume F(x, z , p) = � lpl2 + F(x, z) , i.e. we are in the setting of (POE) . Suppose Q = (p d ql , ' " , P /q ) , n n where Pi E N, qi E N\ {O}, and Pi and qi are relatively prime, 1 < i < n . In particular if Pi = 0, qi = 1 . Let e l , . . . , en denote t.he usual orthonormal basis in Rn. Set Ao = {u E WI�; (Rn, R)lu(x + q,Ci) = u (x) + Pi, 1 < i = n } . -
•
Let Q = rr� [0, qiJ . For u E An , set
1 ( - I'\7uI2 + F(x, n» dx Q 2
and define (4 ) Then we have:
1° NOt == {u E A"IIOt (u) = co} i 4> , THEOREM 2. 2° Any u E No is a solution of (POE) of type (i) (ii), 3° NO/. is an ordered set.. -
PAUL
448
ll.
!tABINOWITZ
Aside from the fact that u is of type (i) , the remaining assertions of 1°-2° follow from standard minimization and regularity arguments. That u is of type (i) is based in part on 3° which follows from (4) and the maximum principle. H aving Theorem 2 for a E Qn , Moser used an approximation argument to prove: THEOREM 3. For each with rotation vector Ct.
a
E Rn, there is a solution of (PDE) of type (i)-(ii)
It is an opcn question as to whether there is a direct minimization argument giving existence for the case of Ct E Rn\Qn There has been some work in this direction by Bessi [13] in the spirit of the minimal measures of Mather [14] . Henceforth we will only study the case of rotation vector a = O. All the relmlt.s for this case have analogues for a E Qn . To fix the notation, let L( u) = � lV'ul2 + F(x, u) {U E W;�'; (Rn , R) l u is I-periodic in Xl , . . . , Xn } , fo
(5)
Jo (u)
Tn
L(u)dx,
inf JO (U) ,
Co
"Ero
and Mo -
{u
E
fo I Jo (u)
=
co} .
Note that since F is I-periodic in Xi , it is also 1!i periodic in X, for any 1!i E N. In fact there are infinitely many choices for an orthogonal family of directions of periodicity for F. For each such choice,
J L(u)dx
(6)
can be minimized over the corresponding class of "Vl�'; functions periodic in these directions. This produces a solution of (PDE) as in Theorem 2. Any member of Mo is a candidate for the variational problem for the functional in (6). In fact Moser showed the set of minimizers in this broader class of functions is simply Mo. Retum.ing to fo , by Theorem 2, Mo i= ¢ and if u E Mo, by (F2) so is u + j for any j E Z. Since Mo is ordcred, either Mo foliates Rn+l or there are gaps in Mo, i.e. there exist adjacent Vo < Wo in Mo .
(*) 0
If (*)0 holds, we refer to Vo and Wo as a gap pair. In [11], it is shown that (*)0 is generic in the sen�e that if it fails, for any c > 0 and Vn E Mo, F can be replaced by F with IIF - Fllcl(T"+l) < c such that if M is the set for F corresponding to Mo for Fo, Nt = { va + j Ij E Z } . Assuming (*)0 holds, (PDE) was studied by Bangert [4] who fmmd a. solution of type (i)-(ii) which is heteroclinic in X I from Vo to 1JJo a.nd periodic in xz, . , Xn. Thus for all xz, . . . , xn , lim U(x) - wo (x ) . lim U(x) - vo(x) = 0 XJ Similarly there is a heteroclinic from Wo to 'Uo . The proof of these statements involves an approximation argument together with estimates derived by Moser. "
....
"'
...
. .
-. - 00
=
X l -OO
SINGLE
A N Ll M U LTI-TRANSITION
SOLUTIONS OF A FA:vIILY O F PDES
449
RBMAItK 1 . More generally, Bangert showed if �i = �aijei (with a'i.1 E Z ,and having no common faclor for fixed 'i) and the vectors Wi are orthogonal < i < n , then there is a solution of type ( i) - (ii) of (PDE) which is periodic in the W2 " " , Wn directions and heteroclinic from Vo to Wo in t he WI direction. Moreover for each
1
fixed set of wi's, as in 3° of Theorem 2, the set of corresponding solutions is ordered. The effect of varying Wi will be discussed after Theorem 6.
(el " ' " en ) , lct Nl denote Returning to the silIlple�t case of (WI , ' " , wn ) the ordered set of heteroclinics in XI from Vo to Wo which are of type (i)-(ii) and periodic in X 2 , " " Xn· If UI E NJ , so is r:j U1 (x) UI ( x + jel l and as for A1 0 , either Nl foliates the region in Rn+l between Vo and Wo or there are gap pairs in /Vl. If VI < W I is such a pair, B anger t showed there iH it Holution U2 of (PDE) of type (i) -(ii ) which is heteroclinic in X2 from VI to WI and periodic in X3, , , . , X n . SiJl(�e each of VI and W I are heteroclinic in Xl from VI) to W(), so is U2 . Furthermore, as above, the set, N2 , of such doubly heteroclinic U2 's is ordered. Thus if there arc gaps in .11/2 , there are triply heteroclinic solutions, et c . Whenever one has a HOlution , U , of (PDE) that is heteroclinic in X l from Vo to WI and likewise, U* , heteroclinic from Wo to vo , t he theory of dynamical systems suggests that there should be infin it ely many multi-transition solutions that shadow formal chains obt ained by gluing together phase shifts Df U and U·. The simplest kind of such solutions are therefore 2-transition homoclinics from Vo to 1)0 in X l which are periodic in X2 , . . . : Xn . Likewise there should be 2-transition homo clinics from 1)1 to VI which are periodic in X3" . . , X n . Such solutions w ill be disc ussed in §5. While not necessarily corresponding to a formal chain, there are aJso 2 (or possibly more) - transition heteroclinics from '1)0 to 1 + 'll!o which wi ll be treated in §6. Variational methods for finding such shadowing solutions have been developed in various settings, See e,g. Mather [15] and references [7]-[11] . These methods require a minimization formulation of the basic solutions which are used as build ing blocks to construct multi-transition solutions. Thus such a characterization is needed for the members of Nl , N2 , etc. However Bangert ' s existence results are not variational i n n ature Therefore the first step in finding multi-transition solutions is to develop a variational approach to obtain the basic heteruclinics. This will be the object of the next two sections. =
-
.
3. Renormalization and minimization A natural way
to find
heteroclinics from Vo to Wo is to minimize
rJRx
(7)
Tn-l
L ( ) d:r -/1,
over a class of functions with the desired asymptotics in X l . Unfortunately in generaJ the functionaJ in (7) will not be finite for any admissible function. E g. This will be the case for F(x, z ) > b where b is a p ositive constant. Thus (7) must be modified or as we prefer to say, renormalized in order to make the resulting functionaJ finite valued on a reasonable class of functions. Towards that end, let i E Z and Ti = [i , i + 1] x T,,-I , The cla.-;s of admissihle functions wi l l be taken to .
,
450
PAUL H. RABINOWITZ
be { u
lI u and Il u
+ Wl�'; (R x
-
VO Il£2(Ttl
Tn+ l R) l vo < u < WO,
--->
'Wo il£2 (T,J
,
0,
i
0,
i
--->
-
00
---> 00
,
}.
implies u is periodic in X 2 , , X n and the asymptotic behavior is achieved in a mild way. For E Z and U E r 1 Vo , wo) set
N ate
that u E
r 1 (vo , 'Wo)
-
•
i
For p
(5).
< q E Z,
J1 ,p,q(u)
[11J
that
r 1 ( Vo , wo) .
JI ;p , q (U)
•
-
Co
set q
L Jl ,i (u) , p
is bounded from below independently of p, q E
With the aid of Theorem in
2
=
.
(
r L (u)dx JTi
Jl ,i(u) =
where CO is defined by
--->
and some of the remarks following (6), it was shown
Z
and U
This suggests defining
J1 (u)
lim
=
p - - oo
J1;p,q (u).
q - OQ
This definition of .11 �eems awkward but as shown in
(I.{)
J1 (u)
lim
p - - oo
(9)
The properties (9) possessed by
.II
[11],
if J1 (u)
< 00,
then
J1 :p ,q(u)
i . e. the lim inf is a limit and moreover
lI u - Vo I I W,,2(T,J
E
0, i ->
0,
---> -00
2 -> 00. .
suggest that some compactness is built into it .
A� further evidence, define
ct (vo, wo)
(10) Then it was shown in
[11]
=
uEr, (vo .wo) inf
J1 (u) .
that any minimizing sequence for ( 1 0) is bounded in a
WI�'; (R x Tn-l , R). This implies it. has subsequence (Uk) converging weakly to some U E WI�'; (R x Tn-I, R) as k -> 00. But actually (Uk) eonverges strongly in WI�': (R x Tn-I , R) . Thus .11 satisfies a Palais-Smale type property for minirni7,ing
sequences . These facts lead to
4. ( [11] ) : (F2 ) If F satisfies (F1) THEOREM 1°
2°
(*)0 holds, M J (vo, 'WI)) {u E r 1 (vo , wo ) I JI (u) = c(vo , 'Wo ) } i= ¢ Any 1L E M 1 (vo , 'Wo) is a classical solution of (PDE) with 00 (a) I I U vo l l c2 (Td -> 0 , i II U wo llc2(Til -> 0, i -> - 00 (b) Vo < U < T�I U < 'Wo -
and
-
-
->
-
and 3° M1 (vo , wo)
is an ordered set.
-
,
SI'
4° U
E
451
Ml (Vo, wu ) implies U is minimal, i.e. of type (i) .
REMARK 2 . (i) The proof of Theorem 4 uses the special propertie:; of J1 ment.ioned above plus compactness and regularity arguments and the Maximum Principle. (ii) The monotonicity property, 2° (b) , of Theorem 4 implies U is of type (ii). (iii) Defining f 1 (wo, vo ) and cd1110, vol in the obvious way, (*)0 yields solutions of (P DE ) in M l (wo , vo l . (iv) The gap enndition (*)0 is not only necessary for the existence of minimiz ing heteroclinics but is also sufficient. Indeed as was shown in [11] , THEOREM 5. If (Ft } - (F2 ) holds, v < w belong to Mo, and there is a U [ 1 (v, w) such that inf J1 (u ) , .h (U) Cl (1J , W) _
E
uEr,(v,w)
=
then ("')0 holds with v and W a gap pair in Mo.
Theorem 5 is proved via a kind of "curve shortening" argument. The next re�ult �hows that although Bangert's class of heteroclinics and the sets Ml (vo, wo) U M l (wo, vo l are obtained rather differently, in fact they are the same. THEOREM 6. ( [1 1] ) : Let F satisfy (FI) - (F2) and suppose that u is a solution of (PDE) of type (i) - (ii) with rotation vector a = 0 and u(x+ei) = u(x), 2 < i < n. Then 1 ° '/1, E )lv/o or 2° u E },/h (vo , wn) u M 1 (wo , vo) for some gap pair va < Wo in Mo. Again comparison arguments play a key role in the proof of Theorem 6.
REMARK 3. As was mentioned in Remark 1 there are solutions, U - UW1 , ... of (PDE) of type (i) - (ii) heteroelinic in any direction WI as described there and periodic in the remaining directions. The same sort of arguments that. showed Mo depends is independent of such directional considerations also prove that UW1, only on W I . Thus far only solutions of (PDE) that are of type (i) - (ii) have been discussed. There are also solutions that are monotone in the sense of 20 (b) of Theorem 4 (and therefore of type (ii» but are not minimal. To see this, first consider 10 on fo. If (*)0 holds, then for any gap pair Vo < Wo, ,wn
_ _ _
Jo (vo )
=
Co
=
1o Cwo )
,wn
and 10 (u ) > Co for any u E fu\Mo. Geometrically this suggests that the functional Jo is of mountain pass type. In fact using the minimality properties of Vo and wo, one can show that there is a minimax solution u* E ro with Vo < n* < wo o Although 'll* i:; periodic in X l , . . . Xn and therefore of type (ii) , it cannot be minimal via Theorem 6. If (*) , holds, a similar analysis can be carried out in the setting of Theorem 4. Thus if VI and 7171 arc a gap pair in M 1 (Vo, wo) , J1 (111 ) = 11 (WI ) and an argument of mountain pass type - see [16] - shows there is a solution U* of (PDE) between VI and WI ' It is monotone since VI < U• < WI < _
'II 1 -
1 11 1 < T- 1 U' .
452
PAUL H. RAI3lNOWITZ
Again by Theorem 6, it cannot be minimal. \Ve have learned that
E.
Valdinoci and K.. de La Llave have obtained a related
result in the setting of an Allen-Cahn model equation.
4. Doubly heteroclinic solutions M I (vo, 1110)
As mentioned in §2, either
Wo
and
or there are gaps. In
[11],
foliate� the region in
it was shown that generically
Ml (vo, 1110 )
=
Rn+ l.
between
Vo
{ T':j Vl l j E Z},
i .e. it consists of the phase shifts of a single heteroclinic. In any event , suppose There is a gap pair Then as Bangert did in in x. from
VI
to
WI
VI < WI
in
M 1 (vo , 1110 ) '
[4], one can seck solutions of (PDE) that are heteroclinic
and periodic in
X3, . . . , Xn .
In this section, we seek to show
that �uch solutions can be obtained via a minimization argument. Following the template of §3, for
i
R x Ii, i + 1 ]
=
E Z, set Si
X
T,,-2 .
between the current setting and that of §3: the ' cubes' in contrast to the sets
Si here.
There is a major difference
Ti
of §3 are compact in Rn
This loss of compactness complicates the treatment
of the present case .
As class of admissible functions, we take
{
U E �1loc,2 (R2 x T,, -2) lv1
I l u - Vl l l £' (s, ) -t 0 , i -t lI u WI II £ ,(Si ) -t 0, i -t
-
-
and
<
_
U < W1 , _
00 ,
00 ,
I I V (u - vd l l £2 (Si) < 00, i E Z } . The new conditions on just mentioned. Let
-
J1 ( r2 iU) is not defined. However it is not difficult to show J1 extends to T� i u for all i E Z . Nevertheless in general :LiEZ Jl (T: iU) will
periodic in that
U
V(l/, vIJ required here are due to the loss of compactness E r2(VJ , Wl ) and set T:;U(X) = u(x + ie2 )' Since U is not
X2 ,
a priori
not be finite. Thus a second renormalizatioll is required. Fortunately it follows the same lines as in §a. For and for p < q
iEZ
and
h,i (U )
E Z,
U E r2 (Vl , W I ) . Let J1(T:iU) CI (VO , WO)
=
-
set
h ;p ,q(u) Then as in §3, there is a
1(2 > 0
=
q
L Jz,i (U), p
such that for all such p,
J2 ;p,q( U) > -K2
and thus allows 111) to define
Jz (U)
=
lim p _ _ ou q - oo
Jz;p ,q (u).
q, u,
SINGLE AND MULTI-TRANSITION SOLUTIONS OF I\. FAMILY OF PDES
453
As in §3, if h C u) < 00, the lim becomes a lim and the L2 (S;) convergence in the defiuition of r2(Vl l Wd becomes W l , 2 (Sd convergence. Moreoyer J2 possesses the same Palais-Smale type property for minimi7.ing sequences that J1 has. Thus if (11)
then the following analogue of Theorem 4 obtains:
THEOREM 7. ( [11)): If F satisfies CF1) - (F2) and (*) 0 , ( *h hold, then
1° M2(Vl , Wl ) - { u E r2 (Vl , wd l h (u) = C2 (Vl , wd } f=
The analogues of the remarks after Theorem 4 also apply here.
5. Multi-transition solutions between Vo and Wo Given t.he basic I-transition heteroclinic solutions of (PDE) of §3-4, there are more complex multi-transition solutions which are not minimal but have somc local minimality properties. There are two main classes of such solutions. This section treats solutions lying bet.ween '00 and Wo which shadow members of M 1 (vo , wo) and M 1 (wo, vo) . The simplest case is that of two transition solutions hOIIlodinic to Vo or to wo0 More generally there are k transition solutions alternately shad owing members of M 1 (vo , wo) and M 1 (wo, vo ) . Indeed as will be indicated below, oncc one has the two transition cases in hand, the existence of k and even infinite transition solutions can be proved via an clementary geometrical argument. There are an infinite number of 2-transition solutions of (PDE) that shadow members of M 1 (vo, wo) and M 1 (wo, vo ) . To find them, J1 is again minimized over a class of admissible functions which satisfy more constraints than those controlling asymptotic behavior as in rl (vo , wo ) and ft Cwo , vo ) . To be more precise, for u E M t Cvo , wo) , define p- Cu) = I l u - vo IIL2 (To ) and
p+ (u)
=
I l u - wo l l u (To ) '
Then P± : Ml (vo , wo) --; (0, I l wo - Vo I I VeTo » ) ' Since Ml (vo, wo) is an ordered set, p± arc monotone maps and by ( * h the sets p± (M 1 (vo , wo) ) contain gaps. Choose P I in such a gap for p_ ami (l2 for p+ . Replacing M 1 (vo , wo) by M 1 (wo, vo ) , a similar argument yields a P3 for p+ and P4 for p_ . Next let f E N and m E Z4 wit,h rn l < Tn2 < rn2 + 2£ < m3 < m4 . The class of admissible functions for 2-transition homoclinics to Vo is: Ym,e
Ym,e(vo, wo) - { U E Wl�'� (R x Tn-I, R) I Vo :::; u < Wo and u satisfies ( 12)-(13) }
PAUL H. RABINOWITZ
454
where ( 1 2)
(i) (ii) (iii) (iv)
-e
p_ (r:iu) < PI p+ (r:iu ) < P2 p+(r:;u ) < P3 p- (r:iu) < P4
ml
and
'Tnl
( 1 3) Now set ( 14) Then we have
THEOREM 8. ( [1 1 ] ) : Suppuse (Fd - (F2) are satisfied and ( * ) 0 and ( * h for Ml (VO , WO ) and Ml (wo, vo) hold. Then if e, m2 - m}, m4 - m3 » 1, there is a U E Ym,£ such that Jl (U) = bm,f. Any such U is a solution uf (PDE) and II U - vo l l c 2( Ti ) ---> 0
as
Iii
---> 00 .
A few words about the proof of Theorem 8 are in order. As for Theorem 4 and 7, there is a minimizing sequence for (14) which converges in Wl�'; (R x Tn- I , R) to U E Wl��2 (R x Tn - I , R) with Vu < U < Wo . Moreover as in the earlier proofs, U is a solution of (PDE) away from the c:oIlstraillt regions ( 12). If U satisfies each of these 4 sets of constraints with strict inequality, it is also a solution of (PDE) in these regions via standard regularity arguments. Hence U is a solution in R x Tn- I . To verify the strict inequalities, it is first shown that for each of the 4 sets of constraint regiuns, there is some i such that strict inequality holds for the corresponding Ti . Therefore (PDE) is satisfied in T;, . This fact with a comparison argument shows ( 13) holds and therefore U E Ym, e. Knowing this allows us to prove lI eU) = bm,t Exploiting that U is a minimizer for (14) , further comparison arguments yield strict inequality for all of the constraint T;, 's, completing the proof.
REMARK 4. (i) Applying Theorem 8 with increasingly larger values of e, 'fil2 ml, and m4 - m3 shows that there are infinitely many such 2-transition solutions of (PDE). Thus in some sense this give a 3 parameter family of solutions of (PDE) . It is an open question as to whether 3 parameters are needed. (ii) Clearly these solutions are not minimal or monotone. However they are interior minima of 1I in Ym,e and Lhey also have an asymptotic minimality property, i.e. there is an R > 0 such that (3) holds for all 'P with 'P(x) 0 for IXI I < H. =
Having established the existence of a 2-transition homoclinic solut.ion, V, of (PDE) in Ym/(vo , wo) and similarly of W in Ym l (wO, vol , we will sketch a rather , geometrical approach for findillg 3-transition solutions. A related argument has been lIsed Lo find solutions of a second order Hamiltonian system in R2 [17]. Due to their respective asymptotic behaviors in Xl, for .il E Z with jl « - 1 , TlI W ( X) > V{x ) for all x E R x Tn- I . Likewise for h » 1 , Ti2 W > V and for is » h, Tj, V < Til W. Set 4>(x)
= max(V, rJV) ; >;3
SINGLE AND MULTI-TRANSITION SOLL"TIONS OF A FAMILY OF
and
PDBS
4.5.5
Then if
A(
b(
=
inf
UEA(<J>,w)
J] (u) .
Then the arguments of Theorems 4 and 7 show there it> a minimizer of Jl in A( cP, W ) , it is a solution of (PDE) , and it is heteroclinic in X l from Vo to woo The above construction works whenever jl « 12 « h. Thus there are infinitely many such 3-transition solutions of (PDE) . Moreover the construction extends to arbitrary k E N with a corresponding k-tuple jl < < j2 < < . . . < < jn yielding a k-transition homoclinic or heteroclinic lying between the associated
6. Monotone Illulti-transition solutions The second main class of multi-transition solutions are heteroclinics between nonadjacent members of Mo. These solutions arc monotone, but by Theorem 5 cannot be minimal. The main result here is:
THEOREM 9. ( [11] ): Let F satisfy (Fl ) - (F2 ) and suppose that ( * ) 0 is satisfied. Let vo, Wo and V, w be gap pairs in Mo with Wo < V. If ( * h holds for M1 (vo, wo) and M 1 (v, w), then there exist infinitely many monotone solutions of (PDE) in r 1 (VO, W). It is surprising that even if there are continua in Mo between Wo and v or continua in M1 (v* , w*) for some gap pair v*, w* again between Wo and 'fI, there arc still heteroclinic solutions from Vo to w.
•
PAUL H . RABINOWITZ
456
As in §5, these HolutiollS of (PDE) can be obtained by minimizing Jl over an appropriate class of admissible functions. Indeed this can be done very much in the spirit of Theorem 8. Howeve r such an approach rloes not seem to yield the monotonicit.v of the soluLions. Therefore a different class of admissible functions which are already monotone will be employed. The price paid for this morc direct approach is that it becomes morc difficult to prove the regularity of the resulting . l mm mlzers . To introduce the class of admissible functions here, let ,.
,
.
30
Then and by (*h for
with
s,
= { r hdxlh E Ml (vo, wo n
.ITo
¢ =I 30 c
(.IrTo vod:r,
M 1 (vo, wo), 30 contains gaps. s,
tE
(r
iTo
vodx,
t in a pair of different gaps. Therefore
Similarly choose
.
r 1JiO d.T ) .ITo Choose
To
wodx) hdx < t} =I
To
s, l in gaps in •
J = { r hdxlh E M1 (v, w)}
.ITo
so
C = C(s, i) = {h E M1(v, w) l.s <
r
.ITo
hdJ; < [} =I ¢; .
The goal is to construct a solution of (PDE) that shadows some ho E Cu and Towards that end, choose m = (mb m:.d E Z2 with ml + 4 < m2 and set
(16)
Ym
{ U E Wl�;(R x Tn-l , R) lvo U satisfies ( 17) ( 19) }
< < 111 1L
where
( 1 7) ( 18) and
s<
( 1 9) Now define
(20)
1T"�2 max(u, li)d:r; < t. b", = inf J1 (u). •
uEYm
Then Theorem 9 and more follow from the TIIore precise result ;
and
h E C. A
A
SINGLE AND M U LTI-THANSl'1'ION SOLUTIONS OF A FAMILY O F
PDES
457
THEOREM 10. ( [11] ) : Suppose F satisfies (Fd - (F2 ) , (*)0 holds, (vo, wo) and
(il, 'll;) are associated gap pairs wi th 11]0 < il, and M 1 (vo , wo), M l (ii , w) satisfy (*h, , Then there is a U E Ym such that :lI (U) = bm . If m2 » ml , any such U is a solution of (PDE) , II U - vo l l c2 (Ti l II U - wo llc2 (To )
(21)
and
0, 0,
i -; i -;
- :Xl ,
00,
1 U' < w. Vo < U < 7_1
(22)
A
Moreover for any p, R > 0
(; E 6(,5, t) such that
and
'rrL2 - 'lnl possibly still larger, there is a Ua E Go (s, t),
(28)
and
(24) To prove Theorem 10, as earlier it can be assumed there is a minimizing se quence for (20) which converges in Wl�'; (R x Tn- I , R) to U E Wl�'� (R x Tn-I , R) with U < 7:'IU and J1 (U) < 00. Using (18) and ( 1 9) leads to (2 1 ) . Therefore U E Ym and earlier arguments show J1 (U) = bTn• Next comparison arguments like those of Theorem 8 show for m2 > > m l , the constraints hold with strict inequality. This fact with a local minimization argument from [7] shows U is a solution of (PDE). Use of the maximum principle yields (22) as in earlier results. Lastly to prove the shadowing estimates (2a)-(24) requires a result of indepen dent interest. A
PROPOSITION 1 . For any E > 0, there is a 0 = 0 (E) such that if u E r 1 (va , wo) with J1 (u) < Cl (vo , wo) + 0, then there is a W E M l (Vo, wo) such that for all i E Z, Ilu - w ll wl .2(Ti_1LJTiLJT,+1 ) < f.
(25)
Theorem 10.
Proposition 1 and comparison arguments then yield the rest of t he proof of
RRMARK 5. (i) Using th at U as given by Theorem 10 sati sfies U < T:'1 U, it is shown in [18] that there is a solution of mountain pass type between U and 7:' 1 U . (ii) One can construct formal chains of solutions of (P DE) from those given by Theorem 8 and 10. The arguments of Theorem 8 can then be employed to get actual such solutions that shadow the chains. Nothing is yet known about corresponding solutions of mountain pass type. (iii) As was mentioned in §2, all of the results of this paper for rotation vector (Y = 0 have analogues for lY E Qn and B anger t has done some work when a E Rn\ Qn. However essentially nothing is known about solutions of the type constructed in §5-6 when a E Rn \ Qn aside from a paper of Kessi
[13] .
(iv) Another completely open question concerns analogues of the results of t his paper for systems. •
PACL H. RABINOWITZ
458
References [IJ
Moser, J . : Minimal solutions
of a variational pr o blem on a torus.
AIHP, Analyse
Nonlineaire ,
3, 229-272 (1986) [2J Aubry, S . , LcDacron, P.Y.: The discrete Frenkel-Kantorova model and its extensions, 1. Exact results for the ground states . Physica D, 8 (1983), 381-422 [3J Mather, J . N . : Existence of quasi-periodic orbits for twist homeomorphisms of the annulus. Topology, 2 1 , 457-467 (1 982) [4J B angert , V.: On minimal laminations of the torus. AIHP, Analyse Nonlincaire, 6 , 95-138
(1989) [5J
Alessio,
F.,
Jeanjean,
L.,
Montecchiari,
Stationary layered solutions for
Var .
,
11,
.
[7J
[8J [9J [10J [11 J [12J [13J [14J [15J [16J [17J [18J
a class of nonau
277-302 (2000) Alessio, F., Jeanjean, L . , M ontecchiari, P.: Existence of infinitely many stationary layered solutions for a class of periodic Allen-Cahn equations. Comm. P D . E , 27, 1537-1574 (2002) Rahinowit.z, P.H . , St.redulinsky E.: M ixed states for an Allen-Cahn type equation. Comm. Pure Appl. Math., 56, 1078-11 34 (2003) Rabinowitz, P.H., Stredulinsky, E.: M ixed states for an Allen-Cahn type equation , II. Calc. Var. , 2 1 , 157-207 (2004) Rabinowitz, P.H . , St redulinsky, E.: On some results of Moser and B angert. AIHP, Analyse Nonlineair e, 2 1 , 673-688 (2004). Rabinowitz, P. I I . , Stredulinsky, E.: On some results of Moser and Bangert, II. Adv. Non linear St.ud. , 4, 377-396 (2004) . Rabinowitz, P.H . , Stredulinsky E.: In progress Giaquinta, M . , Giusti, E . : On the regularity of the minima of variational integrals. Acta. Math., 148, 31-46 ( 1 9 82 ) Bessi, U.: Many solutions of elliptic problems on Rn of irrational slope. Comm. P. D.E., 30, 1773-1804 (2005) M ather J.: M ore Denjoy minimal sets for area preserving cliffeomorphisms. Comm. Math. Helv., 60, 508 557 ( 1 985) M at her J . : Variational construction of connecting orbits. Ann. Inst. Fourier (grenoble ) , 43, 1349-1385 (1993) Bolotin, S . , Rabinowitz, P.H . : A note on heteroc1inic solutions of mountain pass type for a class of nonlinear elliptic PDE's. C ontributions to nonlinear analysis, 105-114, Progr . Nonlinear Differential Equations Appl. , 66, Birkhauser, Basel, 2006 Rabinowitz, P.H . , Solutions of a Lagrangian system on T2 . Proc. Nat!. Acad. Sci. USA, 96, 6037-6041 (1999) Bolotin S., Rabinowitz , P.H . : In progress tonomous Allen-Cahn equations. Calc.
[6J
P.:
.
,
,
,
,
DEPARTMENT OF MATHEMATICS, l' NIVERSITY OF WISCONSIN, MADISON, WISCONSIN 5370 6 RABINowr@MATH . WI SC . EDU
Contempora.ry Mathematics Volume 446, �OU7 •
Some Methods and Issues in the Dynamics of Vortices in the Parabolic Ginzburg-Landau Equations Sylvia Serfaty
Dedicated to Haim Brezis with best wishes on the occasion of his 60th birthday. Thank you Haim, for constantly sharing your enthusiasm for mathematics and pushing them forwU1'd, and for the encouragement and a.ttention you prOVide to many and in particular to young mathematicians.
1. Introduction We are interested in the parabolic flow for the Ginzburg-Landau energy ( Ll )
E,, (u)
=
1 2
n
l \7u l 2
+
( 1 - l u I 2 )2 2c:2 '
.
where 0 is a smooth bounded (simply-connected) domain of � 2 and u maps 0 to
The first main study of this functional was done in the book of Bethuel-Drezis Helein [BBB] . A large literature has followed. This energy, which can also be seen as a complex-valued version of the Allen Cahn model of phase-transitions, is a simplified version of the Ginzburg-Landau model £mm the physics of superconductivity: 1 1 ( 2 r 2 2 2 ( 1 .3) GtCu, A) 2 n 1 \7 A ul + Icurl A - hex l + 2 2 1 - l u i ) . c I Here 0 corresponds to the cross-section of the sample (assuming everything is translation-invariant in the third direction) . The function u : 0 ->
�
�
-7
=
=
2000 Ma.thematics Subject Classification. 82D55, 35840, 35K55, 35B25, 35J20, 35Q40. •
45D
©2007 American Mathematical Society
·160
SYLVIA SERFATY
sample. The real parameter hex is the intensity of the external applied magnetic field. Eo corre�ponds to a version without magnetic field of Go : it can be obtained by simply setting all the magnetic field terms equal to O. The parameter II,:: is call1Ki the Ginzburg-Landau parameter, it is a dimen sionless parameLer depending on the material (ratio of two characteristic lengths) . vVhen lie is large enough, we are in the category of "type-II" �uperconductors, when ,:: -- 0, they arc sometimes eaIled "extreme type-II" (or this is also called the "London limit" ) . This is the asymptotic regime we will be interested in. The Ginzburg-Landau energy was introduced by Ginzburg and Landau in the 50s as a modcl for superconductivity. It was first a phenomenological theory, but it was later derived (in a certain limit) from the microscopic (quantic) theory of ilardeen-Cooper-Schrieffer. It is now a widely accepted model, which has earned its inventors the Physics Nobel Prize (in 2003 for Ginzburg) . Another motivation is (,he modelling of superfiuidity, a phenomenon very close to superconductivity, both mathematically and physically, and of Bose-Einsl.ein condensates. All these physical phenomenH have in common the appearance of topological vortices, which are the main object of our study. For further reference, we refer to the physics literature, e.g. [T, DeG] . . . For more mathematical references on (1 .3), and results on its minimizers, their vortices, critical fields, etc, we refer to the monograph [SS5] and the references therein. 1.1.
Vortices. A vortex can be defined as a point where the order parameter u vanishes, around which I�I has a nonzero topological degree. It ca.n also be seen as the object surrounding the zero. For example, a typical vortex centered at a point Xo may "look like" u = pe' 'P with p ( xo) = 0 and p = f( ' X7°') where 1 (0) = 0 and f tends to 1 as 'f' +00, i.e., its characteristic core size i s c , and --
� �� J 2
=
d
E
Z
is an integer, called the degree of the vortex. For example ip = dO, where 0 is the polar angle centered at xo , yields a vortex of degree d. We have the important relation curl Vip = 27l' di c5a;
L ,
where the a; 's are the centerb of the vortices and the d; 's their degrees.
1.2. Goal of these notes. The goal of these
is to present some results on the parabolic flow of (1.2), in particular to present one method to derive the vortex Illotion law. Along the way we will also present several useful t,ools and related results. It turns out that (for energy reasons, as we shall explain later) , in order to see vortices move, one needs to consider the parabolic flow of ( 1 . 2) acceleraterl in time: (1 .4)
OtU I I og c I
=
notes
u 2 + u u (1 - l l ). � 2' e
As we mentioned, we focus on the asymptotic limit E ---> 0, and mainly on the situation of a finite number (bounded as e 0) of vortices. There have been several works studying these dynamics (mainly [Lil, JS2, BOSI, BOS2, BOS3] ) . Here --
DYNAMICS OF VORTICES IN GINZBURG-LANDAU
461
we wish to derive the dynamics through an abstract 'and energy-baHed method which we called ''l'-convergence of gradient flows" and which is based on the idea that ( 1.4) is a gradient-flow ( the L 2 gradient-flow) for Ee and that E, itself f-converges. We will present the abstract method in details.
1.3. r-convergence of Ginzburg-Landau. Before stating the result, let us specify the notion of f-convergence that we will use.
( f-convergence) . A family of functionals Ee (defined on M e) f-converges to a functional F (defined on H) if DEFINITION 1 . 1
(1)
If E (u ) < C then up to extraction, u" u E H we have lim Ee ( ue) > F(u). ,
,,
s
--'
U E H,
amI
for
s
every u" --'
e�O
(2) For every u E H, there exists u" E Me .3 11. E H such that lim E,, (u,;) < F(u) .
e--+O
The sense of convergence S is to be specified beforehand. It can be a weak or strong convergence of u" , it can also be a convergence of a nonlinear function of 1l" . Recall that this notion of f-convergence ensures that if Uc minimizes Ee on M" then 1l" .3 u which minimizes F over H. 1.4. Tracking down the vortices. The easiest way to track down the vor tices is to introduce the vorticity as in fluid dynamics, 3.'l the curl of the cur rent (iu, yu) ( or the "superconducting current" (iu, YAU) for the case with mag netic field) where ( . . ) denotes the scalar product in C as identified with IR2 , i.e. , (iu, yu) = (u x al u, u x a2 U) with x the vector product in IR2 . Writing 'U = [lei", we have (at least formally) 2 (iu, yu) = p yrp aJld since p is close to 1 on lengthscales 10 , the quantity ,
( l .5)
curl (iu, yu)
=
curl (p2yrp)
c:::
curl Vrp
=
27f
L dioa, •
can b e used to trace the vortices. This is also called the Jacobian determinant if written (with differential forms) Ju = d (iu du} = (-idu, dll) = 271."" X U"'2 dXl /l dX2 . The approximation is justified as a limit as 10 0 through the following analysis (derived from the ball-const.ruction method of JeITard/Sandier ) : ,
......
THEOREM 1.2 (see [JS2, SS5] ) . Assume E,, (ue) < Cl log c J , then for any r <
1 (possibly depending on f) there exists a finite collection of disjoint closed balls Bi = B(ai , r;) with I: ri < r, such that 1 l u I < -2 ,. U · B · (the B; 's caver' the zeroes of 71,,), �
c
-
.
,
d;
(l .6)
•
=
deg (ue, BEi),
46 2
SYLVIA SERFATY
Combining the upper bound E,, (u) < Cilog EI and the lower bound (1.6) we deduce that L:, l di l < C for a constant C independent of E and thus the number of vortices of nonzero dp.gree remains bounded independently of c: . Thus, if u" is a family of such configurations, once the af's are found, we may extract a subsequence such that L:i dfoa� L:i dioa, in the weak sense of measures. These fixed points a 1 , ' . . , an are the limiting vortices. From ( 1 .7), taking for example r = Ilog EI- 2 , we obtain that curl (iu" , 'Vu,, ) 27f L dioa" ->
---"
t
thus the quantity curl (iu. , 'V u ) which is an intrinsic quantity (independent of the ball-construction above), does trace the vortices. WP. will sornetimes write u (ai , di ) for the limiting points+degrees configura tions in (!1 x z) n . In the study [BBR], they imposed some Dirichlet boundary condition Ue = 9 on an, deg 9 # 0, so that there have to be vortices. This is in contrast with the case of the full functional ( 1.3), for which the natural boundary condition is Neumann, and vortices appear due to the applied magneti c field. I n what follows, we Lreat both eases of this fixed Dirichlet boundary condition, and of the homogeneous Neumann condition. Let us now recall one of the main results adapted from [BBR] . ,,
,
=
([BBR]) . Assume Ee (ue) < Cllog c:1 and ue = 9 Then, up to extraction, n 27f L dioa, di E Z curl (iue, 'Vue )
THEOREM 1.3
deg 9 #
O.
---"
and
on
an, with
,=1
Here, Wd is a funcLion on nn called the renormalized energy in [BBR] . It contains the (logarithmic) interaction energy between the vortices -7f L:i;O!j log lai aj l , plus terms corresponding to the interaction with the boundary: we see that vortices with degrees of same sign repel while vortices with degrees of opposite signs attract. The cost for vOl'Lices of Id; j > 1 is in fact higher (like dt l log c l ) so for a given degree d > 0 on the boundary, in order to minimize the energy, one needs to choose d vortices of degree + 1, and then to minimi7.e the remaining lower order interaction term Wd , which is independent of c: and governs the locations of the limiting vortices. From now on, we will reduce to t.he case di = ±l and will use the following re�ult
(1) Assume E. (u,,) THEOREM 1 .4 (Gamma-convergence of Ginzburg-Landau). $ Cllog c:1 and u. = 9 on an o r �� = 0 on an, then, up to e:r.tmction, curl (iue 'Vue )
and if V i, di
=
±l
lim (E,, (ue)
0-0
-
->.
27f
n
L diOn; i=1
7fnllog EI) > Wd (a 1 , ' " , an ) ,
DYNAMICS OF VORTICES IN GINZBURG-LANDAU
463
(2) For all (ai, d;) , di = ±1, there exists ue s'l.u;h that lim (E. (ue ) - 1rnllog cD <
a:-O
Wd (al , ' "
,
an) ·
Phrased this way, it is a result of r-convergence of Ee 1rnllog d. We thus reduce minimizing Ee to minimizing Wd which is a nontrivial finite-dimensional pro blem (interaction of point charges) . -
2. The abstract result for r-convergence of gradient-flows Let us now present the abstract met.hod of passing to the limit in gradient-flows of energies which r-converge.
2.1. The abstract situation. Let Ee be again a family of functionals defined on Me (see [SS4] for an idea of what kind of space Me should be . . . ) and F be a functional defined on ]o! such that Ee r-converges to F for thc sense of convergence S, in the sense of Definition 1.1. If we consider a solution of the gradient-flow (or steepest descent) of Ee on ME i.e.,
[hUe = -VEe (ue ) ,
does ue(t)
� u (t) for some u (t ) and more importantly, does u(t) satisfy OtU
- VF(u)? An example with a positive answer is that of the convergence of solutions to the Allen-Cahn equation to mean-curvature flow (which is the gradient flow of the r-limiting perimeter functional) , a delicate result proved by PDB methods (see
[eh, IJ).
Let us point out that the answer in general is negative without further as sumptions. Indeed, a necessary condition is that critical points of Ee converge to critical points of F, but under only the assumption that Ee r-converges to F, this is not necessarily true, because it is easy to perturb Ee by a. sma.ll oscillating func tional, perturbing VEe at order 1 without changing the r-limit, possibly adding new critical points which will not converge to critical points of F. We are thus searching for - an abstract result - an energy-based method - new extra conditions for convergence to occur. Observe that these have to involve the C l (or tangent) structure of the energy landscape, i.c . , be conditions on the derivatives of the energy and not only of the energies themselves. We have been sloppy until now, by writing Dtue = - VE, (ue) and calling this the gradient.-flow of E" . Since we are in infinite dimensions (in general), we need to specify what we mean by gradient, i.e., gradient with respect to which structure. There are many possible choices, each leading to a different gradient-flow. For example, the Allen-Cahn equation is the gradient flow for Me for the L2 structure. We could consider other structures, for example the gradient-flow with respect to the H- 1 structure, whieh then yields a totally different dynamics, called the Cahn Hilliard equation. So, when looking for a result of convergence, we need to specify what the structure for the limiting flow should he (recall that the limiting flow is }I! i- Me') Another element that not taken in the same space, UE E Me should corne into play is possible time-rescalings as we pass to the limit c -t O. s
---"
. U E
SYLViA SRRFATY
464
2.2. The result. For simplicity we will reduce to the following case: Ec is a family of CI function als defined over M , an open set of a B anach space B con tinuously embedded into a Hilbert space Xc (or of an affine space associated to It Banach) . We assume Eg r ) F, with F a CI functional defined over N, open set of a finit.e-dimensional vector space B' embed ded into a finite-dimensional Hilbert space Y. sen s
DEFINITION 2 . 1 . 8g r-l:OIlVerges along the trajectory ue(t) (t
[0, T» in the
e S to F if there exists u(t) E N and a subsequence (still denoted 'ug) such t hat s ) u(t) and "It E [0, T), ug (t) "It E [0, T)
E
lim Eg (u, (t» > F (u( t» . E-'O
the differential of Eg aL u., is linear continuous on Xc , it is represented by a vecLor in X" denote it by Vx, E, (u) ( gradient for the structure Xg) , ch aracterized by
DEFINITION 2.2.
If dEc ( uc) ,
ddt It=O
b'g (U + t Qi) = dE. (u) . ¢ = ('9 x. Eg(u ) , rj;)x . ,
'
<
If this gradient does not exist, we use the convention 119 x, Ec(u) II x, = 00.
For example, for the dynamics of Ginzburg-Landau, recall we wi�1t t.o study the equation (2.1 )
GtU u+ fl I log E I =
u
E
2 1 ). ui l 2(
To fit into the previous framework, Eg is defined on H I (n, q, we take B H1 (0, q c L2 (0) and we define the Xc structure by
�
I 1 = · 2
I IO� E I II . 1112 (0) '
II · lit = 1 10 E I n Le., a rescaled version of L2 Then B embeds continuously in X. , and the gradient for the structure Xg is 'Vx, Eg (u) = J log cl (flu + ; ( 1 - luI2») .
Then the PDE ( 1 .4) is indeed exactly GtU = - Vx, Ee (u) i.e . , the gradient-flow for t he structure Xc ' Recall that for a solution of
(2.2)
we have (with enough smoothness)
(Gt U" , Gt U,,)x, = - ( 'Vx, E,, (11e ) ,Eltug)x. = -OtE,, (u,,)
foT 1 1 f)t'uE II �, dt
OIl
of the
gradient-flow for Ec with respect to the [0, T) is a map UE E H l ([0, T), X,) such that
DEFINITION 2.3. A solution structure Xc
= E. (u,,(O» - E£ (uE(T» .
OtU, = -Vx, E, (u, ) E X"
for a.e. t E [O, T) .
DYNAMICS OF VORTICES IN GINZBURG-LANDAU
465
Such a solution is said to be conservative if Vt E [0, T) ,
[ IIOtUe (S) llt ds, "
E,,(u,,(O» - Ee (u,, (t»
=
(this is true if UE is smooth enough) . If Uo is such a family of solutions in [0, T) and Eo r , F along uE (t ) (in the sense of Definition 2) , we define the energy-excess D(t) by D(t) = lim D,,(t) > 0, ,,-->0
where DE (t) = E, (u, (t» - F(u(t» > o(l ) . A family of solutions of the gradient flow is said to be well-prepared initially if D(O) = 0. Recall
that F is always a lower bound for E" . Also, it is always possible tu construct well-prepared initial data from part 2) of the r-convergence definition, Definition 1 . We define similarly the gradient-flow for F for the structure Y . THEOREM 2 ..1 (Abstract result [SS4]) . Let E" and F be C 1 functionals over r
M and N respectively, E, of the flow of E.
-
,
F, and let u" be a family of conservative solutions
(2 . 3)
with u,, (O) � U o , along which E, r-converges to F in the sense of Definition 2. 1 . Assume moreover thai 1) and 2) or 2 ') below are satisfied:
1 ) (lower bound) For a subsequence such that uo (t) ..:i u(t), we have u E H l « 0, T), Y) and Vs E [0, T) , lim
(2.4)
,,-->0
2) For any t E [0, T) E--+O
lim IIV'E,, (1t,, (t) II�< > IIV' F (u(t» II�.
(2.5) 2 ')
(construction) If u, � u, for any V E Y, any v defined in a neighborhood of ° satisfying = v (O) U Dtv (O) = V there exists v,, (t) such that v,(O) = u" ,
lim l I atv, ( O) II� < l I at v (O ) IIi-E-O �
lim
,_0
-�
dt I t=o
E, (v,) >
-
�
dt It=o
=
F(v)
11V 1li-- , =
- (V'F(u) , V)y.
Then if D ( O) = 0 (i. e., it is well prepared) we have D (t) inequalities above are equalities and Vt E [0 T), uo (t) ..:i u(t)
=
0 Vt E [0, T), all
,
- V' yF(u) 11.(0) = UJ) at'll
i. e.,
11.
=
is a solution of the gradient flow for F for the structure Y .
SYLVIA SERFATY
466
2.3. Interpretation and further remarks. This theorem means that under conditions 1 ) and 2), or 1 ) and 2') (since 2') implies 2)), solutions of the gradient flow of E" for the structure X, converge to solutions of limiting gradient-flow (for the structure Y) if well-prepared. Let us make a few additional comments: (1) The difficulty is not in proving this theorem but in proving that in specific cases the conditions hold. (2) The limiting structure Y is somehow embedded in the conditions 1) and 2). The time rescalings are embedded in X, . (3) In general we expect 1 ) and 2) to be satisfied for any u, 3. u or u,,(t) 3. u(t) not necessarily solutions (here we required it only for solutions) (4) 1) and 2) do provide the extra Cl-order couditions on f-convergence. 2) in particular implies that critical points converge to critical points. (5) We can weaken conditions 1) and 2) to
lim
t 1l8tu,, 11 2 > t 1 18t u ll 2 Jo
.-+o Jo
-
O (D (t ))
lim IlVx, E. ( u,J llt > IlVF(u) ll� - O(D(t) ) I
$--+0
where D(t) is the energy-excess, and handle the terms in D(t) in the proof via a Gronwall's lemma (finally obtaining that D(t) == 0 if D(O) = 0). (6) The method should and can be extended to infinite-dimensional limiting spaces and to the case where the Hilbert structures X. and Y (in partic ular Y) depend on the point: such as Y" = L� forming a sort of Hilbert manifold structure. It would thus be interesting to see how, through r-convergence, the structures underlying the gradient-flows can become "curved" at the limit, even though they are not curved originally at the [ level, and also become possibly nonsmooth and nondifferentiable. In fact we can write down an analogue abstract result using the theory of "min imizing movements" of De Giorgi formalized by Ambrosio-Gigli-Savare [AGS] , a notion of gradient flows on structures which are not differen tiable but simply metric structures.
2.4. Idea of the proof. Since the proof is elementary, let us see how 1) and 2) imply the result. We assume
8
8
467
DYNAMICS OF VORTICES IN GINZBURG-LANDAU
Then, for all t <
T
we may write
E,, (u,, (O)) - E,,(u,, (t))
-
=
o
(\1xo Ee (ug ( s )) , th u. (s) )x, ds
� [ 11 \1x, E,, (ug ) I I �, + I I Otu" II�. ds
I 1t l I \1 yF (u) lIy2 + 1I0t Uil 2y ds - 0 ( 1 )
> (2.6)
t
2 0 t
> o
- (\1yF(u(s ) ) , Otu(s))y ds - 0 (1)
F(u(O)) - F(u(t) ) - 0(1)
and hence
F(u(O)) - F(u(t)) < E,, (uE (O )) - EE (U� (t)) + 0(1 ) . But by well-preparedness, E. (ue (O ) ) = F(u (O) ) + 0(1) and thus
E,,(u,,(t))
�
F(u(t)) + 0( 1 ) .
But, E" r ) F implies lim,,�o E,, (u,,(t) ) > F (u(t)) . Therefore we must have equal ity everywhere and in particular equality in the Cauchy-Schwarz type relation (2.6), that is or
1
2
0
t
II\1y F(u) lI� + lI atull} ds
=
1t0 (-\1 yF( u(s) ), atu( s))y
lot 1 1 \1F(u) + atull� ds
=
O.
Hence, we conclude that at u = -\1y F(u), Vi E [0, T). The idea is thus to show that the energy decreases at least of the amount expected, on the other hand it cannot decrease more because of the f-convergence hence it decrease exactly by the amount expected, all along the trajectory.
PROOF OF 2')
;-
2). (2') is a constructive proof of 2)). Observe that here
u. does not depend on time. For every V E Y we may pick vet ) such that
v(O)
at v (O)
i.e., pick a tangent curve to V at u . \Ne v" (t) such that
v. (O)
u
=
V
=
ume there exists (we can construct)
a..�s
u. lim,,-->o l I at v (0) l it < IIVII� =
.
lim,,-->o - ft 1t=o Ee (ve ) > - :ft 1t=o F(v)
=
- (\7yF (u), V)) y
that is a curve v,,(t) along which the energy decreases by at least desired amount. Then, choosing V = - \1y F (u) , we have ( - \7 E. (u,,) , atv,,)x,
=
d . -d E,, (ve) = t 11 0
�
- (\7y F (u , V )y
=
II\7F(u) lI�
468
SYLVTA SERFATY
thus
lIV'y F (u) lI} < (-V'x. Ee(ue ) , atVe (O)))x, < IIV' x. Ee(Ue ) II x. ll 8t ve (0) IIx. < II V'x, Ee (ue) IIx. (I Wlly + 0(1)) . Recalling that V = - V' yF (u), we conclude that IIV' Ee(ue ) lIx. > II V'F(u) lIy + 0(1).
o
The idea was to rely on the fact that steepest descent is characterized as the evolution which maximizes the energy-decrease for a given II Bt ue ll �. We compare it to a test-evolution obtained by "pushing" Ue in the direction V (and in fact choose the steepest descent direction V = V' F ( u)), i.e., find a curve v(t) and "lift it" to a curve Ve that pushes Ue in direction V' y F(u) with a decrease of energy of at least the expected one, and a cost I I Bt v. 1I 2 which is at most the expected one. We can in fact achieve this in such a way that OtU,(O) depends linearly on V . In "pedantic" terms, we show that there exists a linear embedding -
-
Ie : TuN V
) 1-+
Tu, M
atve (O)
which is an "almost-isometry" in the sense : lim I IIe (V) llx£ e---+ O
=
lim I; V' x,Ee(u.) = V'y F (u) . IWlly and e---+ O
2.5. Application to Ginzburg-Landau. In order to retrieve the dynamical
law for vortices, we need to prove that conditions 1) and 2') of Theorem 2.4 can be proved for Ginzburg-Landau. As seen in Theorem 1 .4, we need to consider the energIes 1 = (2 .7) Fe (n) = Ee(n) 7rnl log .01 2 •
-
and F = W (the renormalized energy) so that Fe r ) to define are 1 2 = 11 . 11 £2(0) II . II �, I log .01 N = nn\diagonals (2.8) 2
II . l I y
(2.9)
=
1
7r
F. The structures we need
2 I I . 1 I(&2)n
where n is a prescribed number of vortices of a priori fixed degrees ± l . Applying Theorem 2.4, we retrieve the dynamical law that the vortices flow according to some rescaled gradient-flow of the renormalized energy: THEOREM 2.5 (Ginzburg-Landau vortex dynamics - [Lil, JSl, SS4]) . Let UE
be a family of solutions of
BtU
_.
I log .01
with either
=
Uo
u €
ll.1L + -2 ( 1 =
9
�; = 0
an an
2
- lu i )
DYNAMICS OF VORTICES IN GINZBURG-LANDAU
such that curl (i�LE ' V'uE) (O) -" 27r with a? distinct points in n, di = ±1, and
'"
L d, 8a� i=1
as c
-->
469
0
(2. 10) Then there exists T* > 0 such that Vt E [O, T' ) , curl (iu, V'u) (t) -" 211" as
c -->
(2 . 1 1 )
0, with
dai dt
n
L di 8a,(t) i=1
-
ai (O)
where T* is the minimum of the r;olli.�ion time and exit time of the vortices under this law. Moreover D(t) = 0 for every t < T* . Thus, as expected, vortice� move along the gradient flow for their interaction TV, and this reduces the PDE to a finite dimensional evolution (a system of ODE's) . This result was obtained in [Lil, JS2] , but with PDE methods, it is reproven in [SS4] with the r-convergence energetic method exposed here. By the same method, we obtained the dynamics of a bounded number of vortices for the full Ginzburg-Landau equations with magnetic field, i.e., the gradient-flow of ( 1 .3) , for large applied fields (the result for bounded applied fields had been obtained by Spirn [Spi]) . We will assume that
hex = ..\Ilog EI 0 < ..\ < 00. In this regime we have obtained various results about the minimizers and critical points of .I, see [885, Sl, SSl, SS2, SS3] . The (heat flow) Ginzburg-Landau equations as proposed by Gorkov-Eliashberg are (2. 12)
(2. 1 3)
atu + iu = V'�u + ; (1 - luI2) in n E
atA + V' = V'l.h + (iu, V' AU) (iu, V'AU) . n = 0 h = hex
in n on 8n on 8n.
These are the gradient-flow for essentially the same £2 structure as in the case without magnetic field. Observe that here there is no need to rescale in time to see motion of vortices. The quantity makes the equations invariant under the gauge-transformations : (2.14)
'U
ue1W A I--> A + V'w if> H
Let �o be as in [SI, S2, SSI; SS3] the solution of (2.15 )
-Ll�o + �o + 1 �o = 0
=
0
in n on on
SYLVIA SERFATY
470
and
Jo =
(2. 16)
1
2
1 !1 1'V�0 2 + �5·
From [SI, SSI, SS3] (see also [SS5]), if curl (iue, 'Vue) degrees di, then
.....l.
21T Li dioai , with fixed
G. (u, A) - h;xJo - 1T Li I di l i log c l > \ '"' .c . 2 1TA L.- d, <,0 ( a, ) , lim l'nf Ilog cl and we have the r-convergence in the sense of Definition
,
1 .1. Theorem 1.2 yields
THEOREM 2.6 (Vortex dynamics for Ginzburg-Landau with magnetic field [SS4]). Let (ue, Ae,
G.(u.(O), A,(O»
< h;xJo + 1Tnllog cl + 21Thcx L di�o(a�) + o(llog c D ·
Then, there exists a time T* > curl
with
,
,
a such that, for all t E [0, T* ) , n
« iue, 'V A,ud + A,,) (t) .....l. 21T L diOai(t)
i= l
Vi
and T* is the minimum of the collision time and of the exit time from 0 for this law of motion. 2.6. Remarks. ( 1 ) The result holds as long as the number of vortices remains the initial one (so that the limiting configuration u = (al , ' " , an) belongs to the same space N) . It ceases to apply when there are vortex-collisions or some vortex exits the domain under the law (2.11), even though these can happen. Then a further analysis is required, see Section 5 below. (2) Under the same hypotheses, if u� is a solution of the time-rescaled gradient flow Otu. = -Ae 'Vx,Ee (u,) with D(O) = a then if Ae « 1, 'ue (t) 3. Uo , 'It, i.e., there is no motion, while if At » 1 , u. (t) 3. u, 'It, with 'Vy F(u) = 0 i.e., there is instantaneous motion to a critical point. Thus, we see that the structure Xe and the relation 1) in Theorem 2 .4 contain the right time rescaling to see finite time motion in the limit. For Ginzburg-Landau without magnetic field, it is necessary to accelerate the time by a Ilog cl factor in order to see motion of the vortices (this is due to the fact that the renormalized energy W which drives the motion is a lower order term in the energy). (3) The method works for Ginzburg-Landau with or without magnetic field as long as the number of vortices remains bounded. It is more difficult to apply to other models such as Allen-Cahn, or 3D Ginzburg-Landau, because what is missing is a more precise result and understanding on the profile of the defect during the dynamics. For example, for Allen Cahn, we need to know that the energy-density remains proportional to
471
DYNAMICS OF VORTICES IN GINZBURG-LANDAU
the length of the underlying limiting curve during the dynamics (which is true a posteriori) . It is also an open problem to apply it when the number of vortices is unbounded as I:: --; O.
3. How to prove 1) and 2') for Ginzburg-Landau 3.1. A product-estimate for Ginzburg-Landau. The relation 1) which
relates the velocity of underlying vortices to OtU, can be read (3. 1 )
1 lim IOtu, 1 2 ds > ,-0 IIog I:: I 10, tl x n
71'
L
0
t Idt ai l2 ds
assuming curl (iu, V'u,) (t ) ---' 2rr L.,i d;ba, ( t) , as I:: --; 0, 'lit. This turns out to hold as a general relation, without asking the configurations to solve any particular equation. It is related to the topological nature of the vortices. It can be embedded into the more general class of results of lower-bounds for Ginzburg-Landau functionals. The setting is now 0, a bounded domain of ]Rn (n � 2) (we will need n = 3), and Ee (u)
=
1 2
i.
1 2 ( 1 - 11.1 1 2 ) 2 . l V'ul + fl 21::2
We define the "current" ju associated to 1.1 as the I-form ju = (iu, du) Then the Jacobian Ju is the 2-form Ju =
� d(ju)
=
=
L., (iu, Ok U}dXk.
�d(iu, dlt) .
It can be identified to a (n - 2)-dimensional current through J'u((/J)
=
� J Ju
1\
for
=
XIY2 - X2 YI 2
D
The total variation of Ju is then 1 11.1 1
=
•
rr l Dlr[l (")' ) the length of the line.
THEOREM 3.1 ( "Product estimate" - [SS3]) . Let such that E,, (u. ) < C llog 1:: 1
1.1"
be a family of H I (O, q
then up to extraction J1.1" ---' J in (Cgo" (11))* "i"! > 0 with ; a rectifiable integer- multiplicity current (see [JS2]) and 'iX, Y continuous compactly supported vecta7' fields, we have (3.2)
1 I.1m -;:--� .-0 I log cl
As a first corollary, in.2D, taking X
=
el, Y
=
e2
an orthonormal basis, we find
472
SYLVIA SERFATY
This estimate implies the estimate lime �o J lO� 'I In 1V'�c12 > 7r L i Idi I but is sharper. It implies in particular that if � In I V ue l 2 < 7r L i I di l ilog 101 and di = ±l , then VX
(3.3)
1 I Iog 10 I
n
'\' • 7r L.J
I VUE ' X I 2
IX(a;) 1 2
.
,
as 10 -7 0
i.e., there is isotropy in the repartition of the energy along different directions. In 3D, taking X .1 Y and maximizing the right-hand side of (3.2) over I X I , W I < 1 we find 2 V I l 1 r ue > l.l l (n ) , lim , �o Ilog 10 1 in 2 an estimate that was previously proved in [JS2] . Our result extends to higher energies EE < N, l log EI with N, un bounded. In that case we just need to rescale by N, and replace .1 by the limit of REMARK.
JUE . N�
3.2. Idea of the proof. The method consists in reducing to two dimensions.
By using partitions of unity, we can assume that X and Y are locally constant. We may then work in an open set U where X and Y are constant. If they are not parallel, they define a planar direction (if I.hey are then .l(X, Y) = 0 and there is nothing 1.0 prove) . \Ne then slice U into planes parallel to that plane. Assume X = el and Y = e2 orthonormal vectors. In each plane we have the known 2D lower bounds of the type
�1
planenu
I V u, 1 2 >
7r
L i d; I llog 101
where di is the degree of the boundary of the balls, constructed with the ball construction method (see [Sa J, 885] ) . This is possible as long as there is a good bound on the energy on that planar slice, and the number of balls can be unbounded. The main trick is to observe that this is true for any metric in the plane, and use the metric >"dx + � dy, leading to ,
�
2
I D1Ue l 2 +
planenU
�1
planenU
I 02U 1 2 2
7r
..
L I di l l log E I ·
Integrating with respect to the slices yields
fu 2�
I Vue X I 2 + .
�
U
2> YI Vu I E •
Optimizing with respect to >.., we conclude that lim 1 1
.. �O
1
og
10
I
U
I Vu, . X I 2
J
U
I V u, . Y I 2
J(X, Y) Ilog 10 1 .
>
J .l(X, Y)
and we may finish by adding these estimat,es thanks to the partitions of unity. u
3.3. Application to the dynamics. In order to deduce a result for dynamics
in 2D, the idea is to use this theorem in dimension n = 3 with 2 coordinates corresponding to space coordinates and 1 coordinate corresponding to the time coordinate (this can be done in any dimension, but we restrict to 2D here for the sake of simplicity). The vortex-lines in 3D are then the trajectories in time of the
DYNAMICS OF VORTICES IN GINZBURG-LANDAU
473
vortex-points in 2D, and clearly the length of these lines is somehow related to the velocity of these points. Doing the coordinate splitting, we write .
) 11,
THEOREM
3.2 ([883]) . Let ue(t, x) be defined over [O, T]
n = 2) and such that
x
n (n
c
]Rn, here
Vt E,,(u,, (t)) < C l log EI [O,TJ x n
IOtu,, 1 2
< Cllog E I
then v"
with
z
--'
V,
Moreover, VX E C� ( [O, T]
x
dt J.L + div V = O. n, ]Rn) , and f E C� ( [O, T]
x
n) ,
1 . hm .,-----, ,,�O I log E I In 2D, the vector V
=
(Vl , V2) really is 1r L, di (Ot ai )Oai(t) , such that
Ot COROLLARY
1r
L d; O",(t) ,
3.2. 1 . If in addition di
=
+
div V
=
O.
± l andVt , � In l \7u,, 12 < 1r ( L I di D I log EI ( 1 +
0(1)), then for all intervals [tl , h) on which the ai 's remain distinct, we have 1 lim IOtu,, 12 > ,,�O I log E I n x !t" t,J
1r
L i
t2
t,
IOtai l 2 dt.
This is the desired estimate 1 ) in Theorem 2.4. To prove this corollary, recall that from (3.3) if E,,(u,,) � lrn l log EI then Ilo� "I In IX · \7u,, 12 � lr Li I X(ai ) 1 2 and optimizing over X and f gives the L2 bound on V. Theorem 3.2 allows one to bound from above the motion of the vortices for solutions of (1.4) . Using the fact that for them 10; " I IoT IOt u l2 is the variation of 1 energy between time 0 and T, it implies the crucial relation (3.4)
-
where l::.p denotes the variation in position of the vortex, l::.T the increment of time, and l::. E the variation in energy.
SYLVIA SERFATY
474
3.4. Proof of the construction 2') . We wish to prove that 2') holds for
Ginzburg-Landau so that we deduce 2) i.e., if UE .!!" U then limE--->o IIVx, EE ( UE ) II X, > II V W ( u) Ily· Observe that this is a static result. We thus assume that curl (iuE, VUE ) ----'27r Li difJai and may consider disjoint balls B(ai , p) of fixed radius p. If IIVE(UE ) I l x, +00 there is nothing to prove. If II Vx, E., (uE) l l x, = 0(1) then we can prove that DE = 0(1) where DE = EE (uE) - 7rn l log 0: 1 - W(u) is the " energy-excess" . For a proof of thifl nlflUlt, see [S3] , it relies on the fact that IIVEE (u) II x, < C means 2 < IO� . 1 = 0( 1) and one can take advantage of the fact u u n + I l .6. :, ( 1 - l I2) I that UE is thus an "almost-solution" . Once this is proved, we may deduce
(3.5) (3.6) where ,
with the appropriate boundary conditions. The rough idea is that V'PE ::::: V � o outside of the vortex balls. Through these relations, everything is well-controlled outside the balls and inside the balls we shall only perform a pure translation. Given V = (VI , . . , Vn) , we want to push each ai in the direction V; . For that purpose, define Xt (x) = x + tV; in each Bi, and extend it in a smooth way outside of the B; into a family of smooth diffeomorphisms that keep an fixed and are independent of 0: . Choosing the deformation .
v. (x, t)
=
uE (X;l (x »
does the job of pushing the vortices ai along the direction V; . However it is not enough, and we need to add a phase correction 'ljit : i (3.7) v. ( Xt Cu), t) = uE (x) e ,pt(x)
so that for every t , the phase of harmonic conjugate of
.6.t
=
VE
is approximately the optimal one, that is the
2 7r � d; clad t)
ai(t)
=
ai + tV;.
,
It is possible to construct 'ljit single valued, independent of 0:, so that V�o + V'ljit
�
V � (t 0 Xt) .
We will now check that the VE constructed this way works. First, 1 2 at v 1 (0) l I log 0: 1 in
r
,,
,
--->
.
DYNAMICS OF VORTICES IN GINZBURG-LANDAU
475
'
because Xt achie\'es a translation of vector V; in the Bi'S while the contribution outside of the Bi 's is negligible; then, we use the relation (3. 3). The first requirement for 2') is thus fulfilled. Let us check the second requirement, i.e. , the energy-decrease rate, by evaluating : t =o Ee (ve (t)) . With a change of variables,
q
Ee (ve(t))
� 10 l \7ve l 2 + 2�2 (1 _ 1ve 12) 2 2 2 2 � 10 I DX;-l \7(ve Xt l l + 2�2 (1 _ lu I ) 0
Now, recall that Xt is a translation in Ui Hi hence I Jac Xt l of U,B, there is almost no energy, hence
=
l Jac Xt l ·
cst there, while outside
d d i e Ee (ve(t» = 1/> t ) I 2 I Jac Xt l + o (l). D Xtl \7 (u e dt I t=o dt It=O Expand \7 (ue ei1/>, ) as \7 ueei1/>t + iUe \7'1Pt, expand the squares, and apply 1ft t=0 ' 1
The crucial fact is that the terms which get differentiated do not depend on c . For the other terms, we use (3.6) , so that there remains
d 1 .L .L DXt \7 <1>0 ' \7 <1>0 fl\UiBi dt I t=o d d .L 1 2 IJac xd + 0(1) \7 t + + I \7 1\7<1>u . <1>0 '1j J 2 dt I t=o fl\UiBi dt It=o 1 d o + \7'l,0t W I Jac Xt l + 0 ( 1 ) IDXt (\7.L 1 dt l t=o 2 fl\u,Bi But observing that 'Ii't was conHtructed in such a way that \7 .Lo + \7 'l,0t = \7 .L ( t 0 Xt), and doing a change of variables again, we find d d 1 2 + 0 (1) Ec(v,, (t)) t 1\7 <1> 1 dt It=O dt It=o 2 O\Ui Bi d t) , . . · , an (t)) + 0(1) Wct (al ( dt I t=o
1
-
i.e., the desired result.
4. Second order approach - stability issues We extended the method to second order in order to treat stability questions for this 2D Ginzburg-Landau equation. Here is the abstract result, pushing the method of condition 2') to second order. The setting is as in Section 2 . 1 . We say a critical point is stable if the Hessian is nonnegative, unstable otherwise.
THEOREM 4 . 1 (Abstract result - [S4]) . Let u" be a family of critical points of Ee with Ue ....,;"S U E N, such that the following holds: for any V E /3', we can find ve (t ) E M defined in a neighborhood of t = 0, such that Bt ve (O) depends on V in a linear and one-to-one manner, and
( 4. 1 )
•
limc _o ft 1t=o E,, (v,, (t))
(4.2) (4. 3)
'U"
lim _o � lt=oE. (v,,(t» ) ..
=
(0) = 1l"
=
ft1t=o F(u + tV )
Jt'2 I t= o F(u + tV)
=
=
dF(u). V
( D2F ( u) V, V) .
SYLVIA SERFATY
476
Then - if (4 . 1}-(4 · 2) are satisfied, then u is a critical point of F - if {4. 1}-{4.2}-(4 · 8} are satisfied, then if UE are stable (resp. purely unstable) critical points of Eo u is a stable (resp. purely unstable) critical point of F. More generally, denoting by nt the dimension (possibly infinite) of the space spanned by eigenvector's of D2 Eo (uo) associated to positive eigenvalues, and n + the dimension of the space spanned by eigenvectors of D2 F(u) associated to positive eigenvalues (resp. n; and n- for negative eigenvalues); then for E' sm.all mough we have n; > n - . Thus, we reobtain that critical points converge to critical points of the limiting energy F (proved in [BBH] for Ginzburg-Landau), but in addition we obtain that under certain conditions, stability/instability of the critical point also passes to the limit. The previous l'esull. (Theorem 1.2), was an analysis of the C1 structure of the energy landscape, thus suited to give convergence of gradient-flow and critical points; while this is the C2 analysis of the energy landscape around a critical point. For ( 1 . 1 ) , the construction done in Section 3.4 can be fe-used, and the calcu lation pushed to second order, to obtain conditions (4. 1 )-(4.3). Thus, from the theorem above, we deduce in [S4] the corresponding theorem for solutions of (1.2) (which was not proved before) : stable/unstable critical points of Ginzburg-Landau converge to stable/unstable critical points of the renormalized energy. An interesting application is for Neumann boundary condition, for which it is known that the corresponding renormalized energy W has no stable critical point. Hence from Theorem 4. 1 there are no stable critical points of Eo with vortices. THEOREM
4.2 ( [S4) ) . Let
UE
be a family of nonconstant solutions of
au = 0 an
-�u = eno'ugh,
(on 12
C
IR2
1Le
�(l lul'l) -
simply connected) such that is unstable.
in 12 on an
£e(u,J <
Cl Io!';
cl;
then, fOT
E'
small
This is an extension of a result of Jimbo and Sternberg [JiSt] for convex domams. •
5. Energy dissipation rates and vortex collisions We now return to the dynamics and address, with the same type of philosophy, the question of understanding what happens during collisions under the dynamical law (2.11) (which dop}; generate collisions when there are vortices of opposite signs) .
5.1. Lower bound on the energy-dissipation rate. For solutions of the
gradient-flow, we have seen that
(5. 1 ) In the context of Ginzburg-Landau, if we write for simplicity
(5.2)
� ( 1 - lu I 2) = fe U = g (r€sp. � = 0) �'U +
in n on an.
,
DYNAMIC:=; OF VORTICES iN GINZBURG-LA1'DAU
we have I" =
477
Ilo� "I '\lxE E,,(u) in the previous notation, and li'\l xEE,, (u) ll i,
= I log
c l l lf,, 1 i12 (fl) · Combining this with (5.1), we see that knowing II f" IIL2 gives the energy-dis.'lipation rate (in time) , or rather - Jlo� "I :it E" . If II f" li £2 is large, then the energy dissipates (5.3)
fast, thus decreasing to a point which allows us to rule out certain configuration� (for example if E" decreases so much that fJ" < C then there can be no more vor tices) . On the other hand, if fE is small, then (5.2) can be seen as a perturbation of ( 1 .2), for which we know certain qualitative facts about vortices. The idea is thus to use this alternative in a quantitative way, in order to obtain information on vortex-collisions or other pathological situations. Observe that vortices colliding corresponds to the more general fact that several vortices converge to the same limit as E -> 0, with possible (but not necessarily) limiting degree O. More precisely, it is known that if u is a static solution of Ginzburg-Landau in the plane, with vortices (ai , d;) then we mw;t have 2 2 d = "' 0 i ,
equivalent to the fact that L:i#j di dj = 0, or to the fact that the forces exerted by the vortices balance each other. This follows from suitable applications of the Pohozaev identity, as in [BMR). Similarly, as seen in [eM! J, if u is a static solution of ( 1 .2) in a bounded domain and has some vortices Ui of degree di accumulating (as € -> 0) around a single point p, then the same rule (L:i d;)2 = L:i d; holds. Now, if u is a configuration with say, t,wo vortices, one of degree 1 , one of degree - 1 , at a distance 0(1) as c -> 0 (which is what happens during a vortex-collision of a +1 with a - 1) then this rule is obviously violated (and it's the same for any situation with (L:i d;) 2 1= L:i df ) , so we can trace how much it is violated in the Pohozaev identity for (5.2), and get a lower bound for Ilf" IIL2 . This is the method we applied to get the result below. In a first stage, we may forget that u solves (1.4), forget time, and jus\' focus on studying the static equation (5.2) which is ( 1 .2) with an L2 perturbation. Before stating the result, let us make a few assumptions. It is natural to assume
( 5 .4) since the energy decreases during the flow ( 1.4) , and (5 . 5)
luEI < 1
M , l '\lu" I < €
which are satisfied at all times for solut.ions of ( 1 .4). We assume in addition that 1 2 (5.6 ) 11 10 1I L2 (0 ) < -".fJ for some (3 < 2 . If this assumption is not true, then clearly we have a large lower bound on II i" I I £2 . If it is true, then after blow-up at the scale c, solutions of (5.2) converge to solutions of Ginzburg-Landau in the plane
-tl.U = U(1 - I U I 2 ) which enables us to define what we shall call a "good collection of vortices" ai with degrees di (depending on 6) for U" . Without going into full details of what
SYLVIA SERFATY
478
it means and how they are found, these are points (depending on c) such that UiBi ;= UiB(ai, Ree) (with some Re < I log el) are disjoint and cover all the zeroes of u" , and d; = deg (u", aB(ai, Ree» .
THBOREM 5.1 (Analysis of solutions of (5 . 2) - [83]) . There exist constants 1o > 0 and Ko > 0 such that, assuming that Ue is as above and that there exists a nonempty subcollection {B;}7 1 of the balls {Bi} which are included in B(xo , 1/2), e Vl log el « I < In as e --> 0, and such that for some K > Ko, either (1 ) B (xo, Kl) c n and B(xo, Kl ) intersects no other ball in the collection {Bi } , and we have 2
(5.7)
•
(2) Xn Then
i= l i=l E an and B(xo, Kl) inter'sects no other ball in the collection {Bi } . C c l 2 110g 10 1 ' 12 10g2 l
( 5.8)
•
All the constants above depend only on /3, M, n and g.
As a byproduct, we retrieve the fact that for 10 small enough, solutions of ( 1 .2) have no cluster of vortices with (l:i d; ) 2 of. l:i d; at mutual distances I < lu. The analysis for Theorem 5. 1 , which is partly inspired by the one of Comte Mironescu [CMI, CM2] for solutions of (1.2), combined with Pohozaev identities, also allows us to obtain the following result. THEOREM 5.2 ( [83] ) . Let u" satisfy (5.2) -(5. 6), then we have,
(5.9)
Be (ue) < 7r
i >7 10g � + Wd (a l , ' " ,=1 .
c
, a n ) + Cllf" 111,2(0) + 0(1),
wheTe C depends only on /3, M, 12 and g.
Observe that under certain conditions, this gives relations of the form De < - I lo�el Jt Ee (t) for solutions of (1 .4) , where De again denotes the energy-excess.
5.2. First application to the dynamics: well-prepared implies very well-prepared. Coming back to the dynamics (1.4), in Theorem 2 . 5, we made the
assumption (2.10) initially, as required by Theorem 2.4, which we can call a "very well prepared" assumption, as opposed to the weaker requirement in [Lil, J82] which was only Ee (uE ) < 1I"n l log e l + C, which we now call "well-prepared" . Our requirement turns out not to be really stronger because within a time 0(1), solutions with well-prepared initial data (or even a weaker requirement, see below) become very-well prepared, as shown in the following result, which is proved through Theorems 5.1 and 5.2. Ug
THEOREM 5.3 (Instantaneous "very-well preparedness" - [83]) . Assume that is a solution of (1.4) such that
(5.10)
curl (iu�, Vu�)
->.
211"
n
L Di6p? i=l
as
e
-.
0,
DYNAMICS OF VORTICBS IN GINZ8URG-LANDAT.:
479
with D; = ±1, and such that (5.11)
Ee( u�)
<
I log 10 1 7rn l log cl + (log I log cl) !'l
for some (3 > 1 . Then, t E [0, T�J , we have (5.12)
such that for every
curl (iue , Vue) (t )
--'-
27r
n
L D;op?
as
c -+
0,
;= 1
and (5.13) Let us sketch briefly how the argument works. The time estimates given by Theorem 3.2 (see also (3.4» ensure that the locations of the limiting vortices do not move in the time Te = 0(1) of the energy-dissipation. For most times t < Te , we can find a "good collection" of (microscopic) vortices ai , and since (5 . 12) holds, each ai converges to some pJ with Li!ai�pO di = Dj = ± l . \Ve may then use Theorem 5.2 to find that for t < Te, J
dEe (ue(t») Ee (Ue (t » � 7rn I log c l + WD (P1, · · · , Pn) + 0( 1 ) . dt - I log cl This is a differential inequality on Ee (ue (t» which can easily be solved, and gives 0
0
C
an exponential decay of the energy excess, and thus the result (5.13) .
5.3. Application to the analysis of collisions. Theorem 5.1 shows that
the energy of a configuration of colliding vortices decreases in general at a rate 1/l 2 where I denotes the distance between the vortices, possibly depending on c. This strongly indicates that the time of collapse/separation should bc of order 12. Indeed, the energy of a configuration with two vortices of degree + 1 and - 1 is roughly 27r log l�) where let) denotes their distance. Integrating in time the relation
(5.14) we are led to
(5.15) Setting M(t) =
l(to ) > 27r log l et)
1t to
C dt F (t)
It: 12�t) dt and solving an ODE, we find eCM(t)
C(t - to ) _ -1 t2 (to ) •
<
which means that this can hold only up to time to + CI2 (to), indicating that the two vortices should have collided in time Cl(to)2. Then, one needs to establish the clearing-out of all the energy-excess they carry. This is done in the same differential inequality spirit as above. The rigorous result we obtain is the following. \Ve assume Ue has a dipole of vortices of degree ±1 colliding, i.e., which are at a distance I « 1 from each other and converging to a point Pdip as c -+ 0, and n other vortices of degree ±1,
480
SYLVIA SERFATY
converging to distinct points p� , . . . , p?" distinct from Pdip - This situation implies that
(5.16 )
curl (iu�, Vu�)
n
--'
27r L Dibp?
as
i=l
c
---..
0,
with Di = ±1. We may al�o assume that there exists Pe: ---.. Pdip such that, consid ering u, (:c) = u� (p, + Ix) we have (5.17)
curl (iu" VUE) --' 27r (bb+ -
bL )
as 0;
---..
0
where I b+ - b_ 1 = 1 . We may also assume that this situation is inherited from a well-prepared data at a previous time, so we may assume that u, is well-prepared with respect to these vortices, i.e., Er(u,) < 7rnllog 0;1 + 27r log ! + 0(1). -
THEOREM 5.4 (Collisions - [83] ) . Let Ug be a solution of (1 . 4) such that at time 0, (5. 5) and (5. 1 6}-(5. 1 7) hold, and EE (U� ) < 7rnllog cl +
I 27r log - + 0(1), c:
z 2 G21 + Cllog 0;14e- v'llog d for
with l = 0( 1). Then the/"e exists a first time Tl < which uE (T1 ) has exactly n zeroes (i. e., the dipole has collided). If I > c:f3 with f3 < 1 , then aL�o Tl > Gl 12 . Moreover, there exists a time Tz < Gl2 + G loTI
o�o;1
(5.18)
curl (iug, VuE ) (t) --' 27r
and
n
L D;bp? i= l
as
0;
---..
0
(5. 19) The relation (5.18) indicates that the vortices not involved in the collision have not moved during the time Tz = 0( 1), and (5.19) that UE has become well-prepared again relative to those vortices within that time. Thus all excess-energy carried by the colliding vortices has dissipaLed in 0(1) time, and the previously known result Theorem 2.5 applies after that time Tz, i.e., one may continue and retrieve the dynamical law with the remaining vortices (this solves a question which remained open). The foJlmving theorem expresses that the dynamical law of the vortices is given by (2. 1 1 ) , where, when two vortices collide, they should be erased from the list, and the law (2.11) should afterwards be understood as the law with the remaining vortices. THEOREM 5.5 (Global in time dynamical law) . Let UE solve (1 . 4) with Dirichlet boundary condition and be such that (5. 5), (5. JO) and (5. .1 1) hold. Setting To = 0, there exist collision times 0 < Tl < Tz < . . . < Tk . . . < 00 such that if either k = 0 or the collisions at times T1 , . . . , Tk are simple {i. e., involve only pairs of limiting vortir:es} then, denoting by pf the distinct points in n and Df = ±1 the integers such that
J.L(t )
-'
2rr
nk
L Df Jp� ;=1
as t
---..
T; ,
,
�l
DYNAMICS OF VORTICES IN GINZBURG-LANDAU
we have curl (iu" , V' u,,) ( t) ----' J1. (t )
n.
=
where the Pi (t) solve the initial value problem (5.20)
dPi = dt
-
1
11"
211" L D�t5p. (t) i= l
as e
--+
0
V'i WO (P1 , ' " , Pn. ) (t )
and Tk+1 < 00 is the first collision time under this law. t E (Tk , Tk+1 )
Moreover, for every
(5.21 ) Finally, nk < nk -1 - 2 hence the number 01 simple collisions is bounded by no/2 n/2.
=
5.4. Exact dynamical law after blow-up. The techniques developed for
the previous results also allow to give the exact dynamical law during the collision. The method consists in blowing up around the collision point, and consider the solution rescaled in time and in space , so that after blow-up the vortices are at mutual distance of order 1 . It is then possible to carry out the f-convergence strategy of Theorem 2.4 and to show that the limiting vortices (after blow-up) evolve according to the flow of a reduced renormalized energy (without boundary effects) ( 5.22)
W (b1 ) . .
·
'
- -11" " bP ) � D'l DJ· log l b · - bJ l ' ii'j
'L
The result of Theorem 5.2 plays a crucial role in the proof because it allows us to bound the excess-energy by 111" I I L2 hence by Ilo� "I Ii V'E" l it , thus allowing to get a bound from below for IIVE" II (for the rescaled in space energy). We also use Theorem 3.2 applied to the rescaled (in space and time) of the solution. Since we arc continuing the dynamics given by Theorem 2.5 just before collision, and since we know that the energy-excess is always zero until then, we may still assume some "very well-preparedness" initially. THEOREM 5.6 ( [S3] ) . Assume u" is a solution to (1 .4), with (5.4) and (5. 5). Assume I = 0( 1 ) with log4 l < Cllog e l and Me points pj Pj j E [1 , nJ UT'e such that, defining 7tj (X, t) = u,, (Pj + lx, /2t), we have --->
,
(5.23)
m
curl (iuj V'Uj ) (0) ----' 211" L Dj, k t5b�.k k=l ,
as e
,
--+
0,
with Dj,k = ±1, Lk Dj,k = Vj , and (5.24) E, ;(u�) < 7r L IDj,k l l log e l +Wv (Ph ' " , Pn)-11" L Dj,kDj,k' log (l l b�,k - b�, k, I ) +re j k#k' j,k
L
SYLVIA SERFATY
482
with ro < 0 (1 ) or r" < "�Ilj3 with /3 > 1 . Then, there exist Hl ( (O, T*) ) trajectories bj,k (t) such that for evenJ t [0, T* ) ,
(Jo�2 1!�:g
curl (iuj , Y'ui) (t)
---I.
E 211" L Dj,k6bj , . (t)
as
k
c ->
0
(5.25) and T' is the first collision-time under this law. Observe that this result allows us to treat a varietv of cases such as collisions of vortices of opposite degrep.s, separations of vortices of same degree. . . "
5.5. Post-scriptum. Since these notes were first written up, t.he study of the dynamics of vortices in the whole plane ]R2 under the general hypothesis E" (u�) <
G l log cl with possible collisions, splittings and recombinations, was essentially com pleted by Bethuel-Orlandi-Smets in [BOSI, BOS2, BOS3] . In particular they derive the dynamical law analogous to (2.1 1) but for arbitrary vortex-degrees, valid away from a finite number of times corresponding to vortex collisions, splittings, or recombinations.
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,
COllRANT TNSTITUTE,
251
MERCER STREET, NEW YORK, N Y
E-mail address: serfaty�cims . nyu . adu
100 1 2 , USA
•
Publications of Hai'm Brezis
[IJ
conj'Ug'Uees relativement a (1967), 284-286.
Fonetions convexes
Sci. 264
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bilineaire degeneree,
c. R. Acad.
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486
PUBf,ICATIONS OF HArM BREZIS
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xxvii
+159p.
Vorticite' quantifiee et energie de Ginzburg-Landa'U, in L"" Grands Systemes des Sciences et
de la 'l'ecllIlologie, hommage it
R.
Dautray,
(J.
Horowitz et J . L. Lions ed. ) , Masson,
1994,
p.
137-148. [189] wit. h F. Merle and T. Riviere, Effets de quantification pour -Ll.u u ( 1 ] u I 2 ) sur R2 , C. R. Acad. Sci. 317 (1993), 57-60. [HID] with F. Merle and T. Rivi ere, Quantization effe.cts for -flu. = u(1 1'1.1]2) in R2 , Arch. Rational Mech. Anal. 12 6 (1994), 35-58. [191) with A. Ambrosetti and G. Cerami , Combined effects of concave and convex non-linearities in some elliptic problems, J. Funet. Anal. 122 (1994), 519-543. [192] Convergence in TJ' and L 1 under strict convexity, in B()undary value problems for partial differential equations and applications, dedicated to E. Magenes, (C. Baiocchi et J . L. Lions ed.), Masson, 1993, p . 43-52. [193) with F . Bethuel and F . RHein, Tourbillons dp. Ginzburg-Landau et energic rcnormalisee, C. R.. Acad. Sci. 317 (1993), 165-17l. [194) with L . Nirenberg , H I versus Cl local minimizers, C. R. Acad. Sci. 317 (1993), 465-472. [195) Remarks on the preceding paper by M. Ben-Artzi, Arch. Rational Mech. 128 ( 1 994) , 359-:�60. [196) with P. Mironescu, Sur une conjecture de E. De Giorgi relative d I 'energie de GinzburgLandau, C . R. Acad. Sci. 319 (1994), 167-170. [197) with L . Nirenberg, J)egree theory and BMO, Part I : compact manifolds without boundaries Selecta Math. 1 (1995) , 197-263. [198] with L. Nirenberg, Degree theory and BMO, Part II : compact manifolds with bo'Undaries, Selecta Math. 2 (1996), 309-368. [199) with Th. Cazenave, Y. Martel and A. Ramiandrisoa, Retour sur les phenomene.s d 'explosion po ur Ut fl u = g ( u) , C. R.. Acad. Sci. 321 ( 1995), 1305-1308. [200] wi t h Th. Cazenave, Y. Martel and A. Ramiandrisoa, Blow-up for 'Ut Ll.u = g(u) revisited, Advances in Di!L toq. 1 (1996), 73-90. [201) with F. Golse and R. Sentis, A naly..- asymptotique de l 'equation de Poisson couplee d la relation de Boltzmann. Quasi-nev.tmliU des plasmas, C. R. Acad. Sci. 321 (1995), 953-959. [202) with T. Cazenave, A nonlinea,' heat equation with singular initial data, J. Analyse Math. 68 (1996), 277-304. [203) New developments on the. Cinzburg-Landau model, Topological methods in N onlinear Anal ysis 4 (1994), 227-236. [204) Degree theory: old and new, in Topological Nonlinear A.nalysis II: Degree, Singularity and Variations, Frascati, 1995, (M. Matzeu and A. Vignoli ed . ) , Birkhauser, 1997, p. 87-108. [205] with A. Bahri, Nonlinear elliptic equations on Riemannian manifolds with Sobolev critical exponent, in Thpics ill Geometry (S. Gindinkin, ed.) , Birkhauser (1996), 1-100. [206) with L. Nirenberg , Removable singularities for nonlinear elliptic equations, Topological Methods in Nonlinear Analysis 9 (1997), 201-219. [207) with J.L. Vazquez, Blow-t,p solutions of some nonlinear elliptic problems, Revista Mat. Univ. Complutense Madrid 10 (1997) , 443-469. [208) with L. Nirenberg, A [,vapunov Schmidt procedure involving a nonlinear projection, in Mul tidimensional complex analysis and PDE (P.D. Cordaro and H. Jacobowitz ed. ) , Contem porary Math. 205, Amer. Math. Soc . , 1997, p. 25-32. [209) with F.E. Browder, Partial differential equations in the 20th century, Advances in Math. 135 ( 1998), 76-1 44. E(fltazioni differcnziali aUe derivate parziali, in Storia della scienza, S . Petruccioli, Editor in Chief, Roma, Istituto della tonciclopedia, vol VIII, 2004, p. 183- 192. [210] Lectures on the Ginzburg-Landau Vortices, (in preparation) . =
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PUBLICATIONS OF HAIM BREZIS
[21 1 ]
with X . Cabre,
Some simple nonlinear
PDE 's
493
without solutions, Roll.
Unione
).fat. Ital .
(1998), 223-262. [212] wilh M. Marcus, Hardy inequality revisited, Ann. Seuola Norm. Sup. Pisa 25 (1997), 217237. [213] Symmetry in Nnnlinear PDE 's, in Proe. SYlIIp. Puce Math. , 65, Florence, 1996, Amer. Math. Soc., 1999, p. 1-12. [21.4] with L. Boccardo. Som e ,,,,marks on a class of elliptic equation., with degenerate coercimty. Boll. Unione Mat. Ital . 6 (2003), 521-530. [215] with M. Marcus and 1. Shafrie, Extremal functions for IIardy 's inequal'ily with weight, J. Funct. Anal. 171 (2000), 177-191. [216] with Y. Li, P. M i roneseu and L. Nirenberg, Degree and Sobolev spaces, Topological methods in Nonlinear Analysis 13 ( 1) )99), 1 8 1-190. [2 17] with J . Bourgain and P. Mironescu, Lifting in Sobole" .'paces, J. Analyse ).fath. 80 (2000), 37-86. [218] with S. Kamin and C . Siva.hi nsky, Initiation of subsonic detonatiun, A�ymptotic Anal. 24 (2000), 7::1-90. [219] I., there failure of the Inverse Funcl'iun Theorem?, in Morse theory, Minimax theory and their Applications in Nonlinear Differential Equations, Proc. Workshop held at the Chinese Acad. of Sciences, Beijing, 1999, (Haim Brczis, ShuJie L i , JiaQuan Liu and Paul Rabinowitz, eds) , International Press , 2003, p. 23-33. [220] wit h Y. Li, Topology and Sobolev spaces, C. R. Acad. Sci . 331 (2000) , 365-370. [221] with Y. Li, Tupology and Sobolev spaces J. Funet. Anal. 183 (200 1 ) , 321-369. [222] Homotopy classes in Sobolev spaces, in Loctures on Partial Differential Equations, Proceed ings of a conference in honor of Louis Nirenberg's 75th Birthday, Taiwan, 2000, ( S.-Y. A . Chang, C.-S. Lin, H.T. Yau, eds) , International P ress , 2003, p. 35-41 . [223] The fascinating homotopy structure uf Sobolev spaces, Proceedings of the meeting "Renato Caccioppoli and modern analysis" , Accademia dei Lincei, Rome, Juin 2002, Rend. ).fat. Ace. Lincei (9), 14 (2003), 207-217. [2241 with J. Bourgain and P. Mironescu, On the structure of the. Sobolev space Hl/2 with values into the circle, C. R. Acad. Sc. 331 (2000), 119-124. [225] with J. Bourgain and P. M ironescu Another look at Sobolev spaces, in Optimal Control and Partial Differential Equations (J.L. Menaldi, E. Rofm an et A. Sulem , eds) a volume in honor of A. Bensoussan's 60th birthday, lOS Press, 2001, p. 439-455. [226] with F. Bethuel and C . O rlandi, Small energy solutions to the Ginzb'U7y-Landau equation, C. R. Acad. Sci. 331 (2000), 763-770. [227] with P. Mironescu, Composition in frac/'iunal Sobolev spaces, Discrete and Cont. Dyn. Syst. 7 (2001 ) , 241-246. [228] with J . Boul'gain and P. Mironescu, Limiting embedding theorems for W',p when s i 1 and applications, volume dedicated to the memory of T. H. \Volff J. Analyse Math. 87 (2002), 77-101. [229] with J . B o urgai n, Sur l 'equation diu u = f , C. R. Acad . Sci. 334 (2002) , 973-976. [230] with J. Bourgain, On the equation di'u Y = f and application to control of phases, J. Amer. Math. Soc. 16 (2003) , 393-426. [231] with F. Bethuel and G. Orlandi, Asymptotics for the Ginzburg-Landau equation in a1'bitmry dimensions, J. Funct. Anal . 186 (2001) , '132-520. Erratum , 188 (2002), M8-.549. [232] with P. Mirone"cu Gagliardo-Nirenberg, composil-io'll and products in fractional Sobolcv spaces, J. Evolution Equat.ions 1 (2001), 387-404. [233] with S. Seefaty, A variationul furmulation for the two-sided obstacle problem wi.th measure data, Comm. Contemp. Math. 4 (2002), 357-374. [234] with P. Mironescu, On some question., of topology for 51 -valued fracttonal Sobole'u spaces, Rev. R. Ac ad . Ciencias, Madrid, 95 (2001), 121-143. [235] with F. Bethllel, J. B ourgain and G. Orlandi, W1,p "timate for solutions to the Ginzburg Landau equation with boundary data in H l / 2 , C. R. Acad. Sci. 333 (2001), 1069-1076 . [236] with Ph. Benilall, Nonlinear problems related to the Thomas-Fermi equation, J. Evolution Equat ions 3 (2003), 67::1-770. [237] The interplay between analysis and topology in some nonl'inear PDE 's invited lecture at the AMS meeting "Mathematical Ch alleng€8 of the 21st Century , Los Angeles, 2000, Bull. Amer. M ath . Soc. 40 (2003 ) , 1 79-201 . 1-B
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PUBLICATIONS OF HAIM BREZIS
494
[238] with A . Ponce, Remarks on the strong maximum principle, Dill. Int. Equ. 16 (2003), 1-12. [239] How to recognize constant functions. Connections with Sobolev spaces, Volume in honor of M. Vishik, Uspekhi Mat. Nauk 51 (200 2), 59-74 (in Russian) . English translation in Russian Math. Surveys 51 ( 2 002), 693-708. [240] w ith J. Bourgain and P. M ironescu , Hl/2 maps into the circle: minimal connectiuns, lifting, a.nd the Ginzburg-Landau equation, Publications mathematiques de l ' I1IES 99 (2004) , 11 15. [241] with L. D upaigne and A. Tesei , On a semilinear elliptic equation with inverse-square po tential, Selecta Math. 1 1 (2005) , 1-7. [242J with J. Bourgain and P. Mironescu, Lifting, degree, and distributional Jacobian, revisited, Comm. Pure Appl. Math . 58 (2005), 529-551 . [243] with J. Bourgai n and P. M ironcs cu, Complements to the paper " Lifting, degree, and distri butional Jacobian, revisited", to be posted on the website http: //ann.jussieu.fr/publications. [244] with P. Mironescu and A. Ponce , �-\fl,l _maps with values into S I , in Geometric Analysis of PDE and Several Complex Variables (S. Chanillo, P. Cordaro, N. Hanges, J. Hounie and A. Meziani, cds.), Contemporary M athematics Series 368, Amer . Math. Soc., 2005, p. 69-100. [245] with P. Mironescu and A . Ponce, Complements to the paper "Wl,I _maps with value.• into SI to be posted on the website http://ann.jnssien.fr/pnblications. [246] with A . Pon ce, Kato 's inequality when �u is a measure, C. R. Acad. Sci. 338 (2004), 599-604. [247] with J. Bourgain, New estimates for the Laplacian, th, dill-c'Lrl and related Hodge systems, C. R. Acad. Sci 338 (2004), 539-543. [248] with M. Marcus and A . Ponce, Non!inea,' elliptic equations with measures revisited, to appear in a volume of the Annals of Math. Studies, Princeton Univers ity Press. [249] with M. M arcus and A. Ponce, A new concept of reduced measure for nonlinear elliptic equations, C . R.. A cad . Sci. 339 (2004), 16l}.174. [250] with L.A. Peletier, Elliptic equations with cntica! exponent on S3,. new non-minimising solutions, C. R. Acad. Sci. 339 (2004), 391-394. [251] with L.A. Peletier, Elliptic equations with critical exponent on spherical caps of 83, J. d A n alyse Math. 98 (2006) , 279-316. [252J New questions related to the topological degree, in The Unity of Mathematics (P. Etinghof, V. Retakh, I.M. Singer, eds.), Proceedings of the Conference in honor of the 90th birthday of I.M. Gelfand , held in Sept. 2003, Birkhiiuser, 2006, p. 137-154. [253J with A. Ponce, Reduced measures on the boundary, J. Funct . Anal. 229 (2005) , 95-120. [254] with A. Ponce. Reduced measures for obstacle problems, Adv. Diff. Eq. 10 (2005), 1201-1234. [255] with J. Bourgain and H.-M. Nguyen, A new estimate for the topological deg"ee, C. R. Acad. Sci. 340 ( 200 5) , 787-791. [256J with M . Chipot and Y. Xi, On Liouville type theorems ( in preparation). 1257] with Y.Li, Some nonlinear elliptic equations have only constant solutions, J. of Partial Differential Equations (volume dedicated to Kung-Ching Chang on the occassion of his 70's birthday), (to appear ) [258] with J . Bourgain, New Estimates for Elliptic Equation., and Hodge type systems, J. Euro pean Math. Soc. 9 (2007), 277-315. [259] with J. Van Schaft ingen Boundary estimates for elliptic systems with Ll - data, Calc. Var. ",
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'
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,
PDE (to appear).
Books of Haim Brezis [1] Op erateurs max,maux monot()nes et semi-groupes de contractions dans les espaces de Hilbert, North-Holland (1973). [2] Analyse fonctionnelle; theorie et applications, Masson (1983) , numerous subsequent edi tions; rep rint ed by Dunod (1999). Translalions: Spani sh (Alianza), English (Springer), Italian (Liguori) , Japanese (Sangyo Tosho) , Greek (Technical Universily of Athens Press), Chinese (Tsinghua Univ. Press ) Korean ( Kyung Moon Publishers), Romanian (Editura Academiei Ramana) . [3] Co-editor with J . L. Lions, Nonlinear partial diffe re nti al equation.. and their applications, Seminaire du College de France, Pitman. VoU (1981), Vol.2 (1982), VoJ.3 (1982), Vol.4 (1983), Vol.5 (1983), VoJ.6 (1984), Vol.7 (1985), Vol.8 (1988), Vol.9 (1988), VoUO (1991), VoJ.l1 (1994), Vo1. 1 2 (1994) . [4] Co-editor with H. Berestycki, Recent contributions to nonlinear partial equations, Pitman (1981). [5] Co-editor with M.G. C randal l and F. Kappel, Semigroups theor1J and applic.ations, 2 vol umes Pitman (1986). [6] with F. Bethuel and F. Helein, Ginzburg-Landau Vortices, Birkhiiuser (1994), xxvii +158p. [7J Haim Brezis, Un math€maticien jui/. Entretien avec Jacques Vauthier, Beauchesne (1999). ,
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