Collection of papers on geometry, analysis and mathematical physics : in honor of Professor Gu Chaohao

jr, with Cauchy data U g C6°(S3) , VV C H2(S3), t h e n there exists e > 0 such that these equations have a global solution on Minkowski space time if its Cauchy data U are such that the following inequalities are satisfied:

21

I! $ - V \\3cUs) < «,

5.2.a

|| V t / - V C / | | H j ( S 3 ) < e ,

5.2.6

where U is the set of two scalars and one vector associated to U_ by the formulas 4.8 Denote by U_ the global solution on Minkowski space corresponding to U . The inequality (5.2.a) is equivalent to, where underlining denotes functional spaces relative to IR3 with the euclidean metric and we have set a = ( 1 + r 2 ) 1 ' 2 , hence u> -1 = \o2, || u i - u ° ' - S - £ '

|| J i - I | | c » < e ,

for t = 0

\\°*(P-~P)\\CZ<£, \\S-S\\&
ua = <5J .

the corresponding global solution on Minkowski space time is

p = n 4 P,

s = c, u = o.

it is a spherically symmetric, non static, flow. Except for the time line of the center of symmetry, all flow lines tend to null lines at infinity. The theorem 3 express that this solution is stable: it remains global under changes of the Cauchy data which are small is the sense indicated in that theorem. References. 1

F. John Comm. Pures App. Maths. 27 p. 377-405, 1974. S. Klainerman and A. Majda Comm. Pures App. Maths. 35 P.629-651, 1982. 3 T. Sideris Preprint Univ. California, Santa Barbara. 4 A. H. Taub Phys Rev. 103, 454-467,1956 5 Y. (Choquet)-Bruhat Bull. Soc Math, de France 86, 1958 6 G. Pichon Ann. Inst. Poincare, n ° l , 1965. 7 J. Smoller and J. Temple Comm. Math. Phys. 156, 67-99, 1993 8 Y. Choquet-Bruhat Comm. Math. Phys.3 p. 334-357, 1966. 9 J. Leray "Hyperbolic Differential Equations" Princeton 1953. 10 J. Leray et Y. Ohya Math. Ann 162 p.228-236 1966 11 K. O. Friedrichs Coll. C.N.R.S. n° 184, p. 177-189, 1970. 12 T. Ruggeri and A. Strumia Ann. Inst. Poincare, X X X I V , p. 65-84, 1981. 13 M. Anile "Relativistic fluids and magneto fluids" Ca. Un. Press, 1989. 2

22 14

A. Rendall J.M.P 1992. Y. Choquet-Bruhat and D. Christodoulou Ann E.N.S. serie 4, 14 p. 481-500 1981 16 Y. Choquet-Bruhat and Gu Chao Hao C. R. Acad. Sci. Paris 308 serie 1 p.167-170, 1990. ! 7 Y. Choquet-BruhatC.R. Acad. Sci. Paris 319 serie 1, p.237-242, 1994. 15

23

ASYMPTOTIC ANALYSIS OF ELASTIC SHELLS Philippe G. CIARLET Academie des Sciences and Universite Pierre et Marie Curie Paris This article is dedicated to Professor Gu Chaohao on the occasion of his seventieth birthday. ABSTRACT We give a complete classification of linearly elastic shells, according to their asymptotic behavior as their thickness approaches zero. Without resorting to any a priori assumption, we establish the convergence in appropriate functional spaces of the three-dimensional scaled displacements toward the solution of a two-dimensional problem as the thickness of the shell approaches zero. The resulting two-dimensional equations depend on the geometry of the middle surface of the shell and on the imposed boundary conditions. In this fashion, we find either flexural equations, or membrane equations. In certain cases, the membrane equations provide examples of sensitive variational problems, in the sense recently introduced by J.-L. Lions and E. Sanchez-Palencia.

1. The three-dimensional problem of a linearly elastic shell Greek indices and exponents (except e) belong to {1,2}; Latin indices and exponents belongs to {1,2,3}; the summation convention is used. The Euclidean norm and inner product, and the vector product of u, v € R 3 are denoted by \u\,u ■ v,u X v. The L2(A)- and Hl(A)-norms of real, or vector-valued, functions are noted || • ||0,/i and || • \\itALet u be a bounded, open, connected subset of R 2 , with a Lipschitz-continuous boundary 7, the set w being locally on one side of 7. Let y = (ya) denote a generic point in w, da = d/dya,da/3 = dad@, and let <9„ denote the outer normal derivative along 7. Let tp : w —> R 3 be an injective mapping of class C3 such that the two vectors aa = daf span the tangent plane to the surface S = ) at all points in 0, define the sets fle = u x ] — e,s[ and Tg = 7o^l — £.e[i where 70 C 7 and length 70 > 0, let xe = (xf) denote a generic point in fi (hence x% = ya), and let df = d/dx?. For each e > 0, we consider an elastic shell with middle surface S and thickness 2e, whose reference configuration is thus the set * ( n £ ) C R 3 , where ${xe) = 0

24

small enough, the vectors g\ = d\$ are linearly independent at all points in Q . We also define the vectors gie by g*'e ■ g\ = b\, the metric tensor by its covariant and contravariant components gfj = gf > R 3 , where the functions components of the displacement field of the shell; u\ = n£ — R are the covariant this means that, for each x£ € if, the vector uE(xe)gl'e(xE) is the displacement of the point #(x £ ). In linearized elasticity, the unknown solves the variational problem: ue €eV[fl V(n££)) := {vE = « ) e Hl{ilE);vE eE E £ E E £ E jI^ A^ ' ¥e(uk¥(u )e lb(vn^dx A^ei y4j(vnV¥dx

= 0 on onTg}, FE0},

E == j I f'f'EevvtV¥ ^

E dx dx*

(1) (2)

for all vE £ V{QE), where ik jt,ec Xgij,€gkl,s + gik,e*g^ g iW,* :._= Xg^gM'* + ^n(g

A Ai3ki,e

it,ek'gejk^ + gu'Egg> ),

+

E ef|3U{v(^):=i(Sf«|+5|«f)~r?/«;, e% ):=\{dtvE + dEvE)-TVfvEv,

(3)

(4)

where A > 0 and n > 0 denote the Lame constants, assumed to be independent of e, of the material constituting the shell, and / 1 , e G L 2 (fl £ ) are the covariant components of the applied body force density. Problem (l)-(2) has one and only one solution: This either follows from the classical Korn inequality in Cartesian coordinates (Duvaut & Lions 12 ), or from Korn's inequality in curvilinear coordinates (Ciarlet 4 ). Remark. Surface forces acting on the upper and lower faces 4>(Te+ UP_) of the shell, where r?j_ = w x {e} and TE_ = w x {—e}, may be as well taken into account. For the sake of conciseness, we assume here that they vanish. D In the ensuing asymptotic analysis, the following "two-dimensional" ana­ logues of the "three-dimensional" functions (3)-(4) will naturally arise: Qff /3T aa0°T = - i ^ i - a Q / V T + 2/i(a a + aaTa0a), ; v A + 2/i A + 2/i

7Q/3(^) + dd0Va) - Faf3V<J r^0ria - ~ bba0a0V3, r]3, 1afi{j}) = = ^(daV/3 X^a^ + 0r)a) ~ Papiv) = daSri3 - Yaa0dar)z + Vp{dai)a - TTaar)T) + baa{d0r)cj - TTpaVr + {d0bi + r^bi

- ra0b°)n„

-

ca0m-

(5)

(6)

(7)

They respectively denote the contravariant components of the elasticity tensor of the surface S, and the covariant components of the linearized strain tensor and

25

linearized change of curvature tensor associated with an arbitrary displacement field rjid1 of the surface S. The functions 7Q,a(Ti) and pap{r)) may be also written in the more condensed forms lapiv)

= ^{Va\0 + l0\a) ~ ba0V3,

Pa0(v) = V3\a0 + bpV
with self-explanatory notations for covariant derivatives. 2. Transformation into a problem posed over a domain independent of e Our objective is to describe the asymptotic behavior of the vector field u£ as £ - > 0 . Since us is defined over a domain fiE that itself varies with e, our first task naturally consists in transforming the variational problem (l)-(2) into an equivalent problem, but now posed over a domain that does not depend on e. To this end, let 0 = wx] — 1,1[ and r 0 = 70 x [—1,1], With a point xE = (zf) € 0 , we associate the point x = (xt) G fi, defined by xea = xa, xe3 = exj, as e.g. in (8), (9), (12), (16), and we let di = d/dxi. We assume that there exist functions /* € L2(Q) independent of e such that the components of the applied body force density satisfy f <e{xe) = e2f'(x)

for all xe 6 Qe

(8)

(a definition of the otherwise loose statement that "the body force is 0(e 2 )"). Then the scaled unknown u(e) : fl —► R 3 defined by ue{xe) = u{e){x) for all xE e Tf,

(9)

satisfies (compare with (l)-(4)): u(s)e

V(Q) := {v = (vi) € Hl{Q);v

/ Aiiki{e)ek\\i{e\u{e))ei\\j{e;v)^/g{e) Jn

= 0 on r 0 } ,

dx = e2 I Jn

fviy/gjejdx fvi~/g(e)dx

(10) (11)

for all v e V(fi), where Ai]ke{e){x)

:= Al]ki'e{xE),

g(e)(x) := ge(xe) for all xe € Q£,

ea\\0(e\ v) := -{dav0

(12)

+ d0va) - Tpap{e)vp,

(13)

- TaaZ{e)va,

(14)

e 0 |j 3 (e; v) := -{dQv3 + -d3va) e3||3(e;«) := -d3v3,

(15)

26

r?-(e)(i) := rfj e (x e ) for all x£ e fle.

(16)

Note that Eqs. (11) are not denned for e = 0. They constitute an instance of a singular perturbation problem in the sense of J.-L. Lions 15 . In order to perform an asymptotic analysis of the solution u(e) of problem (10)-(11) as e —> 0, we use singular perturbation techniques as for plates; cf. Ciarlet 3 . The "geometry" of the shell poses however considerable technical difficulties. 3. Asymptotic analysis of "fiexural shell" The asymptotic behavior of a family of shells is entirely "governed" by the mapping (p :ZJ C. R 2 —> R 3 together with the subset 70 0 / 7 , by means of the space Vp(u>) defined in (17). To begin with, we study the case where this space contains nonzero functions. Theorem 1 (Ciarlet, Lods & Miara 8 ). Assume that the "space of inextensional d i s p l a c e m e n t s " (the functions 7^/3 (77) are defined in (6)): VP{u)

:= {r, =(m)

£ H1^)

x Hl{w)

H2(OJ);

x

Vi = 9vV3 = 0 on70, 7a/?(?7)

(17)

=0inw}

does not reduce to {0}. Then the scaled unknown defined in (9) satisfies u{e)^umH1(Q.)

ase — 0 ,

(18)

where the limit u 6 V(J1) is independent of the "transverse" variable 13. Further1 f1 more, the function £ := — / udx$ belongs to the space Vp(ui) and satisfies the

2 i_i two-dimensional equations of a scaled "fiexural shell" (the functions aa"aT and Papi'n) are defined in (5) and (7)): paT{<;)p dy ==/ J{J { / \Ja/ a0aa0aT °Tp„ ap(Tj)y/a~ T{Qp ap{Ti)^dy

fldxfdx dy 3}r}i^a~ 3}Vl^dy

for all r] € VF(ui).

(19) D

Remark. The de-scaled function (f defined by (f := £ (in view of (9)) satisfies -3 a0ar aAC)pap(ri)Vi aa0ar Ppar{C)Pap{ri)^ j J a,

forallrje

ddy

y == /j {[j'/

f^dx^r,^ f^dxl)^ dy

VF{w) (cf. (8)).

D 20,21

The space Vp{uj) was introduced by Sanchez-Palencia , who also noted that Vp{ijj) / {0} when S is a portion of a cylinder and
27

on the family {u(e))e>0, which themselves crucially rely on a "first" generalized Korn's inequality (20) valid for an arbitrary surface S = 0 and C\ > 0 such that (the functions ej||j(e;w) are defined in (13)-(15)j

N|i,o<^{^||e t | | j ( £ ; V )||^ n }" |wHi.o < ^{X;i|ci|u(c;«)l|g,n}*

(20)

hi

for all 0 < e < £i and all v € V(fi) (the space V(fi) is defined in (10)).

□

4. Asymptotic analysis of "membrane" shells It is remarkable that, in some cases (cf. Theorems 4 and 5), the "constant" C\/e appearing in (20) can be replaced by a constant independent of e, at the expense however of "replacing ||^3||i,fi by ||i>3||o,n" in the left-hand side (cf. 24)). This provides us with a first instance where the space Vf?(w) of (17) reduces to {0}. Theorem 3 (Ciarlet & Lods6) Define the space 22 2 V„(u) o H , % 6 eL£2(u,)} fl*(w) VM(W) = = {{T} {m);va6 e^ H^u),m M) == #oH xfl*(w) * flJMxxLL(u,), (u), V =={m);Va

(21)

and assume that there exists a constant c > 0 such that £ H'tallL I MH^IIL} I ^ } 1 / 2172 ^ c<{ cE 77 1£ eV M^( WM ). - (22) {E Wn°WU+ + { EHW'tfllo.u.} II^WIIL}1721 7^2 foralla11 a

Q,/3

TTierc

{0}, VFP{w) (w) == {0},

(23)

and there exist £2 > 0 and C2 > 0 such £rea£ ££ u 1/2 e ei||j(£;u)||o,n} i||j(^)Ho,n} ; ) llo.n} { E H ^ H ^ + llv3llo,n}1/2
(24)

i }j

/or all 0 < e < £2 and all v € V(H), where V{Q) is the space of (10) luzi/i r 0 = 7X[-1,1]a Theorem 4 (Ciarlet & Lods 5 , Ciarlet & Sanchez-Palencia10). Assume either that 7 is of class C3 and ip is analytic in an open set containing w, or that 7 is of class C4 and

(w) is "elliptic", in the sense that the two principal radii of curvature are either both > 0, or both < 0, at all points of S. Then relation (22) holds. □

28

Remarkably, this sufficient condition is also necessary, even under weak reg­ ularity conditions on 7 and ip: Theorem 5 (Slicaru 23 ). Assume that 7 is Lipschitz-continuous, cp € C2(u>), and relation (22) holds for all functions 77 = (%) € Hl{w) x Hx{u>) x L 2 (w) i/iai vanish on 70 C 7. 77iera 70 = 7 a^d i^e surface S is elliptic. □ When 70 = 7 and the surface 5 is elliptic (or equivalently, when inequality (22) holds by Theorems 4 and 5), the only information Theorem 1 provides is that u(e) -» 0 in Hx{0) as e -► 0, since the limit is independent of 13 and its average belongs to the space Vf (w), which reduces to {0} in this case (cf. (23)). In fact, u(e) is even "of order 0(e 2 )" in this case: Theorem 6 (Ciarlet & Lods 6 ). Assume that the applied forces are still of order 0(e2) in the sense 0/(8), that 70 = 7, and that relation (22) holds. Then the scaled unknown defined in (9) satisfies x —u{e) H\Q) -* 0, -2 «( M tn i7 (n) x H\Q) jg^fj) X x L 2 (fi) as e -+

(25)

where the limit u is independent of the "transverse" variable X3. Furthermore, 1 f1 the function £ := — / lids 3 belongs to the space VM(W) of (21) and satisfies the 2 J-1 two-dimensional equations of a scaled "membrane shell": a T ^ laT{O J aQ/3<7T 7ar(C)7a/3(r/)v^ dy = J {[Jj la0{V)^Ldy

for

fdx^rnyfady /*dx 3 j»hVS dy

all rj € VM(U>).

(26) □

Remark. The de-scaled function (£ defined by (^ := e2C, (in view of (25)) satisfies ej {C)la0a(ri)^dy 0{r])^dy e j aa0^1(JT 7aT(Ch for all 77 € VM(UI).

= J {f

^eedx%}r,^d dx%}r,^dy a

The proof of the convergence (25) hinges again on a priori estimates, which now crucially relies on the "second" generalized Horn's inequality (24). That the constant in (24) is independent of £ now implies that u(e) is the same order 0(e2) as the applied body force. 5. Asymptotic analysis of "membrane sensitive" shells It remains to study all the "remaining" cases where Vp(u) = {0}, i.e. , those that are not covered by the analysis of Sect. 4. In this direction, our results are

29 best illustrated by considering the important class for which the semi-norm (the functions Ja0{v) a r e those of (6))

= (IK) \vC- ■■= { E{ £II^WIIL} II"■1^\if-r,■■ r, =- (IK) |r,|^ := h ^ M I I 1^^} ^

(27)

a,/3 a,0

becomes a norm over the space V[u) := {v = fa) 6 H 1 ^ ) ; rj = 0 on 7 o } .

(28)

If this is the case, the shell is called a "membrane sensitive" shell. This happens for instance if the surface S is elliptic, but length 70 < length 7, or if 5 is a portion of a hyperboloid of revolution. Remark. The definitions given here of "flexural", "membrane", and "mem­ brane sensitive", shells, are closely related to those of shells with "non-inhibited", "well inhibited", "not well inhibited", pure bending, as proposed by Sanchez-Palencia 20 ' 21 . D In order to get a convergence theorem, we need to assume that the applied body forces are admissible, according to the following definition (which may be slightly generahzed; see Ciarlet & Lods 7 ): There exist functions ipal3 € L2(u>) such that the right-hand side of equations (2) can be also written as (the factor e2 expresses that such forces are again "of order 0(e 2 )"): = = ee22 JJ ^^la0 {vee)dy )dy la0{v

Ij E£ f^vf^cbf f^vf^cbf

s e with with tf tf := := ^i J* ^ vtfdzg, dx 3,

(29)

for all ve = (vf) e V[QE). Theorem 7 (Ciarlet & Lods 7 ) Assume that the semi-norm | • |** of (27) is a norm over the space V[w) of (28), and that the applied forces are admissible in the sense of (29). Then the scaled unknown defined in (9) satisfies 1 f1 —^ / u{e)dx3 -> C«" Vj(w) ase -> 0,

(30)

where the space Vf^(u>) is defined by Vff(u}) := V^(w) := completion ofV(u)) ofV(uj) with respect to |\ •■|„\^..

(31)

Furthermore, the function £ satisfies the two-dimensional equations of a "membrane sensitive shell": <(C B* (C rj) V,)==L* L* (77) (77)/or /or att allr,r e? V% e < (ui), H,

(32)

30

where B^ is the unique extension to V^(w) of the bilinear form BM : V(w) X V[ui) —> R defined by BM(C V)= [ aa^TlaT(Cha0(v)y/^

dy,

(33)

J uj

and Liff : V^(w) —* R is the unique extension to Vj^(u>) 0/ t/ie linear form L: V(w) -> R de/med 6?/ (C/. (29);

Lin) = /I
(34)

J UJ

a Remarks. (1) It is precisely in order that the linear form appearing in problem (32) be continuous with respect to the norm \ ■ \ff that the force are assumed to be admissible in the sense of (29). (2) A convergence result also holds for e~2u(e) (in addition to (30)), but it is too technical to reproduce here; cf. Ciarlet & Lods 7 . □ In the "last" case, where Vp(u) = {0} but \-\*f is a "genuine" semi-norm over the space V(u>), a similar convergence result can be established (see again Ciarlet k. Lods 7 ), but only in the completion \^,(u>) with respect of | • \*f of the quotient space V(w)/V0H, VMM{w) (w) = V(w)/VoM, where VQ(UI) = {n £ M W );7Q/3( J 7) =

Omu)}.

If VF(OJ) = {0} (otherwise than when 70 = 7 and S is elliptic), we have therefore justified by a convergence result the two-dimensional variational equations of a "sensitive membrane" shell, so named because the limit variational problem is "sensitive" in the sense of Lions and Sanchez-Palencia16, and its bilinear form is an extension of the bilinear form of a "membrane" shell (cf. 33)). 6. Conclusions and c o m m e n t s 6.1. The convergence results of Theorems 1, 6 and 7, constitute an asymp­ totic analysis of linearly elastic shells in all possible cases. All together, they provide a rigorous justification of the formal asymptotic approaches of Sanchez-Palencia 22 , Miara & Sanchez-Palencia19, and Caillerie & Sanchez-Palencia2. 6.2. For a linearly elastic plate, or for a linearly elastic shallow shell, the asymptotic analysis yields a two-dimensional limit model where both the "bending" and "membrane" terms simultaneously appear 3,9 . For a linearly elastic shell by contrast, the asymptotic analysis yields a two-dimensional limit model that is either the flexural equations, or the membrane equations.

31

6.3. An earlier convergence result was obtained by Destuynder 11 for mem­ brane shells, under the assumption of uniform ellipticity of the middle surface S. However, his analysis was still "partially formal", in that it assumed the existence of a formal series expansion of the "normal" component of U(E) as powers of e. Using T-convergence techniques, Acerbi, Buttazzo & Percivale1 have also ob­ tained earlier convergence theorems, albeit in "weaker" senses, for shells viewed as "thin inclusions" in a larger, surrounding elastic body. As a consequence, the dis­ tinction between flexural and membrane shells is no longer related to the geometry of the middle surface and the boundary conditions as here. 6.4. For nonlinearly elastic shells, two-dimensional "membrane" or "bend­ ing" equations can be likewise identified through a formal asymptotic expansion of the scaled three-dimensional solution 17,18 . As for nonlinearly elastic plates 13 , F-convergence techniques can be also successfully applied that yield a convergence theorem to the solution of a "large deformation" membrane shell model14. References 1. E. Acerbi, G. Buttazzo and D. Percivale, Thin inclusions in linear elasticity: a variational approach, J. reine angew. Math., 386 (1988), 99-115. 2. D. Caillerie and E. Sanchez-Palencia, Elastic thin shells: asymptotic theory in the anisotropic and heterogeneous cases, Math. Models Methods Appl. Sci. 5 (1995), 473-496. 3. P.G. Ciarlet, Plates and Junctions in Elastic Multi-Structures: An Asymp­ totic Analysis, Masson, Paris, and Springer-Verlag, Heidelberg, 1990. 4. P.G. Ciarlet, Mathematical Elasticity, Vol. II: Plates and Shells, NorthHolland, Amsterdam, 1996. 5. P.G. Ciarlet and V. Lods, On the ellipticity of linear membrane shell equa­ tions, J. Math. Pures Appl, to appear. 6. P.G. Ciarlet and V. Lods, Asymptotic analysis of linearly elastic shells. I. Justification of membrane shell equations, Arch. Rational Mech. Anal., to appear. 7. P.G. Ciarlet and V. Lods, Asymptotic analysis of linearly elastic shells: "Sen­ sitive membrane shells", to appear. 8. P.G. Ciarlet, V. Lods and B. Miara, Asymptotic analysis of linearly elastic shells. II. Justification of flexural shell equations, Arch. Rational Mech. Anal., to appear. 9. P.G. Ciarlet and B. Miara, Justification of the two-dimensional equations of a linearly elastic shallow shell, Gomm. Pure Appl. Math., XLV (1992), 327-360. 10. P.G. Ciarlet and E. Sanchez-Palencia, An existence and uniqueness theorem for the two-dimensional linear membrane shell equations, J. Math. Pures Appl, to appear.

32

11. P. Destuynder, Sur une Justification des Modeles de Plaques et de Coques par les Methodes Asymptotiques, Doctoral Dissertation, Universite Pierre et Marie Curie, Paris, 1980. 12. G. Duvaut and J.-L. Lions, Les Inequations en Mecanique et en Physique, Dunod, Paris, 1972. 13. H. Le Dret and A. Raoult, Le modele de membrane non lineaire comme limite vanationnelle de I'elasticite non lineaire tridimensionnelle, C. R. Acad. Sci. Pans, Ser. II, 317 (1993), 221-226. 14. H. Le Dret and A. Raoult, Derivation variationnelle du modele non lineaire de coque membranaire, C. R. Acad. Sci. Paris, Ser. I, 320 (1995), 511-516. 15. J.-L. Lions, Perturbations Singulieres dans les Problemes aux Limites et en Controle Optimal, Springer-Verlag, Heidelberg, 1973. 16. J.-L. Lions and E. Sanchez-Palencia, Problemes aux limites sensitifs, C. R. Acad. Sci. Paris, Sir. I, 319 (1994), 1021-1026. 17. V. Lods and B. Miara, Analyse asymptotique des coques "en flexion" non lineairement elastiques, C. R. Acad. Sci. Paris, Ser. I, 321 (1995), 10971102. 18. B. Miara, Analyse asymptotique des coques membranaires non lineairement elastiques, C. R. Acad. Sci. Paris, Ser. I, 318 (1994), 689-694. 19. B. Miara and E. Sanchez-Palencia, Asymptotic analysis of linearly elastic shells, Asymptotic Analysis, to appear. 20. E. Sanchez-Palencia, Statique et dynamique des coques minces. I. Cas de flexion pure non inhibee, C. R. Acad. Sci. Paris, Ser. I, 309 (1989), 411417. 21. E. Sanchez-Palencia, Statique et dynamique des coques minces. II. Cas de flexion pure inhibee, C. R. Acad. Sci. Paris, Ser. I, 309 (1989), 531-537. 22. E. Sanchez-Palencia, Passage a la limite de I'elasticite tri-dimensionnelle a la theorie asymptotique des coques minces, C. R. Acad. Sci. Paris, Ser. II, 311 (1990), 909-916. 23. S. Slicaru, Sur l'ellipticite de la surface moyenne d'une coque, C. R. Acad. Sci. Paris, Ser. I, to appear.

33

Generalized Solutions Denned by Lebesgue-Stieltjes Integrals Ding Xiaxi Institute of Math.,Shantou University,Shantou, P.R.China, 515063 Institute of Appl. Math.,Academia Sinica, Beijing, P.R.China, 100080

Dedicated to Professor Gu Chaohao on the Occasion of His Seventieth Birthday

Abstract This paper gives a survey of some recent results on generalized solutions defined by Lebesgue-Stieltjes integrals in the theory of conservation laws.

§1. Introduction In recent years, many authors [1], [2], [3], [4], [5], [6], [7], [8] found and studied a new kind of singular hyperbolic waves namely Delta waves from various equations and problems. For these problems, in general, the solutions can not be found in classes of bounded and measurable functions or in Lp(p > 1) classes. So people need to generalize the solution idea. Of course the standard classical one is to adopt the notion of distribution. But it seems that it still has some difficulties for treating some problems. So strictly speaking, it still lacks a strictly mathematical foundation of Delta wave solutions. In 1993, we in [9] gave a new kind of generalized solution by Lebesguestieltjes integral to a class of hyperbolic conservation laws whose solutions contain Delta waves, and studied its Riemann problems and Cauchy problems with piecewise smooth solutions. In this paper, we shall first present the new definition of generalized so­ lutions given in [9], then introduce some results of solutions involving Delta measures of hyperbolic conservation laws with the new definition.

34

§2. T h e New Definition of Generalized Solutions for Hyperbolic Conservation Laws Consider the Cauchy problems of the hyperbolic system

f ut + f(u)x = 0,

(2.1)

(\ vt + {g(u)v)x = 0, t = 0:

u = u0(x),

v — v0(x).

(2.2)

According to [9], we introduce the generalized potential /■(*.*)

u>(x,i)=

f v dx — g(u)v dt, 7(o,o)

(2-3)

Then v = wx,

ut = -g(u)ux.

(2.4)

Therefore u> satisfies u)t + g(u)u)x = Q.

(2.5)

So, we just focus on the following Cauchy problems instead of (2.1), (2.2) f ut + / < « ) . = 0,

^

^

[ w, + g{u)u>x = 0. t = 0:

w

u = Mo(z),

= w0(a:) = /

v

o(s) ds.

(2.7)

Jo

Using Lebesgue-stieltjes integral, we introduce the conception of general­ ized solutions to the Cauchy problems (2.1),(2.2) and (2.6),(2.7). Definition 2 . 1 . The bounded and measurable functions (u,u>) is called a generalized solutions of (2.6),(2.7) and (u,U)x) is called a generalized solutions of (2.1), (2.2), if / / {u
dx dt = 0, (2 8)

ff

-

!

/ / oji/jt dx dt — / / g(u)ip du(x, t) dt = 0 hold for all ip, r/> e C?(R2+). And

{

l i m ( ^ + 0 / u(x,t)ip(x,t) J

dx =

J

u0[x)ip(x,Q)

lim^+o H^bbj^n^k^afencO-

dx, (2.9)

35 This definition is an extension of the classical generalized solution, and such solution involves 6-shock wave introduced in [8]. Of course the above definition of generalized solutions can also easily be generalized to the following system of equations

f M (w), (/(u)«), ==o, 0, < ++ (/(«*)«). \vt + {g(u)v)x = 0. \vt + (g(u)v)x = 0.

(2.10)

§3 R i e m a n n P r o b l e m s Applying the above definition of generalized solutions, in order to solve the Riemann problems of (2.1) with u_,x v_,x < < 0, 0, , / u _ , x < 0, , / v_,x . , U ^°W » == (( »V ++ , x > ^ ^" o ( l » = ( «U + , x > 0 , > 00 , we first consider the following initial value problems of (2.6) , . -_ J j uU_,X 0, ., t„\ - ' x << °'

V-X,X < 0, , _ jJ v~x,x<0,

U

U,=

°W-\u+,X>0,

\«+x,x>0,

, . (3 2)

'

Since u can be solved independently, the problem is reduced to solve the initial value problem of single equation of w. uwtt+g(u)u + g{u)uj x x = 0, \ v_x,x < o,

!

t —0:

UJ

=<

.

3.3) (3.3)

[ v+x,x > 0.

[ v+x,x > 0. We can construct the solutions by the classical characteristic fine method. For example, we consider the Riemann problems of ut + {\u2)x

= 0,

(3.4)

Vt + (uv)x = 0,

t = 0:

u= P - ^ J [[ uu +,x, z > > 0, 0, +

v = \ V^X<^ «+,£ > [[ v+,x > 0, 0,

(3.5)

i.e. f(u) = ^u2,g(u) = u in the Riemann problems (2.1), (3.1). We only discuss the case of u_ > u+. It is well known that there is a discontinuity line L : x = X{t) = "-+"+1, i.e. u contains shock-waves whose propagation speed is a = "~2""1", and u respectively take u_ and u+ on the

36

both side of shock waves. So we can calculate the solution u> by characteristic method. In this case, we have io(x,t) = v(x,t)(x — at) + v_(a — u_)t + { H < r - [uv]}tH{x - at),

l

,„ fis ' '

where _. V ^

. t )

=

\ v_,x < at, \v+,x>at,

. . H = »+-"-'

»(*) = {{ °:' X<1 0 x <0

v ;

H ence

(Heaviside function)

\ \,x > 0,

v(x, t) = LOX = v(x, t) + {[v]a — [uv]}tS(x — at),

(3-6)

where S(x) is Dirac Delta measure. So, we know that whether Delta waves appear is determined by whether the generalized potential u>(x,t) discontinues, i.e. the discontinuity of u>(x,t) produces 6-waves. According to above ideas, C.Z.Li and X.Yang [11] have concretely con­ structed the generalized solutions of the following Riemann problems of

f ut + C-u2)x = 0,

J

n

(A 0)

^

(3 8)

-

[ vt + (Auv)x = (J, t n t = 0 :

f. \ / iu-,v-),x (u.v) = < , ;

< 0. „

/<, o^ (3.9)

The main results of [11] are listed as follows. Except the trivial case u_ — u+, we set our discussions into eight cases according to the different situations of M _ , U + and A, that is: CASE.l: M_ > u + , a > max{Aw_, Au + }, the solution is J + S; CASE.2: u_ > u+, <x < min{Au_,Au + }, The solution is S + J ; CASE.3: M_ > u + , Au_ > a > Au + , the solution is Sg; CASE.4: u_ > u + , Au+ > cr > Au_, the solution is J + S + J ; CASE.5: «_ < u+, Xu_ < u_, \u+ < u+, the solution is J + R; CASE.6: w_ < u+, Xu_ > u_, \u+ > u+, the solution is R + J; CASE.7: i/_ < u + , Au_ < u_, AM + > w + , the solution is J + R + J ; CASE.8: w_ < u + , Au_ > M_, AW+ < u + , the solution is R; where we denote a = "~*" + , S, J, R and Ss respectively as shock speed, shock wave, contact discontinuity, rarefaction wave and 5-shock wave.

37

T h e o r e m 3.1 The solutions of (3.8)(3.9) are described as above list, and satisfy equation (3.8) as well as initial condition (3.9) in the sense of defination 2.1, in additional, 1). solutions of (3.8)(3.9) contain 8—wave iff u_ > u + , 2).

and

Au_ > a > Au + ,

if w_ < 0 < u + , A < 0, then v is unbounded, but in V,

where

i
r M ( + ( ^ = o,

(3io) (3.10;

\ vt + {uv)x = 0. which is the isentropic gas dynamics equations in Euler coordinates with v = PiPip) = const, we find it is a special case of (2.10). The Riemann problem can be treated similarly as (3.4), (3.5).

§4 Cauchy Problems Consider the Cauchy problems

U< + (|u2)* = o,

(4i)

\ vt + (uv)x = 0, t = 0:

u = u0(x),

v = v0(x).

(4-2)

From discussion in §2, we study following Cauchy problems instead of (4.1), (4-2),

f w( + (|w 2 ) I = 0,

(4.3)

{ u>t + uux = 0, t —0 :

u-

u0(x),

w - LO0(X) -

I

vJy)

dy,

(4.4)

where u> = f v dx — uv dt is a generalized potential. 7(o,o) Here we use the conception of generalized solutions defined in §2. i.e. Definition 4 . 1 . The bounded and measurable functions (u,u>) is called a generalized solution of (4.3), (4.4) and (u,u>x) is called a generalized solution of (4.1) ,(4.2), if ul f JJ{ //(«¥>* + \
I If^foffandfafakfai*)dt

=°

(4.5)

38

hold for all v , if> £ C^(R%).

lim

And

u(x,t)(p(x,t)dx= u(x,t)(p(x,t) dx = / uoo(x)
(4.6)

lira \\w{x \\u(x,t)-wo(x)\\ Loa }t)-u0(x)\\ Lfo

(4.7)

(—++0

== 00.. toe loc

Considering the uniqueness of solutions, we introduce the conception of adimissible solution. Definition 4.2. For the system (4.3), (4.4), a generalized solution (u,w) is said to be admissible if u satisfies the (E) criterion, i.e. there exists a positive constant E such that u(x2,t) — u(xi,t) E u(x2,t)-u(Xl,t) (4.8) < E X2 — X\ X2 — X\

~ ~

t t

holds for all —oo < X\ < x2 < +oo and almost all t > 0. Using the above conception, X.Ding and Z.Wang in [10] have proved the existence and uniqueness of the solution for the Cauchy problem (4.1), (4.2). We have T h e o r e m 4 . 1 . Suppose that u0(x) £ L°°(R) and u0(x) £ C(R)nBVloc{R). Then there exists a global solution for the Cauchy problem (4.1), (4.2) and (4.3), (4.4). T h e o r e m 4.2. Suppose that (iti,Wi), (u 2 ,^2) are admissible generalized solutions of (4.3), (4.4), Then U\ = u 2

a.e.,

u>\ = ix>2 a.e.

For all A > 0, Applying the idea developed in [10], F.Huang, C.Z.Li and Z.Wang [12] have solved the Cauchy problem of (3.8) and studied some prop­ erties of solutions. We restate some of them. Consider the system (3.8) for A = | with initial values (22 ( u0(x) - \ xx-2- 2

x < 0, 0 <<xx <<44, , x > 4.

(4.9)

• 2 x <-1, 1 -1 < x < 2 x - 2 2 < x < 4, . 0 x > 4.

(4.10)

I2 v0(x)

By the characteristic method, we have

39

I f=f X(t)<x<2t

«(*,*) = < ' + 1 z

+ i,

' . otherwise.

(4.H)

where X(t) = — 4 0 + 1 + 2(< + 1) + 2 is the discontinuity line of u (shock wave). (i). 0 < t < 3, we have

x—2 u{x,t) = u(x,t) =x-t+( x - t + {X~ -x + t + 2)H{x-X(t)), 2)H(x-X(t)), vt+ 1

(4.12)

(ii). t > 3.

[ 2(* 2(i -- 1 f) ) + 11 ++ 3H{x 3H{x -- Y(t)), Y(t)), xx << X(t) X(t) u>(x,t) = < (j, _ 2)„,. "(*>*)=<{ a , „ _ „ + 2,x>X(t) \ %$ ^ + 2,x>X(t)

(4.13)

where i/(x) is Heaviside function, x = Y(t) = t — \ is the tangent line of the curve x = X(t) in the point (2,3). Therefore, we obtain (i). 0 < t < 3, v(x,t)

= bjx =

1+(75JT-!)#(*-*(*))

(4-14)

+ ( ^ r - z + * + 2)<5(x-x(<)) (ii). i > 3

(f 2 + 3 £ ( x - t + ll),z ) , z <<X(f) X(f)

(4.15

From (4.11), (4.14) and (4.15), we know that L\ : x = X(<) is a Delta shock wave in the sense of [8] as 0 < t < 3; while L2 : x = Y(t) is a Delta wave of contact discontinuity type, and Lx is changed into a usual shock wave as t > 3. (see Figue 1) Therefore the <5-shock wave L\ defined in [8] is separated into a <5-wave L2 and a usual shock wave initiated from the point (2,3) as time increases. It seems that <5-shock wave is a complex wave but not a basically hyperbolic wave similar to shock wave or rarefaction wave. Furthermore, F.Huang [13] has generalized the reaults of [12] to the system

r „,+/(■.), = », I vt + (g{u)v)x = 0,

(4.16)

40

where f(u) £ C2 is convex and q(u) is increasing.

Figure 1 In general, when we study the Cauchy problems of hyperbolic conservation laws like (2.1), we have to meet a multiplication problem of a discontinuous function with a generalized function, which is hard to deal with, if we use the conception of classical generalized solutions satisfying equations in the sense of distribution. But using the definition of generalized solutions introduced in §2, we first consider the generalized potential Lo(x,t) related to Lebesgue-stieltjes integral, and hence avoid the multiplication problem of discontinuous function with generalized function, therefore, we can study the existence and uniqueness of generalized solution to Cauchy problems of the hyperbolic conservation laws.

§5 T h e linear hyperbolic systems with discontinuous coefficients For Cauchy problems of linear hyperbolic equation or equations with dis­ continuous coefficients

I ut + Aux — 0, [ t = 0 :

(5.1)

u — U 0 (:E),

I.Gelfand [14] analysed the existence and uniqueness of solutions as coefficient A only has a discontinuity line, and claimed that there usually is no classical solution. However Z.Wang, C.Z.Li and X.Ding [15] have proved the existence of generalized solutions defined in §2 for single equation. Additionally, the phe­ nomena will still appear in the systems of conservation laws with discontinuous coefficients. In order to study the existence and uniqueness of solutions,I.Gelfand con-

41

structed the L2 energe integral of systems u, + (Au)x = 0,

(5.2)

in the appendix of [14], where A has a discontinuity line. But, there probably has no solution in L2 space. For example, set A(x) = —sgnx, then the Cauchy problem of (5.2) is equivalent to [ vt + (\v2)x

= 0,

I| uutt + + (uv) {uv)xx = = 0, ([ t = 0 :

(5.3)

v = —sgnx,

u=

u0(x).

X.Ding and Z.Wang have proved that (5.3) usually has no solution in the measurable function space, but there exists an unique generalized solution involving A-wave in the sense of §2. According to [15], using the new conception defined in §2, we can prove ex­ istence of solutions for the hyperbolic systems with discontinuous coefficients. Consider Cauchy problems of conservation and nonconservation law

{

ut + {au)x = 0, t = 0:

(5.4)

u = u0(x)

and

(

u)t + a(x)ujx = 0, t = 0 :

(5.5)

u> = OJ 0 (X),

where a(x) is bounded and locally variation bounded function. First, we introduce generalized potential u>(x,t) = f u dx — a(x)u dt, Jfo.o) /(o,o) then u = w x , u>t + a(x)u>x — 0, and ui0(x) = / u0(y) dy. ./o

(5.6)

Definition 5.1. suppose that u(x,t) is measurable to t, locally variation bounded to x. u> is called a generalized solution of (5.5) and ux is called a generalized solution of (5.4), if llwh

JJ

dx dt-

J I a(x)ipduj(x,t)

Copyrighted Material

dt = 0

(5.7)

42 hold for all ij> e C^(R2+).

And

lim llwfi.f) — w 0 (x)||,oo = 0 . The last integral of (5.7) is Lebesgue-Stieltjes integral, in order that a(x) is L-S measurable, we rule a(x) = mid{a(s - 0),a(x + 0),0} where mid is denoted to take the middle value of three functions, In the above sense, Z.Wang, C.Z.Li and X.Ding have proved following the­ orems in [15]. T h e o r e m 5 . 1 . If w 0 (i) <E C{R)DBV,oc(R) or u0(x) e L]0C(R), Then there exists a global solution for the Cauchy problems (5.5) and (5.4), respectively. Remark: The method can be extended to systems. A c k n o w l e d g e m e n t . I am grateful to Professor C.Z.Li and F.Huang for inspiring discussions and their great help in the preparation of this paper.

References [1] D.Tan and T.Zhang, Two-dimensional Riemann problem for a hyperbolic system of nonlinear conservation laws I.II., J.Diff.Eqs., vol.111, 1994, 203-282. [2] D.J.Korchinski, Solution of a Riemann problem for a 2 X 2 system of con­ servation laws possessing no classical weak solution, Ph.D.Thesis, Adelphi Univ, 1977. [3] B.L.Keyfit and H.C.Kranzer,A viscosity approximation to a system of con­ servation laws with no classical Riemann solution, in Nonlinear Hyperbolic Problems, Lecture Notes in Math. Vol.1042, Springer-Verlag, Berlin,New York, 1989. [4] H.C.Kranzer and B.L.Keyfitz, A strictly hyperbolic system of conserva­ tion laws admitting singular shocks, in Nonlinear Evolution Equations that change type, IMA Volumes in Mathematics and its Applications, Vol 27 Springer-Verlag Berlin,New York, 1990.

43

[5] F.Bampi and C.zordan, Higher order shock waves, J.Appl.Math.Phys. 1990, 12-16.

41,

[6] J.Rauch and M.C.Read, Nonlinear supperposition and obsorption of Delta waves in one space dimension, J.Funct.Anat.73, 1987, 158-178. [7] P.LeFloch, An existence and uniqueness result for two nonstrictly hyper­ bolic systems, Preprint, Ecole Polytechnique, Centre de Mathematiques Appliquees.No.219, August, 1990. [8] D.Tan,T.Zhang and Y.Zheng, Delta-shock wave as limits of vanishing vis­ cosity for hyperbolic systems of conservation laws, Journal of Diff. Eqs. Vol 112, No.l. August (1994), 1-32. [9] X.Ding, On a non-strictly hyperbolic system, Preprint, Dept of Math. University of Jyvaskyla,No. 167,(1993). [10] X.Ding and Z.Wang, Existence and uniqueness of discontinuous solutions denned by Lebesgue-Stieltjes integral, Preprint, Institute of math., Shantou Univ.,No.7, 1994. [11] C.Z.Li and X.Yang, Riemann problems of a class of non-strictly hyperbolic equations, Preprint,Institute of math., Shantou Univ.,No. 19, 1994. [12] F.Huang, C.Li and Z.Wang, Solutions Containing Delta-Waves of Cauchy Problems for a Nonstrictly Hyperbolic System, Preprint, Inst. Math., Shantou £/7u'«.,No.23,(1994). [13] F.huang, Existence and Uniqueness of Discontinuous Solutions for A Hy­ perbolic system, Preprint, 1995. [14] I.Gelfand, Some problems in the theory of quasilinear equations, Usp. Mat. Nauk., 14(1959), 87-158; English transl. in Amer. Math. Transl. ser 2, (1963), 295-381. [15] Z.Wang, C.Z.Li and X.Ding, On a problem proposed by I.Gelfand, Preprint, 1995.

44

A U T O M O R P H I S M S OF T H E CIRCLE—AND TEICHMULLER THEORY JAMES EELLS Warwick/Cambridge

Abstract Certain homogeneous spaces of the diffeomorphism group of the circle have wide­ spread physical, geometric, and analytic influence. In particular, they have a natural homogeneous Kahler structure—and are suitably recognized as submanifolds of the universal Teichmuller space. I present an exposition of these ideas—emphasizing contributions of Nag, Sullivan, and Verjovsky.

In this lecture I shall construct and relate various p a r a m e t e r spaces of geom­ etry, representation theory, and m a t h e m a t i c a l physics—placing special emphasis on [Bers]' notion of universal Teichmuller space.

T h e m o n o g r a p h of [Pressley,

Segal] provides a fine reference for those aspects of representation theory close to my m a i n theme.

And a recent article by [Pekonen]—fortunately brought to

my attention by A. Verjovsky while I was preparing the present text—presents a related exposition, specifically addressed to working physicists. From m y own viewpoint, the article [Nag, Sullivan] develops a further wide variety of interactions with geometry—symplectic and algebraic in particular.

1. Homogeneous Spaces of DifF+(S1) (1) Let Diff+(5 1 ) = V denote the topological group of orientation-preserving diffeomorphisms of the circle 5 1 . (We work in C ° ° , although any Cr(r

> 1) would

do.) Consider the conformal automorphisms / : U —> U of the unit open disc U C_(E. These have the form / ( * ) = A * ~° with |A| = 1 a n d \a\ < 1; 1 — az a n d certainly extend to U —> U. They form a 3-dimensional Lie group P S L ( J E 2 ) = V, a closed subgroup of V. And of course V contains the compact s u b g r o u p 1Z of rotations of 5 1 :

n c P c v.

45 (2) Use 71 and V to normalize the X>-action on S1. We are led to the homoge­ neous

fibration V/7l vjn == AfN

iJ. V/V

fibre

V/Ti. = open disc.

= M

T h e inclusion 7Z. —> T> is a homotopy

equivalence;

theory, both M and M. are contractible.

so by elementary homotopy

Indeed, using the homotopy sequence of

(V,1Z) we see t h a t TTi(M) = 0 for all i. T h a t implies that M is contractible, for it is an absolute neighbourhood retract. And then, so is

M.

(3) Consider the Lie algebra L{V) of smooth vector fields tangent to S1.

Com­

plexify it: L<E(V)=W®L(V). {V)=W®L{V). T h a t is generated by the functions Lk = e' M J^> for each k £ Z. According to Gelfand-Fuchs, there is a unique central extension of Z/ff(X>); it is generated by (Lk)kez

and a central element c with commutation rules

f

,rr

i,

v..

rrr3 — m m

\. r

!

[Lm,Ln\ group, — m)L + — —group — d m i T>, - n cwhich , = i(n the There is an associated Virasoro-Bott is a central m+n extension of V. (By way of contrast, there is no group associated with La(T>).) [Ln,c} = 0. (4) T h e dual L*{T>) of L{V) is much studied in various contexts; e.g., There is an associated group, the Virasoro-Bott group T>, which is a central (a) in representation theory: Bakas and W i t t e n have shown t h a t the Liea algebra extension of V. (By way of contrast, there is no group associated with L (T>).) (4) T h e dual L*{T>) of L{V) is much studied in various contexts; e.g., (a) in representation theory: Bakas and W i t t e n have shown t h a t the Lie algebra L(T>) admits no coadjoint actions by nontrivial subgroups of V, other t h a n Ti. and V (see (8) below). (b) Gelfand-Fuchs recognized the L 2 -pairing with the quadratic differentials i0

b(e )d92.

Hu = d\ +

u(z)

T h e complex form for (c) functions u : 51 — ► W.of L*(T>) is identified with the space of Hill operators Hu = d\ + u(z) isospectral of these correspond to KdV flows; i.e., to solutions for The functions u : S 1deformations —»W. of The isospectral deformations of these correspond to KdV flows; i.e., to solutions —QJ- = 3 [ P K , i 7 a ] , vhere

pu = -d3z + udz +

dz-u.

46

That is the Lax form of -£ = d\u + 6ud2u. at (5) Both M and M carry natural complex manifold structures (of infinite di­ mension, of course), relative to which the coset fibration is a holomorphic map. [Pressley], [Bowick, Rajeev]: (a) The tangent space to Af at 1: Ti(Af) = j u = y ^ t t t l n with uk = u_jk|. The endomorphism Ju = —i y

^sgn(k)ukLk

k

satisfies J 2 = — Id, so defines a complex structure on T\(Af). In fact, J is a Hilbert transform. Define J on M by right translations. Then (Af, J) is integrable. (b) Similarly for AA : Ti(M)

= \u =

2 J UfcXt : fit =M_ibj. fc^o,±i Incidentally, L-\,LQ,L+I generate the complex Lie algebra of P . (c) There is a natural homogeneous Kahler metric on both Af and AA [Bowick, Rajeev]. Construction for AA : By homogeneity, it suffices to determine the Kahler form u> at 1; and because we require dw = 0, w([L m , L„], Lp) + u)([L„, Lp],Lm)

+ u([Lp, Lm], Ln) = 0.

Also, w(Io,-) = 0 = w ( £ ± 1 , - ) . The only possibility is w(Lm,L„)

= a(m3 - m)Smi-n

for m,n

eZ\{0,±l}.

Here a = ib with any 6 > 0. Take 6 = 1. The required metric on AA is oo

g(v,w) = 2Re ^

t>,„u)m(m3 - m).

(6) R e m a r k . Let £ = {/ € C""(S\ff) : / embeds S 1 into Z/}.

47 T h e n £ is a b o u n d e d domain in CU(S1 ,W). Segal has identified

its Shilov

boundary

with T>. Denote by £"+ = {/ g £ extending to a holomorphic m a p U —> W with / ( 0 ) = 0 } . Then

M= Af = P/ft V/K = = £/£+. £/£+. (7) R e m a r k . Every f>o €LT> has a unique extension to a harmonic difFeomor­ phism W - t W . (8) R e m a r k . Consider the homogeneous spaces of 27, the Virasoro-Bott group— and, in particular, those arising as coadjoint orbits (i.e., as an orbit of a vector in the dual L*(T>) under the X>-action. [Bakas] and [Witten] have shown t h a t Af, are the only such spaces with homogeneous

Kdhler

M

structure.

Now coadjoint orbits carry n a t r u a l symplectic structures. And t h a t of M. is the Kahler form of the Kahler structure (5).

2. Quasisymmetric Automorphisms of S 1 (1) In m a n y respects, the group of orientation-preserving homeomorphisms of 1

S is the one we should be considering (important steps in t h a t direction have been taken by [Penner]). For instance, t h a t plays a physical role as the reparametrization group of a closed string.

However, for other purposes t h a t group is too

large—and D = DifF+(5 1 ) is too small. We take an intermediate course, following Beurling-Ahlfors: Say a homeomorphism / : S1 —> S1 is quasisymmetric

if it extends to a quasi-

conformal m a p (definition recalled in (3.1) below) of the closed unit disc U —> U. T h a t can be expressed via cross ratios: 3K> 11 : : 3K>

^

< RV(*i)tK**)j(*»),n*4))

for any point with R(zi,Z2,zj,Z4) Let Q = QS(Sl)


^

= |.

denote the group of these. We note t h a t each / 6 V can

be extended t o a difFeomorphism of U\ and therefore / € Q. T h u s we extend our chain of groups:

K r P CV r D r CQ. O. TlCV (2) The automorphisms serving the Hilbert

in Q are characterized

space

£ 2 ,(5 1 ,«)/^ = W. 22

as those homeomorphisms

pre­

48

Write -rf = W+®

W-,

where W± are the ^pi-eigenspaces of the complex structure J on H. In particular, W+ = {/ £ T-F, whose Fourier coefficients of negative index = {/ : (W,0) -> (ff,0) holomorphic, with (3) Set V =

=0}

/ |/'| 2 < oo}. Ju

Cco(S1;R)/R.

On it we have the symplectic form 55 :: V V xx V V -- ». li given by 1If

/■

°° °°

?(/,
|5(/,ff)|< 11/11||ff||. llsll. \s(f,g)\ < ll/ll And on H, its inner product ( , ), the Hilhert transform J, and the symplectic form S are related by

(f,9)=S(f,Jg). (f,9)=S(f,Jg). [Segal] has shown that V acts faithfully and symplectically on V. And [Nag, Verjovsky] have extended that to a Q-action on "H. (4) The Hilbert space G(U) of harmonic 1-forms in C2(U) has the symplectic form S(a,P)= S(a,P)=

/ a A/9, Ju Ju and has a compatible star operator *. [Nag, Sullivan] have established an isometric isomorphism G{U) = g(U) =

C\I2/2{S\R)IR{=H), C* (S\R)/R(=H),

with A corresponding to S; and * to J. G(U) is the basis of a universal Jacohian. (5) We have the induced map

nIT:: M M -> -> Sp(V)/U, sP(v)/u, which [Nag] has shown to be an equivariant Kahler holomorphic isometric im­ mersion. Think of it as a period map. [Nag, Sullivan] have extented that to the universal Teichmiiller space ((3.2) below): II : 3(1) —> SpH/U = Sao = universal Siegel space.

49 Here Sp H = t h e group of all bounded symplectomorphisms of H. And U = the subgroup of those keeping W+ invariant. Again, II is an equivariant

holomorphic

immersion. (6) Take a compact Lie group G and form AG = C\{S\G) T h e n ClG = AG/G;

andSlG

=

C2(S1,l;G,e).

and AG is the semi-direct product CIG oc G.

Now HG has a natural symplectic

structure Js1

which is

AG-invariant.

T h e group T> acts as a group of symplectomorphisms sider L(V)

of (AG,S).

We can con­

as a Poisson algebra ( = Lie algebra of functions with Poisson bracket

on tiG). V acts on t h e almost complex structure Jo o n £IG, transforming it t o f~l o J o o / „ . For it, Jo is fixed iff / is a rotation . And the transformed structure is also invariant on fit? a n d is compatible with the symplectic form S. Thus M = Diff

S1/1Z

parametrizes these at a point. (7) Following [Popov, Segeev], let Z denote the bundle of AG-invariant complex structures on S7G compatible with S :

Z

'[ V ttG

M

And in fact, Z is a AG manifold with Z/ A G = ftf. Thus we have a double fibration of Z. T h a t determines a natural almost complex structure is holomorphic. Points

3 on Z; and p

There is also a real structure on Z; t h a t transforms 0 into — 0-

of QG correspond

to real holomorphic

sections

of p : Z —> A/".

3. On Universal Teichmiiller Space (1) Let £°°(W)i denote the open unit disc in the (nonseparable) Banach space C°°(U).

W i t h each n € C°°(U)i

satisfying Beltrami's

is associated a homeomorphism w^ : U —> U

equation dziv =

/idzw.

T h a t equation characterizes the quasiconformal ize Wp, requiring it to fix + 1 , — 1, — i.

automorphisms

of U. We normal­

50

Every w^ extends continuously toU —> U; and determines the map £°°(W)i —► Q defined by

p-> ft = «vl sl (2) [Bers] denned the universal Teichmuller space

3 = 3(l) = £ 0o (W) 1 /~ with the equivalence fi ~ v iff w^ and w„ have the same boundary values on 5 , modulo V. (3) There is the characterization 03 = Q/V. Elements of the right member are viewed as quasisymmetric automorphisms of S1 fixing + 1 , - 1 , —i. (4) Let $ : C°°(U)\ —> 3 denote the quotient map. Then 3 inherits a complex homogeneous Banach manifold structure, with $ a holomorphic submersion. (The map $ is represented by the Schwarzian derivative in (5) below.) The differential d<3>(0) is a complex linear surjection with Kerd$(0) = { / i £ £°°(W) C°°{U) : f/ \iB tf = 0 for every C £ 'l -holomorphic function 98 : U — —t(Ft. ► (E \. Therefore, T 0 (3) = C°°(U)/Kevdi(0). It has been shown by [Earle, Eells] that 3 w contractible—by first establishing that $ is a locally trivial fibration. (5) Bers has embedded $ as a holomorphically convex domain in the complex Banach space 2 B= U | e : H* tf* -► -►0'
Here H* denotes the lower half plane of (E. If f S 3 is thought of as a quasiconformal map on H*, then $ ( / " ) z = S(f(z) for z g H*, where S(f) is the Schwarzian derivative

«/>-(£)'-£(£)'■ *o-(£)•-!$)• 3 contains the open disc of radius 2 in B and is contained in the closure of that of radius 6. (6) Prom the chain in (2.1), we have the inclusion map A4 Q/V ==20M =V/P-* =V/V-4

51 [Nag, Verjovsky] have remarked t h a t that inclusion

is holomorphic.

(Indeed, M

is a coset in 3 ; a n d the point is to identify the complex structures at 1 in both spaces.) [Nag] has identified metric

the Kdhler

metric

on M (§1,5) with the

Weil-Petersson

onZ '■

«-<>-& U&&-««

T h a t integration is convergent in £ | - p a i r i n g of (i, v (or in C > + £ in any e > 0). 2

(7) It seems n a t u r a l to expect^-using an extended version of t h e Weil-Petersson m e t r i c — t h a t M a n d Af are naturally geodesic spaces of nonpositive curvature (in the sense of Alexandrov, Busemann, Gromov,- ■ ■). W i t h t h a t we would have an existence theory for harmonic m a p s into them. (8) 3 is universal, in the following sense: Any R i e m a n n surface of genus> 2 has the form M = li/T,

where T is a Fuchsian

group ( = a discrete subgroup of the holomorphic automorphisms of U).

The

universal cover of t h e Riemann moduli space of M is 3 ( T ) = {(i £ 3 which is respected by T ( = is ( 1 , l ) - a u t o m o r p h i c ) } . Thus, each 3(T) is holomorphically

embedded in 3 = 3 ( 1 ) .

This is indeed significant in various applications, e.g., where dynamics or string evolution requires consideration of all Riemann surfaces at once. (For instance, if degeneration is permitted!)

52 REFERENCES I I . Bakas, Conformal invariance, the KdV equation and coadjoint orbits of the Virasoro algebra, Nucl. Phys., B 3 0 2 (1988), 189-203. 2. L. Bers, On moduli of Riemann surfaces, Lecture Noles ETH, 1964. 3. M. J. Bowick and S. Rajeev, The holomorphic geometry of closed bosonic string theory and D i f f f S ' y S 1 , Nucl. Phys., B 2 9 3 (1987), 348-384. 4. M. J. Bowick and S. Rajeev, String theory as the Kahler geometry of loop spaces, Phys. Rev. Lett, 58 (1987), 535-538. 5. C. J. Earle and J. Eells, On the differential geometry of Teichmiiller spaces, J. Anal. Math., 19 (1967), 35-52. 6. S. Nag, T h e complex analytic theory of Teichmiiller spaces, Wiley-Interscience, 1988. 7. S. Nag and D. Sullivan, Teichmiiller theory and the universal period mapping via quautum calculus and the HI space on the circle, Osaka J. Math., 32(1995), 1-34. 8. S. Nag and A. Verjovsky, Diff(5 1 ) and the Teichmiiller spaces, Comm. Math. Phys., 130 (1990), 123-138. 9. O. Pekonen, Universal Teichmiiller space in geometry and physics, J. Geo. Phys., 15 (1995), 227-251. 10. R. C. Penner, Universal constructions in Teichmiiller theory, Adv. in Math., 98 (1993), 143215. 11. A. D. Popov and A. G. Sergeev, Symplectic twistors and geometric quantization of strings, Alg. Geo. and Appl. Aspects of Math., N ° 2 5 (1994), 137-157. 12. A. Pressley, Decomposition of the space of loops on a Lie group, Topology, 19 (1980), 65-79. 13. A. Pressley and G. B. Segal, Loop groups, Oxford, 1986. 14. G. B. Segal, Unitary representation of some infinite-dimensional groups, Comm. Math. Phys., 80 (1981), 301-392. 15. E. Witten, Coadjoint orbits of the Virasoro group, Comm. Math. Phys., 114 (1988), 1-53.

53 NONLINEAR RELATIVISTIC WAVE EQUATIONS IN GENERAL DIMENSIONS

M O S H E F L A T O and D A N I E L

STERNHEIMER

Research Institute for Mathematical Sciences, Kyoto University Kitashirakawa, Sakyo-ku, Kyoto 606-01, Japan^ and J A C Q U E S C.H. S I M O N and E R I K

Physique Mathematique, Facultes de Sciences Mirande,

TAFLIN*

Universite de Bourgogne B.P.138, F-21004 Dijon, France

ABSTRACT Given a linear partial differential evolution equation covariant under the action of a Lie group G, one can equate the evolution operator -sz with a representative of an element of the Lie algebra g of G. The same holds for a nonlinear P.D.E. covariant under G. Here the mathematical apparatus is a Lie theory of nonlinear representations. The linearizability of such equations is tightly connected to cer­ tain cohomology spaces. After explaining these facts we shall concentrate on some scalar field equations, for which this framework permits also the treatment of noncovariant interactions, and on a recent result concerning global Cauchy problem, asymptotic completeness and infrared electron tail for the coupled Maxwell-Dirac (= electrodynamics) equations in physical (1 + 3) dimensions. Foreword This p a p e r is dedicated t o our friend GU ChaoHao on t h e occasion of his sev­ entieth b i r t h d a y a n d fiftieth year of educational work, as a person a n d as a scientist. It is based o n t h e notes for a lecture delivered by one of us ( M F ) July 10, 1995 in Sapporo at t h e fourth MSJ-IRI on Nonlinear Waves a n d is intended to b e a presentation of t h e "cohomological" treatement of nonlinear evolution equations, the linear p a r t of which is covariant under some symmetry group, using nonlinear representations of t h a t group a n d t h e spaces of initial d a t a t h a t these represen­ tations suggest. T h e linearizability of t h e time translations, obtained by showing t h a t certain cocycles are in fact coboundaries, will provide global solutions for t h e nonlinear equations. * Permanent address: Physique Mathematique, Universite de Bourgogne, Facultes de Sciences Mirande, B.P.138, F-21004 Dijon, France; e-mail: [email protected] * Permanent address: Union des Assurances de Paris, 9 place Vendome, F-75052 Paris Cedex 01, France

© 1 9 9 5 T h e authors

54 We shall begin (Section 1) with a short s u m m a r y of the theory of nonlinear Lie group representations, started by our group almost 20 years ago, a n d continue with a very short presentation of recent spectacular applications, t o Nonlinear KleinGordon equations (Section 2) and Maxwell-Dirac equations (first-quantized elec­ trodynamics) in Section 3. T h e short list of references t h a t follows in m e a n t only as an indication of first complementary reading a n d source of further references. 1. N o n L i n e a r R e p r e s e n t a t i o n s 1.1 NonLinear

Representations

of Lie Groups and

Algebras

For this part the basic reference is Ref.5, completed by Ref.6, some later ref­ erences t h a t can be found in the review 1 6 by one of us (DS) a n d by Ref.14. 1.1.1. Let £ b e a Banach or Frechet space, Cn(E) the space of continuous sym­ metric multilinear m a p s En —* E. It is isomorphic to the space C(®nE, E) of linear continuous m a p s from the completed (with respect to the projective ten­ sor p r o d u c t topology) symmetric tensor product of n copies of E into E. ( T h e Hilbert-space oriented reader is reminded t h a t , when £ is a Hilbert space, E®E corresponds to trace-class operators on E, with symmetric kernels here, a n d is smaller t h a n the Hilbert space completion which corresponds t o Hilbert-Schmidt operators). To any fn in Cn(E) one associates the monomial / " : E —» E by

f(v)sf(y,...,y),

veE.

We denote by F(E) the space of formal series f = 2 n = i / " wl^h p r o d u c t • given by the composition of the corresponding m a p s X} n =l f" fr°m E t o E, i.e. if h = Y^T 'l™ *= F(E) t h e n the n - t h term of the product is such t h a t :

EBf7hn&)= E l
ft

E

^(v)®-..®^)).

ii ii + + ...+i ...+ipp = = nn

A (formal) nonlinear representation of a Lie group G is a couple (S, E) where S is a homomorphism G 3 g — i > Sg = Yln=i '-'9 °^ *-* m * ° * n e group of invertible elements of J~{E). Similarly one defines the notion of analytic nonlinear representation by assuming t h a t the morphism S is analytic at 0 (when !F{E) is endowed with a n a t u r a l topology). T h e equivalence of nonlinear representations is denned by an intertwining operator A: ( 5 , E) ~ (S',E) (S\E) and S is said linearizable

S'g = ASgA'1,

A e6 T{E) T(E)

invertible invertible

(1.2)

if S ~ S , S 1 linear.

1.1.2. Associated linear maps: Take E = U„ ffiJJ=1 (®PE) ("Fock space without v a c u u m " ) . A one-to-one algebra homomorphism A : J-(E) —> C(E) m a p p i n g • t o the product of linear m a p s is defined on the n-th level by: A

(/)n= E E

E E

i + . . . + ip=n l
(f (f11®---®/ ®---®/ 1 ')-

(1.3)

55 Its differential dA is t h e n defined on the n - t h level by:

dA(/)-= x ) ( E

^®/ n - p+1 ®/ P - ? -i)

(1.4)

l < p < n 0
and m a p s the bracket fxh — hxfoi formal series (which is therefore a Lie bracket) into the c o m m u t a t o r of the linear operators dA(f) and dA(h); f x h is defined on t h e n - t h level (denoting by o the composition of maps) by:

(fxh)n=

Y,

/P

°( E

l
Iq®hn-r+1®Ip-q-,).

(1.5)

0
A nonlinear representation of a Lie algebra g is a Lie morphism of g into J~(E) equipped with this bracket. An i m p o r t a n t result of the theory is t h a t one has a Lie theory, i.e. t h a t on the Frechet space Ex of differentiable vectors of the linear part S1, t h a t can be realized as the linear span of the vectors of the form {(p = JG Slgipa(g)dg}, with ip G E and a G 'D(G) (the space of C°° functions on G with compact s u p p o r t ) , t o any nonlinear representation S of G one can associate its differential dS, a nonlinear representation of g, and vice-versa. This result holds in b o t h 5 the formal and analytic frameworks and in fact 1 4 , for a large class of nonlinear analytic Lie algebra representations dS (including those arising from physical nonlinear evolution equations), the space of differentiable vectors for S coincides with the space of differentiable vectors for its linear part S 1 . 1.2. NonLinear

Representations

and

Cohomology

1.2.1. Cohomology. Let G be a group, U a linear representation of G in E, g, g' G G. We define 1-cocycles as maps R: G —► E such t h a t Rggi = Rg + UgRgi and denote their space by Z1(G,E). 1-coboundaries are 1-cocycles for which there exists a (0cochain) ip G E such t h a t Rg = Ugip — ip and we denote their space by B1(G,E). Then the ("affine") 1-cohomology is the quotient 1 H\G,E) Z1{G,E)/B (G,E). H\G, E) = Z\G, E)/B1(G, E).

(1.6)

1.2.2. An extension (V,F) of (U,E) by (U',E') is an exact sequence of G-modules 0-*E^fF^tE'^><$ and is given by a 1-cocycle of G for the representation T in C{E',E) defined b y C(E',E) 3 A H-> TgA = UgAUJ_, G C(E',E). The representation is equivalent to the direct sum if this cocycle is a coboundary. 1.2.3. Nonlinear representations as successive extensions. Let ( S , E) be a nonlinear representation of G, S = 2 n = i ^ " ' ^ e S r o u P ^ a w g i v e s : s2

S ?gS,'= . =Sl° ^ OSl'+S S J . +2gSo(Sl,®Sl.) JO(5;,®5,M

(1.7)

i.e. i?j = 5 1 o S^_i is a 1-cocycle of G in C(®2E, E) for the representation T given by:

56

C{®2E, E)3A»TgA

SloAo = S\°Ao

(Sj_, ® SS1^. , ) .

(1.8)

Therefore (by A) on E, S is obtained by successive extensions of S 1 by its (symmetric) tensorial powers ®"S 1 , n > 2: first S 1 by S 1 ® 5 1 , then the result by ® 3 S 1 etc. Thus one has 6 : 1.2.4. THEOREM 1. If(S,E) is a nonlinear representation of a Lie group G in a Frechet space of differentiable vectors, and if the (differentiable) cohomology spaces Hx(G,C{®nE,E)) = 0 Vrc > 2, then S is linearizable. Scheme of proof: R„- i = Sj_t o S 2 is a 1-cocycle in ^ ( G , £ ( ® 2 £ , £ ) ) , therefore it is the coboundary of some B2 € C(®2E,E): 2 S29g=S]oB = 5 j 02-B S 2e{® - B 2Sl) o(®2Si)

(1.9)

and S is equivalent to a representation J M ( J - B22)~ ) _11S5g(I9 ( 7 - B 2 ) = S* + [terms of order > 3]

(1.10)

and so on, so that we get the linearizing operator n^=2(^ — Bk). Remark. A necessary condition for the 1-cohomology with values in an irreducible representation to be nonzero is that the center of the enveloping algebra be triv­ ially represented. In the general case it is sufficient to have at least one invertible operator in the representation of the center of the enveloping algebra for the corre­ sponding 1-cohomology to vanish. 2. NonLinear Klein-Gordon equation 2.1. The Equation Throughout this Section we shall mainly follow Ref.13, which solves the global Cauchy problem, establishes asymptotic completeness and contains a very thorough comparison with classical and more recent results. This work relies in part on earlier references11'12 and we include some more recent developments 14 . The nonlinear Klein-Gordon equation (NLKG) can be written as: (D + m 2 )
(2.1)

where m2 > 0, x £ K", n > 2 and P is analytic with no constant and no linear term (P(0) = 0 = dP(0)). We transform it into an evolution equation by setting f{t)(x) = (i, x), (tp(t))(x) = -§np(t, x), and defining ae(t) = tp(t) + ei6ifi(t), where e = ± 1 , 6 = (m 2 - A) 1 / 2 and F(a(t)) = P((t)):

±(a+(t)\=

d* U-(W

(a+(t)\

V-M*)/

(F{a(t))\

WOV

57 In t h e following we write ( a + ) as (/_|-,/_) and assume first, for simplicity of t h e exposition, t h a t P is polynomial and covariant under Poincare transformations. 2.2. Associated

NonLinear 2

Representation n

of the Poincare

Lie Algebra p

2

T h e space is E = L (R ) © L ( R " ) , where the differentiable vectors space will be Eoo = 5 ( R n ) © 5 ( E n ) . We denote by II the usual basis of t h e Lie algebra p a n d by II' t h a t of t h e enveloping algebra U(p) defined by the Poincare-Birkhoff-Witt theorem. T h e linear representation of p is defined by: I^(/+,/_) = «(/+,-/-); Tk;

= (x'di

~ xA)->

T

n.=dj

(l<j
k - (/+• / - ) = ( W * i / + > - « ' % / - ) ■

(2-3)

1

It exponentiates to a linear representation U of the Poincare group V on E ^ which extends t o a u n i t a r y representation (still denoted by U1) on 15. T h e nonlinear representation of p is then defined by T\ = Tx + Tx, X G p with Tjf £ £ ( ^ 0 0 ) a n d T x : £00 —* S « „ where

2>„(/) = (Hfinm

Tp<(f) = 0 = f*«,(/);

tMti(f) = (xjF(f),xjF(f)).

(2.4)

W i t h these notations the nonlinear Klein-Gordon equation takes the form

jtm 2.3. The

= (T1Po+fPo)f(t).

(2.5)

Spaces

We shall introduce the Banach spaces Ej(i £ N) which are completions of Ex for t h e norms \\f\\B. = ( S r e t f \\Tl(f)\\2E)^2. Then fx: E{ -» £ , is analytic a n d |y|<;

therefore ZocaZ solutions are obtained by Ref.5: let V be a neighbourhood of the identity in V; if n > 2 and N > No, there exists a neighbourhood 0 ^ of 0 in •EAT0 a n d a local analytic action U of V mapping V x (E^ n O^v ) into £7^. Global solutions will follow from the linearizability of the time translations which in t u r n will be a consequence of the existence of a solution to a related integral equation. 2.4. Yang-Feldman-Kalien

(YFK)

Equation

THEOREM 2 . 1 . We have linearizability 0/Cexp(tP 0 ) and asymptotic if and only if there exists A solution of YFK (2.7).

freedom

(2.6)

In other words, if there exists an operator A on a neighbourhood O0N of 0 in Efj0 with Em \\Uexp{m)Af

- Vtf\\E = 0,

Vt = e x p ( t T ^ )

(2.6)

58

(asymptotic freedom), which implies that Uexp(tp0)A = AVt (linearizability of the time translations £/exp(fP0))> then this A is solution of the YFK equation: A =I-

f°°

V_3TPoAVads

(2.7)

Jo

and conversely if there exists A solution of YFK, then one has linearizability and asymptotic freedom. The integral in (2.7) is taken in the sense of the improper strong Riemann integral in E. Proof (heuristic). From (2.6) one has h(t) = V-tUexp(tp0)Af — / —> 0 for (-> oo and therefore ffh(t) = V_tfPoUexpitPo)Af, hence h(t) = fiV-tfpaAV,fds + Af - f and the limit t —► oo is the Yang - Feldman - Kallen equation. Conversely, (2.7) gives AVt - VtA = Vt Si V_sfPoAV3ds. Therefore, if g{t) = AVtf, one has ftg{t) = {TPo + fPt)g(t) and g(0) = Af. But Uexp(tp0-)Af is solution of the same equation with the same initial conditions, thus Uexp{tPo)A = AVt. Also, AVtf - Vtf = Jt°° VtV.sfp0AV9fds -> 0. 2.5. Application We can write A = £ t > 1 A * , Ak € Ck(ENo,(l - A)E), N0 > 2 (symmetric continuous multilinear maps) with A1 = Idj5w . Then: /■OO

A2 = - / Jo

V_sT2Po(®2Vs)ds

(2.7 2 )

V_s(TPo+T2Po(I®A2+A2®I))(®3V3)ds

(2.7S)

TOO

A3 = ~ Jo etc.

The problem is to prove convergence of this process (existence of solution of YFK). For this we need to control various norms of Ak(®hVt) for t —► oo. One classically uses (denoting di = d/dxi): \\VtfU-

< |
as

t^oo

with c(/) = C(||/l|il +£ll*/IUi + £ | | W I U 0

(2-8)

Therefore, if n = 4, one can compute A For n = 3 and n = 2 one needs P"(0) = 0 (no quadratic term). But here using the covariance of NLKG one gets convergence for n > 2 (with any covariant polynomial P such that P(0) = 0 =
i£X,Yep, 2[x,y] = W

- T* (T^ 0 7 + 10 T | ) - ( T ^ - T£(T£ 0 7 + 7 0 I* )), (2.9)

59 i.e. T 2 G Z1(p,C2(Eco))

(cocycle). T 2 extends to U(p) with

Tl^TlT^+TldiU^U1)^

x,yeU(p).

(2.10)

For the Casimir y = PQ2 — J2 Pf = P^P^ we have i y = yx Vs. (2.10) then gives T 2 = TKTjR-1) - (TlR-^diU1 ® C/1),: because T^1 = T ^ p „ = m 2 and here 1 2 1 1 1 R- = ( m -d(£7' ® U )^is denned since for n = 2 , 3 . . . : n

\m2 - (p? + p ^ ) 2 + $ > j + P i ) 2 | > m 2 .

(2.11)

i=l

Thus if we take A2 = T^R'1 one has until second order TX(I + A2) = (I + A2)T] and one proves that A2 G C2(EN,(1 - A)E) for TV > TV0. Moreover if W = L°°(R n )ffl/ ^ ( R " ) and / , g G £ w then for < G R: \\A\Vtg ® Vtf)\\E < CgJ(l + | t | ) - / 2 p 2 ( V t f f ® V ( / ) | k < C 9 i / ( l + |f|)-" etc. (and similarly for derivatives). Therefore there exists an analytic function A : ON —* (1 — A)E, solution of YFK, where O^ is a neighbourhood of 0 in E^. This solves Eq.(2.5) for t G [0, oo[ with scattering data in O/v at t = oo. To establish the equivalence between the linear representation U1 and the nonlinear representation U we prove that A has an analytic inverse. The method used originally13 is briefly exposed in Section 3 for the case of Maxwell-Dirac. A slightly different but equivalent track is as follows. For 6 G OAT the func­ tion t — i » A(U* ,tp )U*(3-)6) is a solution of Eq.(2.5) when \s\ is sufficiently small, where g(s) = exp(siXi) ■ • • exp(s^JQ), Xi,...Xi G p. One proves that at s = 0, (d1 /(dsi ■ ■ ■ dse))A(U*xpitp0\U\ ^6) is a linear function of Y = X x • - • Xt considered as an element of U(p). We denote this function Y *-* ay(t): it satisfies an equation which is obtained by successive differentiations of Eq.(2.5) and can be solved ex­ actly like that equation. The crucial property proved for the linear map Y >-> ay (0) is that

\\A{9)\\EN
J2 IM0)|k),

(2.13)

Y£DN

where D^ is a finite subset of elements Y G K{p) of degree not exceeding TV and Fpf a polynomial. Since the right hand side is finite the map A : 0/v —* E^ is analytic and therefore has a local inverse at t = 0. 2.6, General Interactions and Statement of results In the above discussion we limited ourselves to covariant polynomial inter­ actions in order to get (2.10). In this case one not only gets existence of global

60 solutions by showing lineaxizability of (space-) time translations via ( t h e inverse of) some wave o p e r a t o r b u t one also gets 1 3 linearizability of t h e nonlinear represen­ tation. In t h e same p a p e r 1 3 it is however indicated how t h e result can b e extended to a general interaction with P analytic. T h e basic idea is t h a t a noncovariant in­ teraction ( a general element of t h e tensor algebra over p) can be "decomposed i n t o " a combination (with b o u n d e d noncovariant coefficients) of covariant interactions; for more details a n d a generalization t o other representations we refer t o Ref.15. T h e following existence a n d uniqueness result holds: THEOREM 2.2

(GLOBAL NONLINEAR REPRESENTATIONS, ANALYTIC LINEARIZ­

ABILITY, GLOBAL SOLUTIONS AND ASYMPTOTIC COMPLETENESS) There exists N0 > 0 and a neighbourhood Of] (a = — , 0 , + ) of 0 in E^0 (we write 0% = EN n O%o for N > N0 and b^ = E^n O%o): 1) U is an analytic

such

map V x O0N —» O0N and g — i > 17?-i&'g is "■ continuous

from V to the space of Banach

analytic

that map

maps 7 ^ ( 0 ^ , E V ) .

2) There exists a unique invertible analytic Ae : OeN —* O0N (E = ± ) mapping OeN onto O0N for N > No such that UgA£ = A£Ug forgeV (Poincare group) and

V J ™ , 'I * W i W ^ "

Ul

e*p(tPo)9 U= 0,

» € Ofra.

(2.14)

In particular there is a neighbourhood O of zero in S = <S(R") © <S(R n ) such that if ((^QJV'O) S O there exists a unique solution (vdV't) of NLKG in S for all t g R. Remarks, i) A similar result holds also (with t h e obvious modifications 1 3 ) when P is only C ° ° . ii) In t h e case of systems of N L K G with several nonzero masses, there is in general n o analogue of inequality (2.11). Such equations a n d t h e corresponding non-linear Lie algebra representations are formally linearizable since t h e cohomology is trivial 1 7 . Moreover t h e present m e t h o d s can b e a d a p t e d 1 2 t o include such cases. In fact t h e m e t h o d s shown there permit t o include quadratic interactions in physical 1 + 3 dimensions, t h a t h a d n o t been t r e a t e d before. All these a r e scalar field equations. Equations involving spinors are more difficult t o t r e a t , in particular due t o infrared divergencies. Nevertheless, as we shall see in t h e next section, t h e general framework presented here is powerful enough t o permit their t r e a t m e n t . 3. A s y m p t o t i c c o m p l e t e n e s s , global existence and t h e infrared p r o b l e m for t h e M a x w e l l - D i r a c e q u a t i o n s T h i s whole Section is based on a recent m o n o g r a p h 1 0 a n d especially o n its introduction of which this Section is a kind of "digest". Earlier references include Refs.8,9 a n d other references quoted in Ref.10.

61

3.1. Presentation of the problem 3.1.1. The physical framework It is well-known that the construction of the observables on the Fock space of QED (Quantum Electrodynamics) requires infrared corrections to eUminate the infrared divergencies in the perturbative expression of the quantum scattering oper­ ator. These corrections are introduced by hand with the purpose to give a posteriori a finite theory. In this Section we shall present rigorous results which we have ob­ tained concerning the infrared problem for the (classical) Maxwell-Dirac equations. Our belief is that such results can a priori be of interest for QED, especially for the infrared regime and combined with the deformation-quantization approach 1,2,3 . Our results show in particular that, also in the classical case one obtains infrared divergencies, if one requires free asymptotic fields as it is needed in QED. But before continuing the physical motivation of this study we shall describe the mathematical context. 3.1.2. The mathematical framework 3.1.2.a. The Equations. We use conventional notations: electron charge e = 1; Dirac matrices 7*1, 0 < fi < 3; metric tensor g1"1, g00 = 1, g" = — 1 for 1 < i < 3 and g^ = 0 for p. ^ u; 7^7" + f"^ = 2g<"'\ d^ = d/(dx"); D = d^d". The classical Maxwell-Dirac (M-D) equations read:

□A„ = xfif^, (if"dh + m)tp = ApYi>, d^A" = 0,

0 < (i < 3,

(3.1a)

m > 0,

(3.1b) (3.1c)

where if> = 4>+la-, 4>+ being the Hermitian conjugate of the Dirac spinor tp. We write equations (3.1a) and (3.1b) as an evolution equation: ^ ( ^ ( t ) , i 4 M ( 0 ) = (A M (t),AA M (t)) + ( 0 , ^ ) 7 ^ ( 0 ) ,

(3.2a)

^VW = W ) - ^ ( t h V ^ M ,

(3.2b)

where V = - £ * = 1 7 V dj + «7°m, A = "£*=! d 2 , t G R and where A„(i), A^iy.R3 -» R, V>(<):R3 -» C 4 . The Lorentz gauge condition (3.1c) takes on initial conditions A^(t0), A^(to), 0 < n < 3, and t/>(
A°(t 0 ) + ^ a i A i ( t 0 ) = 0, i=l

3

AA°(t 0 ) + |V'(
(3.3)

62

S.l.Z.b. Poincare covariance: linear part and representation spaces. Equations (3.1a), (3.1b) and (3.1c) are manifestly covariant under the action of the universal covering V0 = R4&SZ/(2, C) of the Poincare group. Therefore one can complete the time translation generator, formally defined by (3.2a) and (3.2b), to a nonlinear representation of the whole Lie algebra p = R46r5[(2, C) of VQ. We now introduce the topological vector spaces of the representations and first the Hilbert space Mp for - 1 / 2 < p < oo as the completion of S(R 3 , R 4 ) © S(R 3 , R 4 ) with respect to the norm II(/,/)IIM»

= (IIM<7I|2LW«) + IIIVr7lli.(R..R«))1/2,

1 2

2

3

(3.4a)

4

where |V| = ( - A ) / . Denote D = L (R ,C ) and E" = M" © D, the Hilbert space with norm l l ( / , / , « ) | | ^ = (|K/,/)|| 2 M, + ||a|| 2 D ) 1 / 2 . M°" is the closed subspace of ( / , / ) € Mp, f = (f0,fi),

such that

/o = E */«■• /o = - E i v r 2 ^/i, 1<1<3

(3.4b)

(3.4c)

1<'<3

A solution B^ of DB^ = 0, 0 < p, < 3, with initial conditions (/, / ) G Mp satisfies the gauge condition d^B" = 0 if and only if ( / , / ) € M°". We define Eop = Mop®D. Let now n = {P,,,M Q/? |0 < p < 3,0 < a < /J < 3} be a standard basis of the Poincare Lie algebra. There is a linear continuous representation Ul oiVg in Ep, with differentiable vectors Ej^, leaving E°p invariant. Its differential is the following linear representation T 1 of p in ££,, where we take 1 < i < j < 3, 1 < i < 3, Cij = l/27i7j € su(2), riij G so(3), and 1 < i < 3, aoi = 1/2707,, noi € so(3,1):

T1Po(f,f,a)

= (f,Af,Va),

(3.5a)

T/,i(/,/,a) = a!(/,/,a), x

(r*r (J (/./.«))(*) = -i A

(3.5b)

- *A)V, /.«)(*) + (««/.n«/."«*)(*) (3-5c) 3

(TkjfJ,

<*))(*) =

(xif(x),^2dj(xidjf(x)),xiVa(x)) j=o

+ (naif, n0if,cr0ia)(x),

(3.56)

We introduce the graded sequence of Hilbert spaces Ef, i > 0, E£ = Ep, where £&, C E? C Ef for 2 < j . Ep is the space of C'-vectors of the representation U1. Given an ordering X] < Xi < ■ ■ ■ < X^o on n , in the universal enveloping algebra

63

U(p) of p, the subset of all products X" 1 • • • X?J°, 0 < a „ 1 < i < 10, is a basis II' of U(p). Ef, i € N is the completion of E^ with respect to the norm

N k = ( E HTy(")fe)1/2'

(3-6a)

yen' \Y\
M°* = Mf n M°", Ef = Ef n f^', M ^ = M£, n M°" and El? = E^n E°", where Mf (resp. M£,) is the image of Ef (resp. E^) in Mp under the canonical projection of Ep on Mp. Let D; (resp. D^) be the image of Ef (resp. ££,) in D under the canonical projection of Ep on D. D^ = 5(R 3 ,C 4 ). M^ contains long-range potentials. S.1.2.C Nonlinear Poincare covariance. A nonlinear representation T of p in E^ is obtained by the fact that the M-D equations are covariant: TX=TX+T%,

Xep,

(3.7)

1

where T is given by (3.5a)-(3.5d) and, for u = (f,f,a) Tp0(u) = (0,a-ya,-iflll'y0ji'a),

7 = (70,71,72,7s),

T£. = 0 for 1 < i < 3, (T2M0j(u))(x) = x}(T2Pa{u)){x),

€ ££,,

Tl[.. = 0 l<j<3,

(3.8a)

(1 < i < j < 3),

(3.8b)

z = (x1;x2,x3).

(3.8c)

The gauge condition (3.1c) takes on initial data u = ( / , / , a) the form

/o = E */<>

A

h = E ^ - H2-

l
(3"lc')

1<'<3

The subspace V£, 1/2 < p < 1 of elements in E^ which satisfy these gauge conditions is diffeomorphic to E%g. The problem to integrate globally the nonlinear Lie algebra representation T therefore consists of proving the existence of an open neighbourhood W^ of zero in V^ and a group action U: VQ X U^ —> W^, which is C°° and such that Z79(0) = 0 for g G V0 and ^ ^exp(<X)(") l<=o= TXu,X e p. We extend T from p to to the enveloping algebra U(p) by defining inductively: Tj = 7, where I is the identity element in U(p), and TYX

= DTY.TX,

Y e U(p), Xep.

where (DA.B){f) is the Frechet derivative of A at the point / in the direction It maps Ef^ to Ef^. Moreover jTYW(u{t)) where

= TPoY(t)(u{t)),

YeU(p,)

(3.9) B(f).

(3.10)

64 Y(t) = e x p ( i a d P o ) F ,

a d P o Z = [P0, Z],

(3.11)

if jtu{t)

= TPo(u(t)).

(3.12)

We also define fy, Y g U(p) by TY = TY + fY and denote by T M 1 or T 1 M (resp. TD1 or T 1 D ) the linear representation of p which i s . t h e restriction of the linear representation T 1 to Mp (resp. D), etc. 3.1.3.

The Infrared

Problem

On the classical level the infrared problem consists of determining to which extent the long-range interaction created by the coupling A^j^ between the elec­ tromagnetic potential A^ and the current j ^ = tpf^ip is an obstruction for the separation, when |i| —> oo, of the nonlinear relativistic system into two asymptotic isolated relativistic systems, one for the electromagnetic potential A^ and one for the Dirac field ip. It will be proved here t h a t there is such an obstruction, which in particular implies t h a t asymptotic in and out states do not transform according to a linear representation of the Poincare group. This constitutes a serious problem for the second quantization of the asymptotic (in a n d out going) fields. Indeed the particle interpretation usually requires free relativistic fields, i.e. at least a linear representation of the Poincare group (Ul in our case). Therefore we have to introduce nonlinear representations f/' - ' a n d f/'+' of VQ, which can give the transformation of the asymptotic in a n d out states under V0 a n d permit a particle interpretation. T w o reasons permit to determine the class of asymptotic representations U'e\ e = ± . First, the classical observables, 4-current density, 4 - m o m e n t u m and 4angular m o m e n t u m are invariant under gauge transformations. Second, if the evo­ lution equations become linear after a gauge transformation one can use the freedom of gauge in the second quantization of the fields. It is therefore reasonable to pos­ t u l a t e t h a t the asymptotic representations g — i > Uf are linear modulo a nonlinear gauge transformation depending on g a n d respecting the Lorentz gauge condition. Moreover to determine the class of admissible modified wave o p e r a t o r s it is rea­ sonable to postulate t h a t solutions of the M-D equations (3.1a)-(3.1c) should con­ verge when t —> ± o o in Ep t o free solutions (i.e. solutions of equations (3.1a)-(3.1c) but with vanishing right-hand side) modulo a gauge transformation, not necessarily respecting the Lorentz gauge condition; such transformations are admissible since they leave invariant the observables. In m a t h e m a t i c a l terms the infrared problem of the M-D equations t h e n consists of determining diffeomorphisms f2£: O^ —> U^,, e = ± , the modified wave operators, where 0<x, is an open neighbourhood of zero in E^g a n d where U^ is a neighbourhood of zero in V&, satisfying tr<*> = f l - 1 o V, o a„

g£V0,e

= ±,

(3.13)

65

where the asymptotic representations are C°° functions U^:V0 x Eg -+ Eg. In order to satisfy the two preceding postulates, we impose supplementary conditions on U^ and n e , which we shall justify at the end. Denote the Fourier transformation / — i > / . The orthogonal projections Pe(—id) in D on initial data with energy sign E, e = ± , for the Dirac equation are: (Pe(-id)a)A(k)

= Pe{k)a{k) = - ( l + e ( _ £ y y f c i + m 7 < > ) w ( f c ) - i ) a ( f c ) i

(3.14)

3=1

where w(fc) = (m 2 + |Jt| 2 ) 1 / 2 . We postulate that the asymptotic representations have the following form: 17<+>(U) = (UWM(u), D

(U^ (u)r(k)

= £

t/<+»°(«)),

U^M

D

e^'-^Pe{k){Ul *)\k\

= U1M,

(3.15a)

g £ V0,

(3.15b)

e=±

where u = (/, / , a) £ Eg, the function (g, u, k) (-> ¥>,(u, -efc) from P 0 x £ ^ x R3 to R is C°° and if (hg(u))(t,x) = ipg(u,mx(t2 - | i | 2 ) ~ 1 / 2 ) , t > 0, \x\ < t, then nhg(u) = 0. Finally we impose the asymptotic condition

\\ue^(tPo)(n+(u))-u^tPo)(fj)\\MP - £ e>°l<+)^-^Pe(-zd)U£(tPo)a\\D e=±

+ \\Ue°p(tPo)(n+(u))

-> 0 (3.15c)

when t ^ oo, where u = (f,f,a) £ Woo, (u,i, fc) i-+ Se (u,i,fc) is a C°° function from WQO x R x R3 to R, s^ (u,t,—id) is the operator defined by inverse Fourier transform of the multiplication operator k — i > s^ (u,r, fc). It is possible to choose 4 + ) («,t,fc) = -i?(yl ( + ) («) ) (t,-£tfc/a;(/t))),

(3.16)

where A^+\u) is a certain approximate solution of the Maxwell-Dirac equations absorbing the long-range part of A for a solution (A,if>), and where d{H,y) = fL{y)Hll(z)dzi', y € R 4 , F : R 4 -» R4 and L(y) = {z £ R 4 |z = ay, 0 < s < 1}. The function ((, k) — i ► s^ (u,i,fc) was determined by the fact that Ss(u,t, k) = ew{k)t + Se (u,t, k) has to be in a certain sense an approximate solution of the Hamilton-Jacobi equation for a relativistic electron in an external electromagnetic potential:

( ^ S e ( M , *) + Ao(i, ~VfcS£(«,*, fc)))2 3

- ^

(h + A<(t, -V*5«(«,t, k)))2 = m2.

(3.17)

66

3.2. Presentation of the main results 3.2.1. Some more notations. Let u+ = (/,/,<*) € E^ and for t > 0, \x\ < t define Jl+\t,x)

(m/t)3(u>(q(t,x))/>m)S

=

^((P£(-ia)ar(i-g(t,x)))+707,((P£(-za)HA(^?(t,x))(3.18) £= ±

where g(t, x) = -mx/(t2 - | i | 2 ) 1 / 2 . For |z| > 0, 0 < t < \x\, we take j£ + ) (i,:r) = 0. Next we choose a suitable cut-off function x a n d introduce A^+' = A^+' + A^+> by (A<+»(*))(*) (A<+>2(*))(*)

= (B<+W(<))(«) 1 =x((i2-^l2)1/2)(^(+)2(i))(^)J

, te

*^'afl'

, (3 19a)

'

and {AM\t))(X) (A(+>2(i))(x)

= (£<+»(*))(*), \ =0, /

ior

*

(3'19b)

t € R,

(3.19c)

where 4 + > J ( i ) = cos(|V|t)/„ + IVI" 1 MMt)U, oo

/

iVI^andVK*-*))^)^,-)*,

i>0.

(3.19d)

l/(+) is defined by formulas (3.15a) and (3.15b) where the phase function tpg is (pg = 0 for g in SL(2,C) and ¥>,(u,-e*) =
(3.20a)

for ff = exp(a''P fl ), t e R 3 , where ,oo

dco{H,y)

5.2.5.

=

I

y»H,(sy)ds,

y GR\

and p(t,x) = (i 2 - l^t) 1 / 2 .

(3.20b)

Statements.

THEOREM 3.1. Let \ < p < 1. If N > 4 then [/<+»: P 0 x £^f -+ £^" ; 3 a continuous nonlinear representation of Vo in E°n9 and, in addition, the function [/<+>: V0 x El* -» ££? w C°°. AfoT-eouer £/<+> u n o ( equivalent by a C2 map to a linear representation on E^£.

67 THEOREM 3.2. Let \ < p < 1. There exist an open neighbourhood, Ux (resp. 0&) of zero in V£ (resp. E°J,), a diffeomorphism Q+'.O^ -> Ux and a, C°° function U:VQ X U^ —> Ux, defining a nonlinear representation ofVo, such that, foru = (fJ,a)eO(+):

a) ^ e x p ( t x ) O ) = Tx(Uexp(tx)(u)), 6) ft+ o U<+) c

) ^

X e p, teR,

u 6 ^ ,

=UgoU+

( l K W < M « ) ) - U^(tPo)(fJ)\\MP + \\Ue%itPo)(Sl+(u))

- £

e!3<+)(^-iS)Pe(-^)[/e^(tPo)Q||D)

= 0. (3.21)

This theorem solves in particular the Cauchy problemior small initial d a t a and proves asymptotic completeness. By the construction of the wave operator ft+ t h e solution (A(i, •), A(t, •), ij)(t, •)) = Uexp(tP 0 )( u ) °f the Cauchy problem satisfies su

PxGR3 ((H-kl+*) f "''l^|.(*. a : )l + !>0

(1 + |x| + <)|d„A„(i, x ) | + (1 + |z| + t)f |V>(t, x ) | ) < oo.

(3.22)

3.2.3. Cohomology. These results and conditions (3.13) and (3.15c) have a n a t u r a l cohomological interpretation. A necessary condition for U^+' and fi+ t o be a so­ lution of equation (3.13) is t h a t the formal power series development of Ug , Ug and fi+ in the initial conditions satisfy cohomological equations. In particular t h e second order terms Ug , U2 and fi^_ must satisfy 6D.2+ = C2,

6Ri+)2

= 0,

(3.23)

where 5 is the coboundary operator defined (as explained in Section 1) by t h e repre­ sentation ( Ug A2(®2Ug-x) of the Poincare group Vo on bilinear symmetric maps A 2 from E^ —> £ £ , and where C2 = i ? ' + ' 2 — R2 is the cocycle defined by R2g = LT 9 2 (^_j ® Uj_j) and E < + ) 2 = ^ ( + ) 2 ( ^ 1 _ 1 ® C/^-j), 0 £ Po- Equation (3.23) shows t h a t the cochain i v + ' 2 has to be a cocycle and then t h a t the cocycle C 2 has to be a coboundary. This is equivalent to the existence of a solution U^\ fi+ of equation (3.13) modulo terms of order at least three. There are similax equations for higher order terms: 8Sln+ = C"\ 6R(+)n = 0, (3.24) where C " a n d R^+)n

are functions of Q2+,...,

ttT1

and U^2,...,

L^+K

68

There exist a modified wave operator and global solutions of the M-D equa­ tions for a set of scattering data ( / , / , a ) , satisfying fp{k) = fn(k) = 0 for k in a neighbourhood of zero, i.e. /M and fp have no low frequencies component. It follows that the usual wave operator (i.e. !7 ( + ) = U1,*^ = 0 in (3.15c)) does not exist, even in this case where there are no low frequencies. In fact there is an obstruction to the existence of such a solution of equation (3.13) for n = 3 due to the self-coupling of ip with the electromagnetic potential created by the current ■^jptf). However there exists a modified wave operator satisfying (3.24) for n > 2, with [/'+' = U1. Moreover, as we have already pointed out, the phase function se is given by formula (3.16). Thus in the absence of low frequencies in the scattering data, for the elec­ tromagnetic potential, we can choose Q+ such that it intertwines the linear rep­ resentation Ul and the nonlinear representation U of Vo- Now if ( / , / ) £ M££, 1/2 < p < 1, have nontrivial low frequency part (as for the Coulomb potential (fii(k) ~ |fc| -2 ) and necessary to have asymptotic completeness) then there is a cohomological obstruction already for n = 2 if we want to obtain U{+)2 = 0. We show that the cocycle R2 can be split into a trivializable part — C 2 (a coboundary) and a nontrivializable part i i ' + ' 2 , which defines U^+'2 and therefore the whole representation £/'+'. We call this nontrivial part the infrared cocycle. 4 + ) 2 = 0 for g € SL(2,C), 4 + ) 2 = ( 4 + ) 2 M , 4 + ) 2 D ) , Ri+)2M = 0 and

(4+> 2i3 ( Ul ®u 2 )) A (fc) = 5 E

( < n - B i + U ( - ) - B\+)\-

+ #°°{Bl+)1(-)

- a),(u(k),

-ek))Ps(k)a2(k)

- B[+)\- - a), (w(fc), -e*))P«(k)a,(fc)),

(3.25)

for g = exp(a' i P /J ), u{ = (/;,/;,£*;) 6 E%, i G {1,2}, B^1 being the corresponding free field given by (3.19c). If for t -> oo, the limit tf(B<+>\ (t, -tek/u(k)) -» #°°(B< + >\(w(fc),-eJb)) exists, the infrared cocycle is a coboundary. In particular, this is the case when fin fit a r e equal to zero in a neighbourhood of zero. 3.2.4- Physical Remarks a). The asymptotic condition (3.21) in Theorem 3.2 is (when t —> oo) equiva­ lent to \\(A(t), A(t)) - (AL(t), AL{t))\\MP where (A{t),A(t)^(t))

+ U{t) - (e-'^L)(t)\\D

= Uexj>(tPo)(Q+(u)),

(AL(t),AL(t),^L(t))

-> 0,

(3.26)

= E& p ( t f | > ) u and

for a fixed u = ( / , / , « ) € O L , V ' s some function in C°°(R+ x R 3 ). Therefore (A, e~1{p^)~) converges to a free field (Ai,, */>£,).

69 T h e asymptotic representation {/'+' is not linear, it is equal to U1 modulo a gauge projective m a p Qg = Ug 17* ij i.e. this non-linear transformation of the scattering d a t a corresponds to a (Lorentz) gauge transformation ip —>
= -F"aF\

+ ±gK"FaeFae

+

h$Y{id»■-Av)4,

+ {{id» - A")V>)7*^), (3-27)

where 0 < / J < 3 , 0 < v < 3 and F^ = d^A^ — dyA^; and the current density vector jf = tfryPip are invariant under gauge transformations */>' = e'xtf>, A' = A^ — d^X (not necessarily respecting the Lorentz gauge condition t h a t gives here DA = 0). Therefore they go to their free limits when t —> oo. Condition (3.15c) is therefore natural. A similar discussion can be m a d e for the relativistic angular m o m e n t u m tensor. b). T h e same m e t h o d s can be used for nonabelian gauge theories (of the YangMills type) coupled with fermions. T h e aim here is to separate asymptotically the linear (modulo an infrared problem t h a t can be a lot worse in the nonabelian case) equation for the spinors from the pure Yang-Mills equation (the A^ p a r t ) . T h e next step would then be t o linearize analytically the pure Yang-Mills equation (that is known 7 t o be formally linearizable), and then to combine all this with the deformation-quantization approach to deal rigorously with the corresponding q u a n t u m field theories. c). T h e results presented here will thus give indications how a true q u a n t u m field theory (i.e. not based on perturbative theory) can be developed on the basis of this first quantized (classical) field theory (dealing in particular with the infrared problem a n d the definition of observables). T h e quantization should be based on the m a t h e m a t i c a l facts found here and not on a nonrigorous perturbation theory developed from t h e free field by canonical quantization or using some algebraic postulates which (however interesting they may seem) reflect sometimes a "wishful thinking". In other words the p a t h to follow should be based on " q u a n t u m defor­ mations" (in the sense of star p r o d u c t s 1 ' 2 ) of the "classical" theory presented here. In this context it is interesting to get existence theorems for large initial d a t a and to be able to localise specific solutions corresponding to large initial data, such as of the soliton or instanton type. A result in this direction has recently been obtained 4 . A c k n o w l e d g e m e n t . T w o of us ( M F , DS) would like to t h a n k R.I.M.S., where most of this p a p e r was written, and especially its director Huzihiro Araki, for superb hospitality. References 1. F . Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, Defor­ m a t i o n theory a n d quantization I: Deformation of symplectic structures. Ann. Phys. I l l (1978) 61-110.

70 2. F . Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz a n d D. Sternheimer, Defor­ m a t i o n theory a n d quantization II: Physical applications. Ann. Phys. Ill (1978) 111-151. 3. J. Dito, Star-products a n d n o n s t a n d a r d quantization for Klein-Gordon equa­ tion. J. Math. Phys. 3 3 (1992) 791-801. 4. M. E s t e b a n , V. Georgiev and E. Sere. Stationary solutions of t h e MaxwellDirac a n d Klein-Gordon-Dirac equations. I C T P preprint I C / 9 5 / 6 8 . 5. M. Flato, G. Pinczon and J.C.H. Simon, Non-linear representations of Lie groups. Ann. Sci. Ec. Norm. Super. 4 e serie, 1 0 (1977) 405-418. 6. M. F l a t o a n d J.C.H. Simon, Non-linear equations and covariance. Lett Math. Phys. 2 (1979) 155-160. 7. M. F l a t o and J.C.H. Simon, Yang-Mills equations are formally linearizable. Lett. Math. Phys. 3 (1979) 279-283. 8. M. F l a t o , J.C.H. Simon a n d E. Taflin, O n global solutions of the Maxwell-Dirac Equations. Commun. Math. Phys. 1 1 2 (1987) 21-49. 9. M. F l a t o , J.C.H. Simon a n d E. Taflin, T h e Maxwell-Dirac equations: Asymp­ totic completeness a n d infrared problem. Reviews in Math. Phys. 6(5a) (1994) 1071-1083. 10. M. F l a t o , J.C.H. Simon and E. Taflin, Asymptotic completeness, global ex­ istence a n d the infrared problem for Maxwell-Dirac equations. Monograph (308 pages, February 1995) to be published, available from xxx.lanl.gov as hep-th/9502061.Z by anonymous ftp. 11. J . C . H . Simon, A wave operator for a non-linear Klein-Gordon equation. Lett. Math. Phys. 7 (1983) 387-398. 12. J.C.H. Simon a n d E. Taflin, Wave operators and analytic solutions for systems of non-linear Klein-Gordon equations and non-linear Schrodinger equations. Commun. Math. Phys. 9 9 (1985) 541-562. 13. J.C.H. Simon and E. Taflin, T h e Cauchy problem for non-linear Klein-Gordon equations. Commun. Math. Phys. 152 (1993) 433-478. 14. J.C.H. Simon a n d E. Taflin, Initial d a t a for non-linear evolution equations and differentiable vectors of group representations. In: Modern Group Theoretical Methods in Physics, J . B e r t r a n d et al.(eds.), Mathematical Physics Studies 18 (Kluwer Academic Publishers, Dordrecht, 1995) 243-253. 15. J.C.H. Simon and E. Taflin, to be published. 16. D. Sternheimer, Recent developments in non linear representations a n d evolu­ tion equations. Lecture Notes in Physics 3 1 3 (Springer, Berlin, 1988), 65-73. 17. E. Taflin, Formal linearization of nonlinear massive representations of t h e con­ nected Poincare group. J. Math. Phys. 25 (1984), 765-771. A M S S u b j e c t Classifications: 35Q (primary), 81Q05, 81V10, 22E60, 22E65, 22E70.

71

REMARKS ON THE DOMAIN-DEPENDENCE OF CONVERGENCE RATE OF ITERATIONS IN A CERTAIN DOMAIN DECOMPOSITION METHOD —Analysis by the Steklov-Poincare operator—

Hiroshi FUJITA Department of Mathematics, Meiji University, Kawasakishi, Kanagawa-ken, 214 Japan

ABSTRACT In this paper, we study the convergence rate of iterative schemes of a certain DDM (domain decomposition method) by means of theory of self-adjoint oper­ ators and its fractional powers applied to relevant Steklov-Poincare operators. Focusing on the simplest boundary value problems of Poissbn equations in 2dimensional domains, and introducing some assumptions concerning the mutual relation of the involved component domains under linear or circular reflection with respect to their common boundary (i.e., the dividing curve), we derive clear estimates of the convergence speed and give even some optimal choice of the relaxation parameter.

1. Introduction Recently, in various works to solve problems of large scale for partial differen­ tial equations in spatial domains of complex geometry, particularly, in those of 3dimension, the so-called DDM (domain decomposition method) is actively used. As is well known, DDM is a kind of iterative algorithms where approximate solutions of the sub-problem in each of sub-domains of decomposition are iteratively pieced

72 together so that eventually a good approximate solution of the whole problem (the target problem) is obtained, as is exemplified specifically below. Actually, in practice many researchers are currently more interested in finding efficient preconditioners to be used to solve the whole discrete problem, starting from known solvers for the discrete sub-problem and somehow incorporating the discrete interface conditions across the dividing curves which separate adjacent sub-domains. In contrast to these flourishing works, the character of the present paper is rather fundamental and theoretical. Indeed, our purpose is to make an analytical study of the iteration procedure of DDM in the version of continuous variables. Namely, we consider the iteration algorithms of a typical DDM for partial differential equations themselves instead of their discrete analogues. In doing so, we restrict our attention to simplest target problems, namely, to the Dirichlet boundary value problem for the Poisson equation, and to very primitive decomposition of the original domain, while we aim at deriving some clear and rigorous information of the speed ( rate ) of convergence as the number k of repetitions of iteration increases, making certain assumptions on the relative relation of sub-domains. In other words, we are going to study the domain-dependence of the rate of convergence of DDM. In this paper, we restrict our analysis to the 2-dimensional case with the inten­ tion of presenting our basic idea clearly, although most part of our result here is valid also for the 3-dimensional case and the rest can be generalized there with some due modifications. Including the latter, various generalizations and technical details will be published in the forthcoming paper, Fujita-Kobari-Nagasaka 3 ', together with convincing numerical examples, and in some other papers to follow. As to the method of our analysis, we mention here that it is based on (1) the theory of self-adjoint operators and their fractinal powers applied to Steklov-Poincare oper­ ators arising from the interface conditions combined with (2) the relevant variational principle and with (3) 'linear' or 'circular reflections. This paper is composed of four sections. In §2, we formulate our problem, de­ scribing the target problem, the way of decomposition of the domain into two parts, and a certain iterative schemes DDM ( Dirichlet-Neumann type) to be studied. In §3, we present our main result in several theorems concerning the rate of convergence of the iteration, namely, we give estimates of the error in a suitable norm under some assumptions on the relative relations of component domains in terms of the reflections with respect to the dividing curves. §4 is devoted to outlines of proof of theorems. Let us note that the proof is operator-theoretical but is not very sophisticated due to a convenient choice of Hilbert spaces employed. Indeed, intorduction of the inner product defined through the i-power of one of the S-P operator is a crucial step in our proof. However, statements of the main theorems do not involve any operator-theoretical concepts at all.

2. Formulation of the Problem

73 2 . 1 . Target

Problem

Our original target problem is the following Dirichlet boundary value problem in a bounded domain ft £ R2 with piecewise smooth boundary dQ. = T; Find u such that f - A u = / in ft, ,,, l j \ u = /? on r . We assume that / € L 2 (ft),

/? €

Hl/2(T).

In this paper, the boundary values of functions defined in ft are mostly regarded as elements in the i / 1 / , 2 ( r ) . Hence, the assumption fi g Hll2{T) is essential, while we could assume / € H~l(D.) instead of / £ L 2 (ft) without any change of our argument. We shall write u = the exact solution. 2 . 2 . Decomposition

of the

Domain

We introduce simple partition ( = decomposition ) of ft by dividing ft into two parts by a piecewise smooth simple curve 7 connecting two points on T . Then we use the following notations;

r\ = 9 ft, \ 7 ,

r2 =dn2

\7.

(2)

n is the outer unit normal to the domain in consideration. v is the outer unit normal on 7 outgoing from ft, .

2 . 3 . Generation

of Approximating

Sequences

In this paper, we only consider approximate solutions generated by the follow­ ing iteration schemes, which might be called the Dirichlet-Neumann DDM iteration. To be specific, our approximating sequence of functions (the fc-th approximation ) u\ , u2 and A'*' respectively stand for the following approximation and are defined through the schemes below with an arbitrary initial guss A' 0 '; 1.

uj

is the k-th approximation of u in ftx .

2.

u2

is the A:-th approximation of ti in ft2 .

74 3.

A'*' is the Jfc-th approximation of A on 7 , A being the restriction of u to 7.

Starting with A*0', we generate u[h), u[h\ and A<*> (k = 0,1,2, • ■ •) as follows;

I

-Auf> ui*J

= =

/ /5

in Qi , onT,,

uf»

=

AW

(3)

on 7 .

[ -Au[k) = f infi2, u[k) = p on T2 , i (<••) (*) (

-£r

on

= —ir

A<*+i) = (i -e)\W

+

(4) i ■

6u(k).

(5)

Here 8 is the relaxation parameter in 0 < 8 < 1. Since we are concerned only with the rate of convergence of the iteration, it suffices to estimate the error, particularly, the error on 7 . Thus we put

£(*) = \W _\

fx = u] 7 ).

(6)

and intend to estimate ||£'*'|| with some suitable norm. Let us recall here that it is known under general circumstances that if 8 is sufficiently small, then the iteration converges with the speed of (1 — ff) . In contrast, we are concerned with a more definite choice of 9 and with the resulting definite speed of convergence, which can be derived under some particular relations between Oi and f!2 . In other words, we want to find out conditions on the way of decomposition which enable us to make such a choice of 8.

3. Main Result 3 . 1 . Cases with a Linear Segment as 7 For the time being, we suppose that the dividing curve 7 is a linear (straight) segment. Firstly, we introduce C o n d . (/). Let 7 be a (straight) segment, and let 0 2 be the image of H 2 by reflection with respect to 7 ( or the extension of 7 ). We say that Cond. (7) is satisfied if ft2 C ftj .

(7)

3 . 1. 1. Results under Cond. (I) Our first theorem is a special case of Theorem 3 below, 9 being fixed as 9 = -. Nevertheless, it is separately stated in advance to exemplify the nature of our result.

75 T h e o r e m 1. Let 0 = | , and suppose that Cond. (I) is satisfied. Then there exists a positive constant cx = ^($7,7) such that

MW\\HuHl) < c ^ H ^ ' I U ^ , ,

(fc = l , 2 , - )

(8)

holds true. ! The following theorem is again a corollary of Theorem 3; however, it deserves a separate writing since it refers to the optimal speed of convergence, which is brought about by the optimal ( = the largest) choice of 0 in the sense that it is the largest among all possible values of 0 for which the convergence with exponential ( power ) rate is guaranteed for general circumstances under Cond. (7). T h e o r e m 2. Under Cond. (I), the optimal choice of § is 0 — | . Then with the same constant Ci as above,

lieWll^W
(fc = l,2,-..)

holds true. Our principal result under Cond. (I) now reads; T h e o r e m 3. Define r = f(6) (0 < 0 < 1) as follows.

. _ f 1-0, 1 20-1,

r _

for 0 < 6 < f for | < 0 < 1

(9) I

(10)

Then, under Cond. (I) we have the following estimates for 0 < 0 < 1 ;

H^lltf^w < <*f*IK<°>||tfi/»Wl

(* = i , 2 , - ) -

(ii)

A few remarks are in order. Remark 1. From the theorems above we can deduce the corresponding rate of con­ vergence for approximate solutions in Ox and H2 . For instance, from Eq. 9 follows Corollary. Under Cond. (I), and with 0 = 1, ||i4 — u||#i( a i ) and ||t4 — «|j(jjj) converge with the speed of ^ . Remark 2. As confirmed by numerical experiments to be reported in a forthcoming paper by the author and his collaborators A. Kobari and Y. Nagasaka 3 ', we expect to observe the corresponding speed of convergence of iterations under Cond. (I) when we carry out numerical experiments (e.g., by the finite difference method or the finite element method) with a fixed "mesh size". As a matter of fact, for the discretized problem, which is of finite-dimensionality, the sophisticated / / f - a o r m is equivalent with familiar norms, say, sup-norm or L 2 -norm. Keeping the assumption that the dividing curve 7 is a linear segment, we now introduce a modification of Cond. (I), which involves reflection with respect to 7 as well as contraction along the direction vertical to 7 . Without loss of generality, we

76 suppose that 7 is a segment on the i/-axis. Now we state Cond. (Im) where m is a positive constant larger than 1. C o n d . (7 m ), Let 7 be a linear segment on the y-axis, and let m > 1. By Tm we denote the contraction along the x-axis defined by

Tm:(s,y)-»(-,y). m

(12)

r „ , n 2 means the image of fi2 by Tm. And, let (Tmn2)' be the image of Tm H 2 by reflection with respect to the y-axis. Then, we say that Cond. (Im) is satisfied if

(Tmn2y c n , .

(13)

3 . 1. 2 . Results under Cond. (Im) T h e o r e m 4 . Suppose that Cond. (Im) is satisfied. For 0 < 0 < 1 and 1 < m, we define f = f(6,m) by

„_ f 1 - 0, ~ 1 (m + \)6- 1,

r

if m < f - 2, if m > | - 2 .

(14)

Furthermore, assume that m < I — 1. Then 0 < f < 1 and UW\\^h)
(i = l,2,--)

(15)

hold true. If we fix m and chose an optimal value of 0, then we have T h e o r e m 5. Under Cond. (Im), and with (the optimal choice of) 0 = ^ r ^ , W^WH'^)

< cA^)kMi0)\\HU^

(* = 1,2,.--)

i

(16)

hold true. Conversely, for a fixed value of 0, one might be interested in finding out the range of m for which convergence with the exponential ( power) speed is guaranteed. For convenience to compare the result with Theorem 1, we only state the result with

e = \T h e o r e m 6. Fix 0 = | , and suppose that Cond. (Im) is satisfied for 1 < m. Then r above is expressed as TTl — 1

l<m<2=>f=l-0,

2<m=>f =

.

(17)

Hence, the exponential convergence is guaranteed for 1 < m < 3. I Remark 3. When we deal with an individual case with 0 = | , it can happen that the exponential convergence takes place, notwithstanding Cond. (Im) is not satisfied

77 for any m < 3. Seemingly, this is due to the stabilizing effect of the Dirichlet boundary condition on Y\ and r 2 ■ In fact, by numerical experiments we can confirm that when we fix the width in the x-direction and expand the domains in the indirection, the convergence is eventually lost unless Cond. (Im) is satisfied with some m < 3. 3 .2. Cases with a Circular Arc as 7 We now suppose that the dividing curve 7 is a circular arc. To fix the idea, we assume that 7 is a part of a circle with radius R and with its center at the origin of the coordinate plane, and that H2 is adjacent to 7 from the outside. Then we introduce Cond. (R). Let 7 be a circular arc as above, and let Of be the image of £12 by circular reflection with respect to this circle. Then we say that Cond. (R) is satisfied if ilfC flj . (18)

3. 2. 1. Results under Cond. (R) T h e o r e m 7. Under Cond. (R), the conclusions of Theorem 1 and Theorem 2 hold true. 1 We can derive some analogues of Theorems 4 - 6 , after introducing an analogue of Cond. (Im). Namely, we state Cond.(.ft™). Let 7 be the same circular arc as above and let m > 1. By T£ we denote the following contraction in the radial direction defined, by using polar coordinates, as T£:(r,6)-,(R+r-^,B). (19) m T£Q2 means the image of H 2 by T^. Moreover, let (T£0.2)' be the image of TJJ H 2 by circular reflection with respect to 7 (or the whole circle). Then, we say that Cond^-ffm) is satisfied if

(r*n2)' c n , . 3. 2. 2. Results under Cond.

(20)

{Rm)

T h e o r e m 8. Theorems 4 —6 are true if Cond. (Im) is replaced by Cond. (Rm) there. I To conclude this section we give a remark regarding an immediate generalization of the result. Namely, Remark I. Generalization of Theorems 1 - 3 (Theorems 7 ) to the case where 7 is composed of a number of separate linear segments (circular arcs) is possible, if Cond. (7) (Cond. (R)) is duly modified.

4. Proof of Theorems

78 In this section we describe outlines of proof of theorems stated in the preceding section. 4 . 1 . Preliminaries

concerning S-P

operators

We begin with a preliminary consideration of the Steklov-Poincare operator S = S(fl,i) pertaining to a single bounded domain D. in R2 and a part 7 of <90, which is assumed to be piecewise smooth. We put r = dfi \ 7. Before making rigorous definition, let us introduce a formal version of definition ( i. e. restricted expression of ) S which we denote by So = So(Cl,"/). For any (. G C^il), we are going to define 5 0 f in terms of the following harmonic function h = h^ ; ( Ah = I h = [ h =

0 0 (

in 0 , onT, on 7.

(21)

Actually, the harmonic function h = h^ is uniquely determined, and if we put dh c , Sot, = - 5 On

on

7,

then we get a linear operator from CQx(f) into X = L2(*]), which is our basic Hilbert space. The inner product and the norm in X will be denoted by (•, -)L2 and || • H^ or simply by (■, •) and || • ||. In the Hilbert space X, we define a quadratic form Jo in X with its domain V(J0) = Cg°{ 7 ) by setting Mt] = [Sot,£)x

=l^hd~t

= \\Vh\\h{ny

(22)

We want to apply the theory of positive quadratic forms developed by T. Kato 4 ' in order to deal with S from the view point of the operator theory. To this end, we note that there exist positive constants c' and c" depending on fi and 7 such that c'll
(23)

hold true for the harmonic functions h defined above by Eq. 21 and their boundary values. This is a consequence of the trace theorem and the elliptic estimates. Actually, Eq. 23 is extended to the case of weak harmonic functions h £ H1(Q.) satisfying the boundary conditions in the trace sense. In view of Eq. 23, it is easy to check Lemmas 1, 2 below. L e m m a 1. J 0 is a positive quadratic form and is closable (Kato's sense). Its closure J has V = Hhl) as its domain T>(J).

= closure of C ^ h ) under || • \\H^b)

(24) |

79 L e m m a 2. ( Kato-Friedrichs theorem) There exists a (unique) self-adjoint extension S of S0 such that S is a positive self-adjoint operator in X with its domain V(S?) = V,

(25)

and the identity \\SlH\\x = \\Vh\\Lm)

(UV),

(26)

holds true, where h 6 if 1 (f2) is the weakly harmonic function in 0 , which satisfies the boundary conditions h = 0 on T and h = f on 7 in the trace sense. i! Definition of S-P o p e r a t o r The operator S = 5 ( f i , 7 ) in Lemma 2 is called the Steklov-Poincare operator (in the rigorous sense) for fi with 7 as the chargeable portion of the the boundary. Remark 5. The word 'chargeable portion' is used in association the capacity of f2 . Remark 6. Henceforth, saying the Steklov-Poincare operator or we mean the operator the S = 5(fl, 7 )-operator. Often, the chargeable portion 7 are understood without specification. Remark 7 By virtue of Eq. 26 combined with Eq. 23, | | S J £ | | I I M I H ^ M a s t n e n o r m s °f Hilbert space V = H^Kf). bounded and compact in X, and so is 5 _ 1 .

with the notion of simply 5-operator, domain fi and the is equivalent with

Hence follows also that S~ 2 is

Furthermore, we have L e m m a 3 . If two bounded domains f^ and fi 2 share the chargeable portion of the boundary, then 5j2 S2 2 is a bounded operator in X, while Sj 2 S | admits of a bounded extension. Proof. T h e first half of the conclusion is obvious in view of Remark 7. To obtain the _1

1

1

_1

second half, just note that the adjoint operator of 5 j 2 5 2 2 in X is S2 Sr 2 . Q.E.D. For our argument below, the following variational principle plays a crucial role. L e m m a 4. For each £ £ V, set Ui = {4>e Hl(Q.)\4> = 0 on T, = ( on f}.

(27)

Then it holds that ||5^||2 = mm{||V^||2;^et/a.

(28)

Proof. Obvious from the Dirichlet principle for harmonic function subject to the imposed boundary condition. Q.E.D. As a corollary of the preceding lemma, we have immediately L e m m a 5. Assume that two bounded domains Hj and fl2 share the chargeable portion 7 of the boundary and that J72 C fl1 . Then,

H^fH < l|Sf£H (Vfev). holds true.

(29) I

80 Remark 8. We shall not go into details, but it can be shown that under a certain assumption S is a pseud-differential operator of order 1. 4 . 2 . Recursive Expressions

of the Error

We return to the Dirichlet-Neumann iterative schemes described in §2. We want to derive an estimate of the norm of the error- generating operator below, which is expressed in terms of the Steklov-Poincare operators S\ = S( Oj , 7 ) and i>2 = S(fi-2, 7 )• To this end, wefirstlyp u t o\k) = u[k)-u

in n , ,v(k)

= u(k)-u

in 0 2 ,

(30)

and £(><) = \W = u[k) - u

on 7.

(31)

Then we see f At^1

= 0

v[k] = 0

1

in H, , v[k) = f «

on r , ,

on 7,

and note

Similarly, from ( A^'

= 0

in H 2 ,

(

= 0

on T 2 ,

vi

-T£-

=

f—

on 7,

we notice that on 7

S2V^

=-

^=

-S^\

whence follows v2 | 7 = —5 2 1 Si^*^. Consequently, we have

^k+i) = (i-e)^k)-es^s^w

(k = 0,1,2,-■■).

(32)

We state the following lemma which is a little formal at this stage but will be understood in a rigorous way soon. L e m m a 6. Putting A, = (l-6)I-6St1S1, (33) we have (formally) £W = A*fW, 4 . 3 . Self-adjointness

(fc = 0 , l , 2 , - - ) .

(34)

of Ag in V

First of all, we verify that Ag can be re-defined as a bounded operator in Hilbert space V. Namely, the expression Eq. 33 must be understood accordingly. Incidentally

81

as a general notation, let us write £(X, Y) to denote the set of all bounded linear operators from a Hilbert space X to another Hilbert space Y. Moreover, C(X) stands {OTC(X,X).

Obviously, in order to verity that Ag £ C(V), it suffices to show that S^Si £ C(V). To be exact, we are going to show that Sj*Si admits of a unique bounded extension in V. In fact, putting B = ST* • (S?S;*)* ■ Sf,

(35)

we note that

52_l g c{x,v), sfsp £ c(x), (sls^y £ c(x), si e c(v,x), where X is the basic Hilbert space L2(f), and * means the adjoint operator in X. Therefore, B £ C(V). On the other hand, if (, £ D(Si), then ££ = S^S^

(36)

holds good, because we have for any $ £ X

(Bi,4) = (sfHS!spysk,4>) = (s?ushf*)s;h$) = (sh>sls;1) = (s1(,s;1) = (s;1s1(,lj>).

(37)

Since T)(Si) is dense in V, B £ C(V) is the unique bounded extension £ C(V) of Sj Si- Henceforth, when we have to argue rigorously, we use Ag £ C(V) thus re-defined in use of B. Namely, Ae = (\- 8)1 - BB.

(38)

Furthermore, as a new inner product in V = H0 (7), we adopt ((,v)v = (sk,slt,)

(Vf.ijeV).

(39)

Clearly, V is a Hilbert space with this inner product and the corresponding norm is equivalent to the usual //2(7)-norm. Then we claim Lemma 7. The operator Ag is self-adjoint in V under the inner product defined above. Proof. It suffices to verify that Ag is symmetric, namely,

(M,v)v = (t,A,T,)v (Vf.i/ev), which is in turn implied by (B(,r,)v = (t,Br,)v, (Vf.ijeV).

(40)

82 Eq. 40 is seen from

(Btv)v = (shcslr,) = ((slspysH,sln) = (sk.sh^slr,)

= (sk.sh)-

Q.E.D. Since Ag is an self-adjoint operator G C(V), we have only to examine its numerical range, that is, it suffices to examine the quadratic form (Ag(,()v in order to estimate the operator norm \\Ag\\ = \\Ag\\r(v^. In this connection, by virtue of the calculation in the preceding proof, we have L e m m a 8. It holds that

(M,Ov = (i-e)\\sk\\2-8\\sk\\\

((el/).

(41) I

4 .4. Proof of Theorems under

Cond.(I)

Here we assume that 7 is a linear segment. L e m m a 9. If 7 is a segment and 0 2 is the image of fij by reflection with respect to 7, then

Si = St. Proof. Let 7 be a segment on y-axis. Suppose that fii is adjacent to 7 from the side x < 0. For an arbitrary £ 6 V, let hi = hi(x, y) be the harmonic function defined in fit with hi = £ on 7 and hi = 0 on Ti so that Si( = —— on 7 . If we introduce ox h* by setting h'(x,y) = hi(-x,y), then h* is harmonic in H 2 and plays the same role to define 82^ as hi played for Si£. Thus it is clear that ^

- ~ dx ~ dx -

SlC

Q.E.D. L e m m a 10. Under Cond.(Z), we have \\SU\\ < \\sk\\,

(V(eV),

(42)

and

[l-20)U\\v<{M,Ov<(l-9)U\\v,

(V£6V).

(43)

Proof Eq. 42 is obtained immediately, if Lemma 5 and the preceding lemma are combined. Eq. 43 follows from Eq. 41 in virtue of \\sk\\

= U\\v

and 0 < \\sk\\

<

U\\v.

83 4. 4 . 1. Proof of theorems under

Cond.(I)

In view of the self-adjointness of A^and Eq. 43, we have ||A«||y<max{|l-20|,l-0}

(44)

for 0 < 8 < 1. Hence we get Theorem 3. Evidently, Theorems 1, 2 are immediate corollaries of Theorem 3. 4 . 5 . Proof of Theorems under

Cond.(Im)

L e m m a 1 1 . Suppose that 7 is a segment on the j/-axis. Let fi be the image of fi2 by the map Tm. And let S be the S-P operator pertaining to 0 . Then we have \\Sk\\2 < m\\sk\\\

(V(£V).

(45)

Proof. Let h be the harmonic function in fi2 with h = £ on 7 and h = 0 on T 2 . Therefore ||5 2 2 ^|| = | | V / J | | L 2 ( Q ,. Then we define h infiby h(x,y)

=

h{mx,y).

h is no longer harmonic but belongs to H1(fl) and is subject to h = £ on 7 and h = 0 on the rest of the boundary of f!. It is an easy calculation to derive

HVA|&(fl) = -||g||i J ( Q 2 ) + ^ l l f lll^) <

m||VA||k(na) = mllS^H2,

(«) (47)

whence follows the inequality 45 with resort to the variational principle, Lemma 4, applied to fi. Q.E.D. If Cond.(7 m ) is satisfied, we obtain by Lemmas 6 and Lemma 11 \\sh\\2<m\\sk\\\

(VfeV).

(48)

Making use of Eq. 48 instead of Eq. 42, we apply the same argument as for the case of Cond.(7), we can prove theorems under C o n d . ( / m ) . 4 .6. Proof of Theorems under Cond.(R)

or

Cond.(Rm)

We just make use of the following lemma in place of Lemma 9 and apply the previous argument to obtain the corresponding theorems under Cond.(/?). L e m m a 12. Let 7 be a circular arc and let f!2 be the image of Hi of the circular reflection with respect to 7. Then we have Si = S2.

(49)

Proof. Suppose that 7 is an arc of a circle with radius R and with center at the origin. We assume that fij is adjacent to 7 from the inside of the circle. Now, let

84 £ be an arbitrary element in V and let ft =ft(r,9) be the harmonic function defined in Q.x , being expressed in the polar coordinates, which is subject to the boundary conditions ft = ( on 7 and ft = 0 on I \ . Thus Sif = §*■ on 7 . Then we defineft*in 0 2 by ft'(r,0) = A( — , 0 ) . (50) r an We can see easily that ft* is harmonic in il2 d satisfies ft* = £ on 7 and ft* = 0 on F 2 . On the other hand, it holds that Sol = --^— = TT on 7 . Consequently, we dr or have the lemma. Q.E.D. In order to derive theorems under Cond.(i? m ), we just apply the following lemma, which is analougous in nature as well as in the arguments for proof with Lemma 11. L e m m a 13. Suppose that 7 is an circular arc as stated in the definition of Cond.(./?„,). Let 0 be the image of Q2 by the map T£. And let S be the S-P operator pertaining to fl. Then we have \\Sk\\2<m\\sk\\\

(Vt-eV).

(51)

References [1] H. Fujita: Remarks on the domain-dependence of convergence rates of a certain DDM, Invited Lecture at Intern. Symp. on Parallel Algorithm for Science and Engineering Computations, Chiba University, May 22, 1995 [2] H. Fujita: On the domain-dependence of convergence speed of iterations in a certain domain decomposition method, Invited Lecture a t Fourth Colloq. on Numer.Anal., Plovdiv, Bulgaria, August 13-17, 1995. [3] H. Fujita, A. Kobari and Y. Nagasaka: An operator- theoretical and numerical study of the domain-dependence of convergence speed of some domain decompo­ sition methods, to appear. [4] T. Kato: Perturbation 1976.

Theory for Linear Operators, 2nd ed., Springer-Verlag,

[5] A. Quarteroni and A. Valli: Theory and application of Steklov-Poincar'e op­ erator for boundary value problems: The heterogenous operator case, Proc. of Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM 1990, pp.58-81. [6] K. Yosida: Functional

Analysis, 6th ed., Springer, 1980.

85 A n a l y s i s of G i n z b u r g - L a n d a u M o d e l for S u p e r c o n d u c t i v i t y

Department

J i a n g Lishang of Mathematics, Suzhou University, Suzhou, 215006, China

Suzhou,

and

Department

Yu W a n g h u i of Mathematics, Suzhou University, Suzhou, 215006, China

Suzhou,

Abstract The main purpose of this paper is to introduce some recent works of the research group of partial differential equations in Suzhou University on the mathematical anal­ ysis of the Ginzburg-Landau model for superconductivity. These works are on the free boundary problems in type I superconductors, models of layered superconductors and the vortex pinning problems respectively.

1. I n t r o d u c t i o n In 1911 when he discovered the sudden disappearance of the current resistance in Mercury at the low temperature about 4.2 K([l]), Onnes created a new field of science—the field of superconductivity. About forty years later Ginzburg and Lan­ dau established a macroscopic model of superconductivity (GL model) in 1950 ([2]), based on phenomena observed in experiments. The cause of superconductivity had not been found until in 1957 when Bardeen, Cooper and Schrieffer set up the micro­ scopic BCS theory of superconductivity ([3]). Two years later Gor'kov pointed out that the macroscopic GL model can be deduced from the microscopic BCS theory under some appropriate limit process ([4]), which laid the GL model on a firm basis. In 1968 Gor'kov and Eliashberg extended the GL model to the time-dependent case, also based on the BCS theory ([5]). Following the recent discovery of high critical temperature superconductors, the study of the GL model becomes a considerable in­ terests for researches on both physics and mathematics and the mathematical analysis of the model has been made a remarkable progress. The main purpose of this paper is to introduce some recent works of our group in Suzhou University on the mathematical analysis of the Ginzburg-Landau model. In the meantime, we shall illustrate some elementary concepts of the GL model and related problems.

86

2

Ginzburg-Landau Models

2.1

The stationary GL model

Let an open bounded domain ft C M3 be the region occupied by superconducting materials. According to the Ginzburg-Landau theory, the Gibbs free energy of the materials is given by (non-dimensionalized):

I [\(W2-l)2 + \(--V-A)\2 + \cur\A-H\2}dn, (2.1) K Jn z where A is the magnetic potential, curlA is the magnetic field, H is the applied magnetic field; ip is the complex Ginzburg-Landau order parameter, \tp\2 is the density of superconducting charges, 0 < |i/>| < 1; K is the Ginzburg-Landau parameter which depends only on the material; and i = yf—1. In steady-state case, tji and A must minimize the Gibbs free energy (2.1), that is, =

E(ip,A)=

min

E(
(2.2)

Berf(f!) where ft^ft)

= {ip : ft -> C | Im^, Re

ff^ft)

= { 5 : ft -> iR3 | B E (fi^ft)) 3 }.

and

The Euler equations of (2.1)—(2.2) is given by the Ginzburg-Landau equations:

( - |,v - A ) V j = curlcurM

- ^ + I^IV = o,

= -^(*V - VW-*) - \i>\2A + curlA,

in ft,

(2.3)

in ft,

(2.4)

and the natural boundary conditions: ( - V + A)4>-n = 0 (curM - if) x n = 0

on 9ft, on 9ft

(2.5) (2.6)

where j is the density of superconducting current and )/>* is the complex conjugate function of ip, and n is the unit normal vector of ft. (cf. [6], [71).

87

2.2

The non stationary Ginzburg-Landau model

Based on the BCS theory, Gor'kov and Eliashberg extended the Ginzberg-Landau equations (2.3) and (2.4) to time-dependent case whose non-dimensionalized form can be written as follows: ft i

r]~ + iriK$il> + ( - - V - A)9i/> -if>+ |V>|2V> = 0 inQx(0,r),

(2.7)

j = curlcurL4 dA i = - — - V# - — (* V - ipViP') - \i/>\2A + curLff, inflx(OJ),

(2.8)

where n is a positive constant and $ is the electric potential (cf. [5], [7]). More general conditions than (2.5) and (2.6) can be posed on the boundary: ( - V + A)4> ■ n = -i-/iP (curlA - H) x n = 0

on 5 0 x (0, T), on dfl x (0, T),

(2.9) (2.10)

where 7 is a known positive function. 7 = 0 corresponds to a superconductor-insulator (or vacuum) interface while 7 / 0 to a superconductor-normal metal interface. (2.9) is determined by requiring that the normal component of the current be continuous at the boundary (cf. [6]). 2.3

Gauge transformation

An important feature of the Gibbs energy (2.1) is gauge invariance, that is, (2.1) is invariant under gauge transformation. Let x De an arbitrary smooth function. The gauge transformation of (2.1) is defined by Gx : [4>,A) —> (C,S), C = i>exp(iK,x) and Q = A + Vx- Of course, the system (2.3)-(2.6) is also invariant under the gauge transformation. For the non stationary Ginzburg-Landau system (2.7)-(2.10), the gauge transfor­ mations are defined in a similar way: Gx : (4>,A, ((,Q,'P), C = */ , e x P(^x)i Q = A + Vx and & = ,A) and ((,Q), or (ip,A,
88 (1) Coulomb gauge For Coulomb gauge transformation, the function x is selected so t h a t the magnetic potential A satisfies

{f divA div/1 = 0

in i n fni xx (0,T), (0,T),

An = = 00 )\ A-n

on aann x (0,T). (0,5").

Consequently, (2.9) splits into

- ^ + Kflp K7 = = 0, O, — on an

A-n A-n = 0o

on aan n xx (0,T), (o,r),

and (2.4) and (2.8) become an elliptic equation and a parabolic equation with respect to A respectively. Moreover, the electric potential $ also solves an elliptic equation (see 2.4). (2) Lorentz gauge. For Lorentz gauge transformation, \ is such a function that the magnetic potential A and the electric potential $ satisfy |[ d\vA (0,T), dWA +

1y A-n A ■ n=-0 o

ononan. an.

Under the Lorentz gauge, (2.7) and (2.8) are transformed into a system of two parabolic equations with respect to A and ip (cf. [6]). 3 3.1

T y p e I s u p e r c o n d u c t o r s and t y p e II s u p e r c o n d u c t o r s The distinction

between type-I and type-II

superconductors

Superconductors can be divided into two classes—type I superconductors and type II superconductors according to whether the mixed state can exist in the superconductors. The mixed state corresponds to 0 < \ip\ < 1, while the idea superconducting state and the normal conducting state correspond to \tp\ = 1 and ip = 0 respectively. If there exist only the idea superconducting state and the normal conducting state in a superconductor, it is a type I superconductor, otherwise, it belongs to type II superconductors. This classification can be decided by the Ginzburg-Landau parameter K which is a material constant: the superconductor belongs to type I if its K less than 1/V2 and to type II if K greater than l / i / 2 . In a type I superconductor, there is a critical magnetic field Hc which also depends on temperature: when the applied magnetic field H is less that Hc, the superconductor exhibits the idea superconducting state (\tp\ = 0), while it exhibits the normal conducting state (|V>| = 0) when H greater than Hc. In a type II superconductor, there exist a lower critical magnetic field HCl and a upper critical magnetic field HC2, HCl = BCi(tt} = ^ ( l o g / c -f u) (v

89 is an absolute constant), HC2 = HC2(K) = y/2HcK and i/ C] < HC2. The type II super­ conductor exhibits the idea superconducting state, or the normal conducting state, or the mixed state when the applied field H is less than HCl, or greater than HC2, or between HCl and HC2, respectively (see [7] and the references therein). A standing feature of type I superconductors is that there is a sharp interface between the idea superconducting state and the normal conducting state, which leads to a free boundary problem (see 3.2). Type II superconductors are characterized by the existence of vortices of super­ conducting currents. The centers of the vortices are points of the normal conducting state (i.e., zeros of ip) around which are the mixed state and superconducting currents. Movement of the vortices will produce resistance to superconducting currents and, consequently, causes loss of superconductivity. Hence it is of significance to study the behavior of vortices in type II superconductors. In 5 we shall discuss vortex-pinning problems (i.e. pin the vortices at fixed points). 3.2

The free boundary problems in type I

superconductors

As we have stated in 3.1, there is a sharp interface between the idea supercon­ ducting state and the normal conducting state for type I superconductors. According to the famous Meissner's effect, the magnetic field is totally excluded in the region of the idea superconducting state, that is, h = curlyl = 0 for \ip\ = 1. Obviously, \h\ > Hc in the region f!n occupied by the normal conducting state and h = Hc on the interface F. By Maxwell's equations, the magnetic field h in fi„ satisfies — = &h infin. dt The conservation of energy on the interface T leads to curl/i x n = —vnh,

on T,

(3.1) '

v

(3-2)

where vn is the velocity of the interface in the direction of n. Moreover, the critical condition holds on T : \h\ = Hc

on T.

(3.3)

(3.1)—(3.3), together with suitable boundary and initial conditions, consists in a free boundary problems. It was pointed out in [7] that (3.1)—(3.2) can be also deduced from the non sta­ tionary Ginzburg-Landau equations (2.7) and (2.8) via a suitable limiting process for K<

72-

_ ,

Yi established existence and uniqueness of the local classical solution of (3.1)(3.3) with smooth initial and boundary conditions under the assumption that fi n is a planar domain. When fi„ C M3, the problem is open (for more details, see [8] and the references therein).

90 4

M o d e l s for layered s u p e r c o n d u c t o r s

Layered superconductors are very important for t h e study of superconducting phe­ nomena and the design of superconducting devices. In this section we shall discuss the Lawrence-Doniach model for layered superconductors, the superconducting-normalsuperconducting junctions and an approximative model for the S-N-S junction. All of the three are closely related to the GL model for superconductivity. 4-1

Lawrence-Doniach

model

One kind of layered superconductors is made by intercalating layers of organic molecules between layers of superconducting materials such as TaS2,TaSe2, NbS2, NbSe2 and so on. Denote by 0 the planar region occupied by each layer, by N the number of superconducting layers and by 5 the layer spacing. According to LawrenceDoniach theory, the total Gibbs free energy E is the sum of Gibbs free energy F} of each individual layer (j = 1,2, • • •, N) and the Josephson coupling energy SEJJ+I of adjacent superconducting layers (j = 1, 2, ■ • •, N — 1). T h e non-dimensionalized form of t h e total free energy E can be written as follows: N

N-l

^ E ^ + E^+i

= Ej[[-i^i 3 +^r+K-^-^)^r 2 + | c u r U j ; -+|curlA - J Sj|HtfjdQ ]dn N l

~

f

1

2

+ E S / [^"TI'/'J-^+i e x P(- Z K 5 Xj)| ~7 Jn * s + |Vx\V + | 2 ]
(4.1)

where tp} is the complex Ginzburg-Landau order parameter of t h e jth superconducting layer, A, is the magnetic vector potential of the jth layer which is parallel to the layers, \i ' s the magnetic vector potential perpendicular to the layers and 8A} =

(Aj-Aj^/s. The stationary Lawrence-Doniach model for layered superconductors consists of finding the groups of functions {^j}^=1, {Aj}^=1 and {XJ}JLI which minimize the Gibbs free energy (4.1). A 2-d nonstationary Lawrence-Doniach model for the layered superconductors was established in [8], but the system is not close. In the joint work [9] with Z. Chen and K.H. Hoffmann we got the expressions of the currents in t h e each layer and across the adjacent layers and reduced the time-dependent Lawrence-Doniach model in a closed form. We proved that there exists a unique strong solution for the 2-d Lawrence-Doniach model and, furthermore, it tends to the solution of 3-d timedependent Ginzburg-Landau equations in a cylinder domain as t h e spacings goes to zero ([9]).

91 4-2

Superconducting-Normal-Superconducting

junctions

Samples in which a thin layer of a normal conductor is sandwiched between two layers of superconducting materials are called superconducting-normal-superconduc­ ting (S-N-S) junctions (or Josephson junctions). Let D be a bounded domain in M2, e and L be positive constants with e < L. Set Q = D x ( — (L + e), L + e). Qs = {D x (e,L + e))U(D x ( —(L + e),—e)) and fin = D x (—e,e). H represents the region occupied by the S-N-S junction with fis and iln being the domains occupied by the superconducting materials and the normal conductor respectively. Denote by x and y the variables parallel to the planar domain D and by z the variable perpendicular to D. After doing appropriate nondimensionalization, the total Gibbs free energy can be given by E = Es + En

= /

2

2

2

[[K{M A s ( | <-l) /'|2-l)2 + + ^Vs\(--V-A)4>\ |(-1V-A)^|

2

K K

In Jn.

2

+ + I / s |vcs\curlA-H\' u r l A - ^ | 2 ]]dn dn

2 2 + / [X [\nnM W2 + Vn\(--V-A)lP\ ^n\(--V-A)1p\ K Jnnn +v + ^n„\cmlA-H\ | c u r L 42]dn, -#|2]dn,

(4.2)

where Es and En are Gibbs free energy in Cls and fi„ respectively; Xs,fis, vt, A„, \in and vn are positive constants; « is the Ginzburg-Landau parameter of the superconducting layers, ip, A and H are the complex Ginzburg-Landau order parameter, the magnetic potential and the applied magnetic field respectively as before (cf. [11]). The stead state problem for the S-N-S junction is to find the function ij> and vector-valued function A which minimize the Gibbs free energy (4.2) in a suitable functional space (for details, see [11]). In a joint work with K.H. Hoffman and N. Zhu ([12]), we extended the steady state problem to the time-dependent case, for which we proved the existence and uniqueness of the strong solution (ip, A) under a Lorentz type gauge transformation (cf 2.3). This evolutional problem is stated as follows. 4-3

An approximative

model for S-N-S

junctions

For the time-dependent S-N-S junctions model we point out in [12] that ip,Ax and Ay must be even functions in z and A2 must be an odd function in z under suitable symmetric conditions. Then the time-dependent S-N-S junctions model for sufficiently small e can be replaced approximately by the following equations: 8A BA i 2 ^^ -- + V# + aascurlcurL4 + ft [\^\ 2 A + — (V>*V - ^V#»)] + V # + curlcurL4 + ft [\^J\ A + — (V>*V - ^ V # » ) ] s

= ascurLf/,

inn°x(o,r),

= ascm\H,

2

inO°x(0,r),

a s ( ^ + iKi>§) + bsiP(\il>\ - 1) - c s (V - « A ) V a s ( ^ + iKi>§) + M > ( M

2

- 1) - c s (V -

2

IKA) 4>

(4.3)

92

= 0

iinnfOi °°xx((00,,rr)),,

di

/

dt

dlyAy

\-3LA. \-dLAt

(4.4)

~dlyAx\ dxyAy

)

2 2I+^-(*/>*V-|

+/j n [H /t + — (*v - vv
curM +|[( )L - ( c u r W )UJ x e* = o. + | [ ( c u r M ) L = 0 + - ( c u r W ) L o _ ] x e* = o. =0+

inft°x(0,r)

in ft° x ( 0 , T )

(4.5)

2

a n (-^T ++ *KV>«>J i«^#) ++ Kn> M -- C„(,V c„(V -- z/cA) <M"a7 lKA)'lp^

c d±,

_d± 0°° xx( 0(ft.T). iinn O ,r),

(4.6)

Az = (i,i/), A A, = 0, u, A AxI = =A AxI(x,y), Ayy = = A vA(i,j/), 9(x,y), inn°„x(0,r),

(4.7)

ip = %P(x,y), $ = §{x,y) infi°x(0,T),

(4.8)

w h e r e l=(A l = (/U,A,), V=(^,^); W n°n=Dx{z = Z? x {; = = Q}, 0}, 0° = = (D (Z? x (0,1)) (0, L)) U (D (£» x x,Ay), (-L,0)). The approximative problem is just like a concentrated capacity problem in the heat transfer. The interesting feature of the approximative problem is that the sample S-NS is composed of 3-d superconductors and a 2-d normal conductor without thickness and the small constant e plays only the role of coefficients in equations. Consequently, it is more suitable than the original problem in numerical calculations. The existence and uniqueness of the strong solution to the approximative problem is established under a Lorentz type gauge transformation ([12]). 5

Vortex pinning problem

In this section, we shall discuss two kinds of vortex-pinning problems. One is vortex-pinning in superconducting thin films with variable thickness, another is vortex-pinning in superconductors with normal impurity inclusion (see the last para­ graph in 3.1). 5.1

Vortex pinning in superconducting thin films with variable thickness

Let fi x ( — Sa(x),6a(x)) be the domain occupied by a superconducting thin film. Here, Q is a bounded domain in M2, a(x) is a function on H, a^1 < a(x) < a0, a0,<5 are positive constants. For sufficiently small 6, the Ginzburg-Landau's Gibbs free energy of the film can be written as (non-dimensionalized):

93

EM,

A) l

-J^a{x){\Vi>-iA4,\2+1±-2(\-\iP\2)2

=

+ \dA\2}dx,

(5.1)

where c = -g, x = xx + is%, A = Aidxi + A2dx2 (cf. [13] and [14]). If A is sufficiently small, (5.1) can be simplified as

EM) |2N 2 -,

. ^{l-\WY)dx, \L«*w+t+h

(5.2)

(I) Vortex-pinning for minimizers of the simplified energy (5.2). Consider the following minimizing problem: EM)=

min EM),

(5.3)

where EM) ' s given by (5.2), g : 0 —» S1, deg(<7, 50) = d > 0, is a given smooth function and

H](tt) = {v-: o -> c\i> e tf'(O), if = <,on an}

(5.4)

We proved in [15] that, for any sequence of the solutions {ipE} of (5.3), there is a subsequence {ipCn}, £„ -» 0 as n -> +oo, such that, the zeros of ipCtl tend to some minimal points of a(x) in 0 and xp£n tend to a function ip* which is a solution of the system { - V • (a(z)W>„) = a(x)iP,\ViP,\2

\ \H = 1 away from a finite singular points (for details, see [15]). Note that, for e n small enough, the zeros of ipen are located near the minimal points of a(x) which are the thinnest points of the superconducting thin film. This is just the vortex-pinning effect of the problem (5.3) which is also suggested by the physical explanation: the thinnest points are most easily penetrated by the applied field. (II) Vortex-pinning for minimizers of the Gibbs free energy (5.1) Consider the following minimizing problem: EMs,Ac)=

min EM,A)

V = M,A) € W J (0) x H\il)\ \u\ = 1 on 50, deg(w, 50) = d, J ■ r = g on 50}

(5.5)

(5.6)

94 where d is a given positive integer, g is a smooth function on <9fi, J = — IVM A — h(ip"'Vip - V>VV>*) is the superconducting current (appropriately nondimensionalized) and W ( n ) and H\il) are defined in (2.2). Our results for the problem (5.5) are similar to those for the problem (5.3) (see [14]). Let {ipc,Ac} b e a n y sequence of minimizers of (5.5). Then there is a subsequence {i>cn, ACn], e„ —» 0 as n —> oo, such that, the zeros of ip€n tend to at most d minimal Wo

points ( J { a , } [Na < d) of a(x) and {ipc„,A£n}

tends to {ip,,A.}

which satisfies

•=i N0

1 -- V V ■■ (a(x)Vi/>„) (a(x)Vi/.,) == a(x)ip,\Vip.\22,, = a(x)il>,\Vxl),\

!

\

inin fl\U Q \ | J{<*<}. {a,}, ii = = ii Wo

|0,| 1, |<M = = 1.

iinfAUK), nQ\UR}, i1== 1i

and f J

7

T V

,

(

1 | U . ^a(x)V{a{x)h.)>+n'

Wo No

in 0

1=1

on <9f2, ^ g M = -a(x),, onc^, where /i, =curl A,, & is the Dirac function. Note t h a t , an extremely interesting problem is the behavior of vortices for the minimizers of (5.1) with natural boundary conditions and the presence of the applied field H. In our knowledge, it is an untouched problem up to now.

!

5.2

Vortex-Pinning

in superconductors

with normal impurity

inclusion.

Both experiments and numerical simulations point out t h a t , when normal im­ purities are included in superconducting materials, the vortices of superconducting currents are pinned near the impurities. Let fl C IRN (n = 2,3) be a bounded do­ main, fi„ C C fi be the domain occupied by the normal impurities and n s SE 0 \ f l B be the domain of superconducting materials. Then the total Gibbs free energy is given by (4.2). When the magnetic potential A is very small, (4.2) can be simplified as

^ ) = ^n/ W L£(l-|^) ^)^/ / j i - , ^2++ ^ / n J ^ n W ++i iL

(5.7; (5.7)

where e,fi are positive constants, e = i , K is the Ginzburg-Landau parameter of the superconducting material. Denote by BT the balls in JR.2 centered at the origin with radius r. For simplicity, let U = BU H„ = B„ (0 < p< 1). Then (5.7) becomes E(
(1 2+ \j ™ l ^l + hi ^ / a-- W ' P + oi?/ == \l v

A z

JB JB)1

2

-I0| 2 ) 2 + ^ /

4t

4 t

JBlJB,\B \Bp P

t\l

JBP

|0|H22.-

(5.8) (5.8)

95 Consider the following minimizing problem: E(i>)=

min

EM

(5.9)

where 1i}g(B^) is defined in (5.4). Denote the solution of (5.9) by ipc for fixed p and p. We found that to 1 uniformly in fis as e —► 0 (after passing to a subsequence). This for sufficiently small e, all the zeros of ipc lie in the impurities Bp, which vortex-pinning effect due to the normal impurity inclusion for fixed p and

\ip£\ tends means that, is just the p.

A more interesting case is that p. = e, e —> 0 and p —> 0. In this setting, we found that, at least one of the zeros of ip€tP must tend to the origin if ^ —» 0 and all the zeros of ipec,p tend to the origin if

2f-i

~* 0

anc

'

2 f+i

—

* + ° ° for fc = 1,2, ■ • • , d — 1. Here

d =deg(g, dBi) > 0 and ip€jP denotes the solutions of (5.9) with e = p (see [16]). A similar result was also obtained by us for the minimizers of the Gibbs free energy (4.2) in the space V defined in (5.6) (see [17]). Note that, in th absence of the normal region fJ„, the minimizing problem (5.9) was studied in the recent book [18]. 6

Conclusion

We have briefly introduced some recent works of our research group in Suzhou University on some aspects of the analysis for the Ginzburg-Landau model and re­ lated problems To end this paper, we give some more information of the analysis for the convenience of our readers. In addition to those quoted above, we refer to the fol­ lowing references and the references therein: [19]-[20] for the dynamical properties of vortices of the Ginzburg-Landau equations with Dirichlet boundary conditions; [21][23] for the study of vortices of the Ginzburg-Landau model with physical boundary conditions; [24] for the mean field model of Ginzburg-Landau vortices; and [25]-[26] for the Neumann problems of the Ginzburg-Landau model.

7. 1 2 3 4 5

References H.K. Onnes, Akad. van Wekenschappen, 14 (1911). V.L. Ginzburg and L.D. Landau, On the theory of superconductivity, Soviet Phys. J E T P , 20 (1950). J. Bardeen, L. Cooper and J. Schrieffer, Theory of superconductivity, Phys. Rev., 108 (1957). L.P. Gor'kov, Microscopic derivation of the Ginzburg-Landau equations in the theory of superconductivity, Soviet Phys. J E T P , 9 (1959). L.P. Gor'kov and G.M. Eliashberg, Generalization of the Ginzburg-Landau equa­ tions for non stationary problems in the case of alloys with paramagnetic impu­ rities, Soviet Phys. J E T P , 27 (1968).

96 6

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15

16

17

18 19 20 21 22

Q. Du, M.D. Gunzburger and J.S. Peterson, Analysis and approximation of the Ginzburg-Landay model of superconductivity, SIAM Review, Vol. 34, No. 1 (1992). S.J. Chapman, S.D. Howison and J.R. Ockendon, Macroscopic models for su­ perconductivity, SIAM Review, Vol. 34, No. 4 (1992). W.E. Lawrence and S. Doniach, Theory of layer structure superconductors, in Proceedings of the twelfth international conference on low t e m p e r a t u r e physics, E. Kanda, ed., Academic Press of Japan, Kyoto, (1971). F. Yi, Local classical solutions of two-dimensional superconductor free boundary problem, CNS Preprint Series # 0 2 4 , (1994), Suzhou University. Z. Chen, K.H. Hoffman and L. Jiang, On the Lawrence-Doniach model for layered superconductors, DFG Report # 4 9 1 , (1993), Tech. Univ. Munich. S.J. Chapman, Q. Du and M.D. Gunzburger, A Ginzburg-Landau type model of superconducting/normal/junctions, including Josephson junctions, Euro. J. Applied Math. Vol. 6, (1995). K.H. Hoffman, L. Jiang, W. Yu and N. Zu, Models of superconducting-normalsuperconducting junctions, CNS Preprint Series # 2 3 , (1995), Suzhou University. Q. Du and M.D. Gunzburger, A model for superconducting thin films having variable thickness, Physica D, 69 (1993). S. Ding and Z. Liu, Pinning of vortices for a variational problem related to the superconducting thin films having variable thickness, CNS Preprint Series #032, (1995), Suzhou University. S. Ding, Z. Liu and W. Yu, Pinning of vortices for the Ginzburg-Landau func­ tional with variable coefficient, CNS Preprint Series # 0 3 1 , (1995), Suzhou Uni­ versity. S. Ding, Z. Liu and W. Yu, A variational problem related to the GinzburgLandau model of superconductivity with normal impurity inclusion, CNS Preprint Series # 3 3 , (1996), Suzhou University. S. Ding, Z. Liu and W. Yu, Pinning of vortices for a variational problem related to the superconducting thin films having normal impurities, CNS Preprint Series # 3 4 , (1996), Suzhou University. F. Bethuel, H. Brezis and F. Helein, Ginzburg-Landau Vortices, Birkhaiiser, 1994. F.H. Lin, Some dynamical properties of Ginzburg-Landau vortices, Comm. Pure and Applied Mathematics, (to appear). F.H. Lin, A remark on the previous paper "Some dynamical properties of Ginzburg-Landau vortices", preprint. Tang Qi, Remark on a relation between magnetic field and topological degree of order parameter in a type LI superconductivity model, Physica D 6 9 (1993). C M . Elliott, H. Matano and Tang Qi, Zeros of a complex Ginzburg order parameter with applications to superconductivity, Euro. J. Appl. Math. Vol. 6, (1995).

97 23 24 25 26

F. Bethuel and T. Riviere, Vorticite dans les models de Ginzburg-Landau pour la superconductivite, preprint. S.J. Chapman, A three-dimensional mean-field model of superconducting vor­ tices, SIAM J. Appl. Math., (to appear). S. Jimbo, Y. Morita and J. Zhai, Ginzburg-Landau equation and stable solutions in a nontrival domain, preprint. S. Jimbo and J. Zhai, Ginzburg-Landau equation with magnetic effect: nonsimple-connected domains, preprint.

98

COMPACTIFICATION OF MODULI OF VECTOR BUNDLES OVER ALGEBRAIC SURFACES J U N Li Mathematics Department Stanford University

It is a great pleasure and an honor to contribute to this book dedicated to Prof. C. H. Gu on his seventieth birthday. I learned a lot of Mathematics from professor Gu at my young age. It is ideal to contribute this paper studying a space both interesting to algebraic geometers and analysts.

Abstract In this paper, we have studied the problem of constructing Uhlenbeck's compactification of moduli of stable vector bundles over an algebraic surface. This is a gen­ eralization of the author's previous work on rank two case. As a consequence of this construction, one sees that the intersection theory defined using Gieseker's compactification of moduli space coincides with the intersection theory defined using Uhlenbeck's compactification.

The study of moduli spaces of vector bundles over algebraic varieties used to be a problem only interesting to algebraic geometer. In 1960's, Mumford first constructed moduli spaces of vector bundles over smooth curves. In doing so he introduced the notion of stable vector bundles, which morally speaking measure the non-degeneracy of vector bundles. Later, Gieseker constructed moduli of vector bundles over algebraic surfaces. The analogue of Mumford's stability in surface case (called /z-stable) is not fine enough to give a complete moduli space. The stability Gieseker used relies on the Poincare polynomial of the sheaves in contrast to the slope used in the /z-stability. Since we do not need their precise statements in this paper, we will refer them to [5,3]. The only part we need that follows from the definition is that the stability depends on the choice of an ample line bundle on X and a /j-stable bundle is necessarily stable. This research was partially supported by NSF grant DMS-9307892 .

99 When X is a smooth protective surface with ample divisor H, Gieseker constructed the moduli space of semistable rank r sheaves of determinant I G Pic(X) and second Chern class d £ H4(X, Z), modulo S-equivalence [5]. We denote this moduli space by . M H ( A ) , where A stands for the data (r, I, d). Since //-stable sheaves are automati­ cally stable, .Mff(A)'', which is the moduli of /i-stable rank r locally free sheaves of determinant I and second Chern class d, is a Zariski open subset of A/f#(A). Since in most cases MH(A)*1 is not complete (not compact in ordinary topology), .M//(A) can be thought of as a compactification of A^//(A) M . Although in algebraic geometry it is more natural to study moduli of stable (semistable) sheaves, the development of gauge theory provides a new interpretation of Mn(A)11 using analytic tool. To this end, we need to introduce the notion of moduli of Anti-Self-Dual (ASD) connections on principal bundles. Let (X, g) be a Riemannian 4-manifold, G a compact Lie group and P a principal G-bundle over X. We let B * ( P ) A S D be the space of irreducible ASD connection on P and Mg(P)ASD be the space of gauge equivalent classes of irreducible ASD connections on P. In other words, if Q is the space of gauge transformations, then Mg(P)ASD ^B'(P)ASD/g. (For the definition and discussion of ASD connection we refer to books [1, 4].) In the following we let G = PU{r) ( = U[r)/U(l)) and P be a PJ7(r)-principal bundle that can be lifted to a [/(r)-principal bundle P. Then if we let V be the associated complex vector bundle V = P (8>t/co

cr

>

ASD

any connection D 6 B * ( P ) lifts to connections D on V that preserves the obvious hermitian structure on V. Further, if we fix a connection D^v on AnV

= P ®yW A"C,

then there is a uniqxie lifting of D such that its induced connection on AnV is DAnV. Therefore, after fixing an ASD hermitian connection D^-v o n A n V, the space of ir­ reducible ASD hermitian connections on V that induces D A nv o n A"V is naturally isomorphic to B'g(P)ASD. In the following we will investigate Z?*(P) ASD when X is a smooth projective surface and g is a Kahler metric. Now let X be a smooth algebraic surface and H a very ample divisor with an associated Kahler form w. We fix an ASD connection Z7A" y on A r V so that its curvature satisfies Q(D^V)

Aw = CLJ2,

where c is a constant. Then given any connection D g

Bl{P)

100 D lifts to a connection Dy on V whose induced connection on A r V is identical to DAry. Since X is a complex manifold, Dy splits as a direct sum Dv

=

dD@dD,

where dD:Sl°{V) -» Q}-°(V) and d:Sl°(V) -> il°-*(V). Further D is ASD implies that dD = Oand wAQ(Dv) = clvw2, where Iv is the identity endomorphism of V. An easy argument shows that the first identity implies that do induces a holomorphic structure on V and the second identity implies that this holomorphic structure is ^-stable with respect to H. Finally, since the induced connection on A r V is D^ry, the determinant of this holomorphic vector bundle is a line bundle / € Pic(.X') independent of the choice of D. This way we obtain a map

£ : BZ(P)ASD —>

MH{&Y,

where A = (r,I,c2(V)). (This map is largely due to Kobayashi where he proved that D is ASD implies that (V,do) is a /^-stable holomorphic vector bundle.) Since gauge equivalent ASD connections induce equivalent holomorphic vector bundle V, K. factor through a map AC/6 : MU(P)ASD —»jWtf(A)". One important and challenging issue is to find compactifications of MU(P) Since Mu(P)ASD is constructed analytically using ASD connections, one obtains a compactification of A ^ „ ( P ) A S D that includes all "limit points" of sequences in Ma(P). Using the work of Uhlenbeck of weak limits of ASD connections on 4-manifolds [4], Don­ aldson constructed such a compactification . M w ( P ) u h , called the Uhlenbeck's compactification of A " ^ ( P ) A S D . Since MU(P)ASD is isomorphic to MH{&Y and M f f ( A ) is complete (i.e. compact in ordinary topology), we obtain a compactification . M # ( A ) that is the closure (in Zariski topology) of Mn{AY C MH{A). A 4 H ( A ) G I is called the Gieseker compactification of MU(P)ASU. The goal of this paper is to investigate the relation of A 4 H ( A ) G i and Mw(P)m. The relation between MH(A)Gi and A ^ „ ( P ) u h when G is PU(2) is understood completely, by the work of J, Morgan [9] and the author [7]. In terms of algebraic geometry, their relation can be expressed as follows: There is a line bundle L on MH{A)G'' such thatH°{MH(A)Gi,L®m) is based point free for large m and the image scheme of MH(A)GI under the morphism Mn(A)Gi is homeomorphic

to MU(P)G'

—►

PH°(MH(A)Gi,L®m)

as topoJogicai spaces.

In this paper, we will show that the same result holds for the group PU(r) as well. The approach to the general case is parallel to the SU{2) case treated in [7]. We hope that this paper will not only completely answer the problem raised but also clarify some related issues left unanswered there.

101 1.

A L G E B R A I C C O N S T R U C T I O N O F U H L E N B E C K COMPACTIFICATION

We continue to use the convention adapted in the introduction. We fix a positive integer r, a line bundle I on X and an integer d. We denote by A the data (r,I,d). We let P be a principal P!7(r)-bundle admitting a lifting to a t/(r)-bundle whose associated hermitian vector bundle V has determinant I and second Chern class d. Our first issue is to construct the line bundle on MH(&) mentioned in the introduc­ tion. By replacing H with its multiple, we can assume H is very ample and HI divides r. Let I be any positive integer and C 6 \IH\ be a smooth divisor. Let Tun be the moduli functor of MH{&) that sends any scheme S of finite type to the set of families of rank r sheaves Ss on X x 5 flat over S such that d e t £ s = p'xl and C](£,) = d, where px -X x S —► X and £, = £s\Xx{a) ls ^ae restriction of Ss to X x {s}. Now we choose a line bundle 6j on C so that X(6l)

+ i deg(7|C) = 0.

We call such line bundle numerical ^J-theta character of C and will use 8j solely for this type of line bundles. For any £$ £ J-un{S) we define a line bundle Detc(£s,0/)6Pic(S) that is the determinant of the perfect complex R'Ps,{£s\CxS®Pc8i), where ps'X x S —» S and pc-Cx.S—*C are projections and £ s | c x s is the restriction of £s to C x S. When £5 and £g € .Fim(S) are two equivalent families, that is when there is an invertible sheaf V on S such that £s = £5 ® Ps^> then since

x(£.|c®*/) = o, Detc(£s>0/) is canonically isomorphic to Deic{£'s,8j). This way we obtain a line bundle L on J M J / ( A ) if it admits a tautological family. ( 5 on A* x A^//(A) is called a tautological family if for any point 5 £ jVf#(A), £|xx{»} ls isomorphic to the sheaf represented by 5 6 Mu(A).) There are many cases where Mn(A) does not admit tautological family. To remedy this problem, we need to use the machinery of GIT theory and descent theory. Following [5], for some large n the tensor product with 0{H®n) of any £ € MH{A), denoted by £(n), can be expressed as a quotient sheaf 771

®Ox

—>£(«),

where m = x ( £ ( " ) ) - We fix a sufficiently large n and the corresponding m and then form the Grothendieck's Quot scheme r\

j.A(n)

Quot0^'

102 of all quotient sheaves ©Ox—>S(n) such that rank S = r, d e t £ = I and c 2 (£) = d. Q u o t £ L n ) is projective, is acted on by SL(m) and admits a universal quotient family, say T. Following [5], there is an open SL(m) invariant subset U C Q u o t ^ i ? ' such that the GIT quotient of U by SL(m) is the coarse moduli scheme .M//(A): U//SL(m)

=

MH(&).

Applying the previous determinant bundle construction to the universal quotient family T, we obtain a line bundle

Betc(f,6i) on Q u o t u m . Clearly, the SL(m)-linearization on T induces an SL(m)-linearization of Detc(^", #/)• Since MH{&) is a good quotient of U by SL(m), the line bundle Detc(.F, 9l)\u descends to a line bundle on A4//(A) if and only if for any point w g U having closed orbit SL(m) ■ {w} C U, the action of the stabilizer stab(u;) of w on the fiber D e t c ( J r , 8j) g> k(w) is trivial. This can be checked easily as was done on page 426 in [7]. We denote the inverse of the descent line bundle by L(C,0i). Combined, we have proved the first part of the following Lemma: L e m m a 1.1. Let I be any positive integer, let C g \IH\ be any smooth divisor and 8j a numerical ^I-theta, character. Then there is a iine bundle L(C,8j) on MH(&) that is the descent of the line bundle DetciJ7, ®l)7u ■ Furthermore, for any positive integer h, smooth C g \hlH\ and numerical -I-theta. character O'j on C, we have naturally

Lice^^Lice'j). Proof. We only need to prove the furthermore part, which can be done by applying the argument given in §1 of [7]. We will omit the details here. Since H is very ample, we can and will fix a smooth C 0 g \H\ and a numerical - J - t h e t a character 0O on Co, thus obtaining line bundle L = L(Co,00). By Lemma 1.1, for any smooth C g \IH\ and line bundle 9j as before, the line bundle L(C,8i) is isomorphic to L® . Our next issue is to show that for sufficiently large / and k, H°(MH(A),L®>k) is base point free. The technique we will use is as follows: We choose a large / and a smooth C g \IH\. By restricting sheaves E g X / / ( A ) to C we get a rational map RC-MH(A)

>Mc(r,I\0),

103 where Mc(r, I\c) is the moduli space of rank r semistable locally free sheaves on C of determinant 7| C , that sends £ £ MH{&) to £\c if it is semistable. Donaldson observed that there is an ample line bundle K on Mc{r, I\c) such that the pullback RC(K) is isomorphic to L®' over where Re is well-defined. Since K is ample, H" (Mc(r, 7| C ), K®k) is base point free for some large k. On the other hand, for many <j> € H°(Mc(r,I\c),K®k) we will show that the pull-back section Rc(), which is a meromorphic section in the first place, extends to a regular section over MH(A). Therefore by varying smooth C 6 \IH\, we obtain enough sections to guarantee the base point freeness of H° (jVf//(A), L®'k). Now we provide details of this argument. In case the space .M//(A) is normal, then it is relatively easy to show that Rc{4>) extends for each <j> e H°(Mc(r,I\c)>K )i since all we need to show is that Rc(4>) extends to codimension 1 points in .M/j(A). However, A4//(A) is not necessarily normal in general. Thus to attack the general case we should use the descent theory. We consider the moduli scheme Mc(r,

I\c)-

We fix an ample divisor He on C of

degree 1. Let QuotJT^ be the Grothendieck's thendieck's Quot Quol scheme of quotient sheaves ©Oc—*JF(n) such that rank T = r, det T = I\c and m = rh + deg J| C + r x ( O c ) , where n is an integer and F(h) = T ® H®". If we choose h large enough, then there is an open subset U C QuotiT^, such that v c Mc(r,I\c)

=

3//SL(m).

We first define the line bundle we need. Recall that the line bundle L®1 on MJJ(A) is the descent of a determinant line bundle of a complex on U that is constructed based on the restriction of T to C x Q u o t 0 „ . Using the universal quotient family T on C x Q u o t a s , , we can form a similar complex and then form the determinant line c

bundle

KQ = det(iJ-7r Q ,(^igi7rje / ))~ 1 ,

where 8j is as before and TTQ-CX

Q u o t ^ ' —> Quot\T^ . As was shown in [2,5], Kn c ? is an ample line bundle. Further, KQ admits SX(m)-linearization and that for some large k,

LJ^PH°(Quot^\K®k)SL{ih)

104 is well-defined and its image scheme is Mc(r, I\c)- Now we fix such k and let (j> e h SLl h)

h SLW Ha(Quot^i\K§ QUot^i\K$ )

) '

be any section.

Next, we will study how to "pull back" i ^ t o a section over Q u o t £ L n ) . Since CcX is a smooth divisor and T restricts to X x U is a flat family of torsion free sheaves, F\Cxu is flat over U. After covering U by affine open subsets {.4*} and fixing trivialization U :®0 /■ -®0AiAi

®PcH$ ®PcH$nn))

-^PAA^CXAI -^PAA^CXAI

we can represent f | C x / ( . ( n ) as a quotient sheaf of ®mOAi. Here we choose n to be sufficiently large. Then by the universal property of the quotient scheme, the above identity induces morphism ipi ■ :' Ai Pi Ai —» —» QQuuoot t^^' '. . Obviously over Ai C\ Aj ^ 0 we have isomorphism m

a* m

9ij ■ ®0AinAj 9ij ■ ®0AinAj

—> —>

®0Ai(xAj ®0Ai(xAj

such that the diagram ®0AinAj

m

Ml

©04,0.4,

®PcHcn)\AinAj

► PAi*(F\CxAi

I!

Si

,

>

„_

:F

PAi»{ \CxAi®p'cH$n)AAinAj

is commutative. In terms of Quot-scheme, ► GL{m) GL(m) such that H>i\AiC\Aj — = 9ij j l ^ i n!nA ^ j i,i Vi'l^.-n^ 9ij - V■fj\A

where ■ stands for the GL(in) action on Q u o t ^ ' .

(The SL(m) action induces an

obvious action of GL(rh) on QuotlT^ .) We now let LQ be the line bundle D e t c ( ^ " , 0 / ) _ 1 with C £ \H\ and 8{ be a numerical -7-theta character. By the universality of the determinant line bundle, we have natural isomorphism

tf(KQ) = (LQ\Ai)&.

105 Now we consider the pull-back section tp*((f>), where

eH°(Quotl-<;i),K®k)SL{'h). Because (pi\AirUj = 9ij ■ Vj\AinAj > w « have ¥>i()\AinAj= =f>j<{>) ¥>i()\AinAj ° 9ij(4)\A,nAj ° 9*jW\AinAj ■ ■ Since is GL(m)-equivariant, g*j{4>) = - Hence ^iW^nAj

= )\Air\Ai-

This shows that . We now show that <j> is 5T(m)-equivariant. Since the SL(m) action on Q u o t 0 m is an affine action, we can assume without loss of generality that Ai is SL(m) invariant. Then for any a G SL(rn) we can find ga:

Ai A, —►

GL{m) GL[m)

such t h a t Qo 9i = = <7

ipioa. ipioa.

T h e r e f o r e , b e c a u s e g%{<j>) = <j>,

tfW vK4)==«'(p*{) is 5£(m)-equivariant. From the above discussion, it is clear that tj> is independent of the choices made. K^k)

L e m m a 1.2. Let the notion be as before. Then for any <j> G H° ( Q u o t ^ V , there is a natural section G H° (Quot0„ Ai C Q u o t 0 m

, LQ"1)

"* whose restriction

is exactly if*(<j>), where ,• is the morphism defined before.

a point w G Quot 0 m following holds:

to each

Furthermore

is contained in the vanishing locus of <j> if and only if one of the

(1) TM, is not locally free along C, where Tw = J-\Xx{w)i (2) Fw\c is locally free and unstable or (3) Tw\c is locally free, semistable and £w\c £ ^ - 1 ( 0 ) .

106 Proof. We only need to prove the furthermore part. Let w £ Quot 0 ™ n be any point in Ai corresponding to the sheaf Tw. Since the restriction of <j> to Ai is ), w £ ^ _ 1 (0)ifandonlyifv?i(u)) £ ^ - I ( 0 ) - Since ^ is S ' i ( m ) equivariant, . When Tw\c is stable, it is obvious that <j>(w) = 0 if and only if 4>(tfi(w)) = 0. This completes the proof of the Lemma. Now we show that for large / and k, k k H°(M H°(M„(A),L°' H(A),L®' ) )

is based point free. Indeed, for any semistable sheaf £ £ j M / / ( A ) G l and large I, £\c is semistable for general C £ \IH\. Then we can choose a section SL(A) ^eH0(Quot^\K§kk))SL(A) <j>eH°(Quot^\K$

so that 4> does not vanish on the orbit in QuotlT^ associated to £\Q- Therefore by the c previous lemma,

ieHO(Quot^\L%> ^ e J D ( Q n o t g W ,kl)| » ) is non-vanishing at the orbit in Q u o t 0 m corresponding to £. Since <j> is SL{m) equivariant, it descends to a section of L® over MH(A) that is non-vanishing at £. This shows that H°(MH(A),L&k) has no base point. Now let Gi 9,k «* MM :: -M//(A) —> MH(A)Gi —> P^°(^f PH°(M w (A),L®'*) H(&),L )

be the induced morphism and let A 4 / / ( A ) u h be the image scheme of the Gieseker compactification of A4//(A)**, where the Gieseker compactification . M H ( A ) G I is the closure (in Zariski topology) of MH(A)** C MH{A). P r o p o s i t i o n 1.3. For sufficiently iarge / and k, the restriction MH(A)Gi:

of$ik

to MH{&Y

C

$',.. AWA1" —> *,k(Mit(&y) H(A)" is an isomorphism

of schemes.

Proof. First we show that as maps between sets, #' /(t is one-to-one and onto. The onto part follows from the definition. Now we prove the one-to-one part. Let £ and

107 £' £ Mn(A)11 be any two sheaves that have the same images under #;*. Since / is suf­ ficiently large, for general smooth C G \IH| both £\c and £!c are stable. We claim that £\C must be isomorphic to £!c. If not we can find sections <j> in H° ( Q u o t ^ V , K^k) such that 4>{£\c) = 0 while <j>(£!c) / 0. Then the descend of ^ will separate £ and £', a contradiction. Thus £\c = £!c- Further, since / is sufficiently large, Hf11 (Wom(5,f')(-'-ff)) (Hom(£, £'){-lH)) =00.. Thus =

Hom(£,£') — H o^Homc(£ m o ^ c .icf,£[c) fc) Hom(£,£') is surjective. Therefore £\C = £!c implies that Hom(£, £') ^ {0}. However, since both £ and £' are /^-stable and have identical rank and degree, £ must isomorphic to £'. This shows that $'lk is one-to-one and onto. Now we show that -> .MH(A )'1 f \ , 92 Spec MH{&Y

are two morphisms such that their restrictions to S p e c ( A / B ) coincide and #(* o y\ = $lk0lP2, f\ and if 2 are necessarily isomorphic. Let ip\, ; <Etf°(Quot H0(Quot^P,K® extends to QuotpL"', * 0 ^tf§V ik oip2 implies that Re °fi= Re ° f2, where Re is the meromorphic map defined before that sends £ to £\e- Therefore, we must have

£l\CxSpecA £l|CxSpec-4 — ^2|CxSpec/l-

However, since we have assumed that / is sufficiently large, the above isomorphism will force £\ = £2 because Ext*(£1, £2(—IH)) = 0. This implies that $'lk is an isomorphism. We will call MH{A)V^ Since MHW

C MH(&)G'

the algebraic Uhlenbeck's compactification of

MH{AY-

Gi Zariski dense, * tis —» -M H (A) u h t : -M H (A)

* i t : MH(A)Gi

—»

MH{A)m

is a birational morphism. 2. GEOMETRY OF OF UHLENBECK'S COMPACTIFICATION

In this section, we will investigate the geometry of the contraction morphism

#l k : M MHH(A) (&)GiGi

h —► MH(A)uvh —» Mtf(A) .

108 Though the construction depends on the choice of / and k, the scheme A 4 H ( A ) and the contraction A 4 # ( A ) G i —> X ^ ( A ) u h is independent of the choice made because

R(M (A)GiGi,L)= ,L) R(MHH(A)

= © ffi

Gi H°(M ,L®n) H°(M H(A) H(Af,L®

n=0

\L®n)

is finitely generated C-algebra. (It is finitely generated because H°(MH(A) is base point free for some n > 0.) Therefore

M ff f ( A ) c b = P Proj roj ( © ©

ai ai n S H°(M„(A) H°(MH(A),L® ,L®n)))

\n=0

J'

and $ik is simply Gi # :M MHH(A) (A)Gi

Vh —-> —> M MHH(A) (A)

== Proj P r o j f( © ©

GiGi nn H°(M H°(M„(A) ,L®)). )). H(A) ,L®

\n=0

/

Our first issue is to give a precise description of $~1(w) for each w 6 MH(A) . To accomplish this we will introduce a new equivalence relation, called U-equivalence relation, that will be shown to be the equivalence relation associated to the map # . To this end, we need to recall the notion of S-equivalence. For any sheaf £, there is a so called Harder-Narasimhan filtration

0= = £00 cC £i £1 C C •■• •• ■ C £,-!C C C £i_i £ t £, = = I £ such that £{/£i-i direct sum

is stable for each /. Such filtration may not be unique. However, the

Gr(£) = = £xjU £ffi Gr{£) • - • ffi £ l / £ | - i i/£0@---@£i/£,-i is unique. Two sheaves £\ and £2 G .M//(A) are said to be S-equivalent if Gr{£\) = Gr(£2). Since X is a smooth surface, the double dual £vv of any torsion free sheaf £ is always locally free and that £ is a subsheaf of £ v v whose cokernel is a torsion sheaf supported on discrete set. For any torsion free sheaf £, we define u(£,x) to be the length of the stalk of £ v v / £ at x. We now define the U-equivalence that will characterize the preimage of w £ . M w ( A ) u h . Let £] and £2 be any sheaves in .Mtf(A). We say £j is U-equivalent to £2 if Gr(£:)v = Gr(£2)v and ^ ( G r ( £ ] ) , i ) = u(Gr(£2),x) for each x € X. P r o p o s i t i o n 2 . 1 . Let £1 and £2 be any two sheaves in M n ( A ) G i . Then4>(£i) =
MHM (AH(Af' Re ■■

M>c{r,IMc(r,I\c) ]c)

109 is well-defined near £. Not only that, Re is well-defined near any £' that is U-equivalent to £, since then G r ( £ ) | C = Gr(£')\c. Hence the set of sheaves that are U-equivalent to £, which we denote by S{£), is mapped to a point in Mc{r, I\c) under Re- Therefore the restriction to S{£) of L& is the trivial line bundle. Further, S(£) is closed and connected, using the argument in §5 of [7]. Therefore under the morphism

MHH(A) M (A)aiai

GiGi —♦ Proj Proj ( J offi H°(M H°{M„(A) ,L®")), H(A) ,L®")),

S(£) is mapped to a point. This proves that each U-equivalent class is mapped to a single point in A1/j(A) D h . It remains to show that if £\ is not U-equivalent to £2, then ${£\) ^ $(£2). We first check the case where Gr{£\Y ^ Gr(£2)v. Assume Gr(£i) ^ Gr(£2), then for general C £ \IH\ with I large, the sheaves £\\c and £-2\C will be semistable but not S-equivalent. Namely, Gr(£x\c) ¥ Gr(£2\c)- Hence £X\C and £2\c will correspond to distinct points in Mc{r,I\c)Mc(r,I\C). Therefore ${£\) #(£i) ^ #(&). $(£2)Now assume Gr(£i)v = Gr(£2)v. Let x £ X be any point. We will use divisors C passing through x to distinguish numbers u(Gr(£i), x). Let C 3 x and <j> be an SX(m)-equivariant section of K% , then the extension of the pull-back (f> vanishes at the orbit in Q u o t 0 m corresponding to £. We will see that the vanishing order of 0 is related to the number u(£i,x), which enables us to prove the Proposition. Let £ € MH(A) be any sheaf. Because £ and Gr(£) corresponds to identical point in J M H ( A ) , without loss of generality we can assume £ = Gr(£). Now we fix a large / and a Co £ \VB\. We can always find a family of divisor C<, t £ T, in \IH\ that contains Co as its member so that for t £ T other than 0 € T, £\c, is locally free and semistable. (We assume 0 £ T and Co is the divisor we begin with.) We denote by C r the total space of this family and by r: CT —* T the projection. We now consider the relative moduli scheme

Mc /T{r,P*I) McTT/T(r,P*I) over T whose fiber over t £ T is exactly the moduli space Mc,(r, I\c,)- Since the construction of determinant line bundle is functorial, there is a line bundle KT on McT/T(r,P*I) such that its restriction to Mct{r> I\C,) C ^ c T / T ( r j P * ^ ) ' s exactly the ample line bundle Kc, defined earlier. We now choose a family of sections 0 k 4TeH (MCT/TCT(r,p*I),K® ), k),
and the extension a GG k (l>kl j>TTeH j> £Ha{M {MH{&) '>xT,q*L ') ) H(k) ''xT,q*L®

of the pull back RCT{<J>T),

RCTCT

where Gi Gi :M M„(A) H(A)

x xTT

► > MMCCT r /,T(r,P*I) T(r,P7)

110 is the map sending (F,t) to T\c, when it is semistable. Here q : MH(&) ' X T —t M H ( A ) G i is the projection. Now we restrict 4>T and q*L®kl to { £ } x T C MH(A)Gi xT, which has the following interpretation: Let 8T be a line bundle on CT such that its restriction to each C, is a numerical i / - t h e t a character. (Such 6T does exist because we can find sections of w : CT -» T.) Then the restriction of q*L®' to {£} X T is naturally isomorphic to det(fl-7r»(p*£® 0 r ) ) ~ \\ det(RTr.(p*£®0 T)) as line bundles over T. ( p : C 7 - t I and p*£ is flat over T since p*£ is locally free over where p: CT - t l is not flat.) If we replace £ by £ v v in the above determinant line bundle, we get a new determinant line bundle that belongs to the following natural isomorphism v det(fl-7r,(p'£ ® 0)eO)^det(R« £ det(fl-7r,(p*£: det(RTTt(p*evvv®6 T))eT))~\he\h t(p*£®0T)) ® 0 T ) ) " \

where he(Co) he(C0) == S£„i eC g cr 0 v{£, K^> Ex). ) - ^ ' s c l e a r ' h a t there is a section vv _1 <^^ vv ££ ^ ( ^ d H°(T,det(RTr,{p'£ < e t ^ ' T r ^ p * ^ ® ^®6 ) )T)) " 1 ))

such that 4>T is induced by <j>Tv via the above isomorphism. Now assume £' is another sheaf such that £ v = ( £ ' ) v . Applying the same construction to £', we obtain a section

det(R7r,(p*£' ®6TT)))) det(Rwt(p*£' ®8 that is also induced by tf>vv. Thus 0 r ( £ ) 0 e r if and only if

ar

1

'

>d 0 T ( £ ' ) will have same vanishing order at

hMCo) h£,(C0). £(C0) = MCo). = #§(£') T{£)and and^ T Therefore, #P{£) (£) = ( £ ' ) will force 4>T(£) ( £ ' )T{£') have identical vanishing order at 0 € T, and then hc{Ca) ftf(Co) = = /if(Co). fof(Co). Since Co is arbi arbitrary, this implies that v(£,x) = u(£',x) for each x £ X . This completes the proof of the Proposition. From the above proposition, we obtain an easy description of points of .MH(A) u h '. They are pairs

(J» such that T is a rank r, determinant / poly-stable vector bundle (i.e. a direct sum of p-stable vector bundles of identical slopes) and w is a 0-cycle in X such that c

2^) 2^)

+ length(ui) length(io) = d. d,

that are limits of p-stable sheaves in MH{&YFrom Uhlenbeck's theory of weak limits of ASD connections, points of Mw(P)m have exactly identical description. Thus ■M„(P) A S D = A 4 / / ( A ) u h as sets. Indeed, we have the following theorem which shows that they are homeomorphic.

111 T h e o r e m 2.2. Let the notation be as above. Then MU(P)ASD is naturally homeomorphic to MH(A)vli, extending the identity map between MU(P)" C M„(P)ASD and MH(&Y

C

MH(A)Gi.

The proof is a line by line repetition of the proof given in [7] except changing rank 2 to rank r. Hence we will omit the proof here and refer it to [7]. The last issue we will discuss is the singularity of .M/f ( A ) u h . First, by [8] and [5,10], for fixed r and I there is a large N such that whenever d > N then ,M//(A) is normal, has local complete intersection singularity over A ^ H ( A ) M and is irreducible. Now we fix such N and assume d > N in the remainder of this paper. Then since .M//(A) is irreducible, MH{A)G' = - M / / ( A ) . Hence A^//(A) G l is normal and irreducible. Because

7W„(A)Uh is

i P r o j ( JoofT°(A< ^ ° ( ^ w (wA(A) ) GG1 , L,L® ® "B))), ), Proj(J

A ^ H ( A ) U 1 1 will be normal as well. Of course, .M//(A) U1> will have local complete intersection singularity over .Mtf(A)' 1 C MH(A) . Now we study the singularity of general points of A4tf(A) u h MH{&YNow let V C MH(A) be the subset MH(A) - MH(&-Y and let w € V be a general point. Let £ G $(io) be any sheaf. Then by previous Proposition, $~1(w) consists of sheaves that are U-equivalent to an £. Since d is large, £ is the kernel of £vv —> Ox for some x £ X. Let T = £vv. Then $-\w) is isomorphic to P r _ 1 , since points in this set are parameterized by surjective homomorphisms Ot. **"—» " — » Ox. In the following, we will determine the normal bundle of V C MH(A) along S(£). By assuming d > N large, we know that M « ( A ) is smooth at £ [5,10] and the tangent space of MH{£\) at £ is naturally

Ext'(£,£) E x t 1 ^ ) 00-To determine the normal bundle of V C Mfj(A), we need to find the tangent space of V at £. We look at the following exact sequence

0 _ * H\£nd°{£)) H\£n£{£))

— — E Vx\}(£,£f xt1^)0 — — H°(£xt\£,£)) H°(£xt\£,£))

We choose an isomorphism

Fz*of*-V®Ox,x so that

£,sO^",)©1*.

2

—> —► H2(£nd°(£)) (£n(f(£)) = 0. 0.

112 where Tx is the ideal sheaf of x £ X. Then ) 1 l) l {£,£)^£xt\l )®£xt (lx,II).i(lx,lz). £xt1{£,£)^£xt {X )®£xt z,ofy IiO%^~

A straightforward calculation shows that the tangent space of V at £ is the kernel of r l) 1 ExO(£,£)° H10(£xt\l Ext (£,£)" — —». / r 0 ( 5 x < (II,O | | ; ~ 1z-) ))). ). x,O®[

Thus the normal bundle of V C M « ( A ) at £ is naturally

H»{£xt\i ,of;-l))) = (o /i ) ® o®|;-«. XtI x ® O l l r 1 ' . F » ( £ r f 1 ( 2 z- „ 0 ® | r 1 > ) ) = (Ox,x/Ix)

An easy calculation shows that the right hand side of the above identity is (£vvvv/£) /£)

v v vv ® (ker{£ ( k e r { £ vvvv ®k ® fc, }), x-^—> £ £/ £/£}),

which is free of the local trivialization we made. To see this as fibers of the normal bundle of V along S(£), we will first construct a tautological family of S(£). This can be done as follows: Let F be the vector bundle of T and let FT be the fiber of F at x. We let S = PFX and p\ and p2 be the first and second projection of X x S. We consider the obvious surjective homomorphism P*F—>plks®p'2Os(l) and its kernel £$■ Clearly £$ is a tautological family of S{£). Therefore (the sheaf of sections of the) normal bundle of V C Mn(A)G' along S(£) is £ £xtxtxxs/s( xxs/s(£s,£s) s,£s)

,

which is (£r/£s) (Sr/£s)

® (ker{£:s vv ® 0{I]XS {I]XS

— , ^ vv / ^f s } )

as sheaf over 5 = {x} x 5 . Clearly, ^ v / f s = 0S(1) and £ ^ v ® 0 { l } x S = Of. Hence if we let T be the sheaf of sections of the normal bundle restricting to S(£), then T belongs to the exact sequence r

o —* 0 —* Tr —» —»e> O ss((-i)® - l ) ® r —» — » Oc o s —> —> 0.o. Therefore, the normal bundle of V C X W ( A ) along S(£) = P r _ 1 is the cotangent bundle of S ( £ ) . As a consequence, the restriction of the normal bundle of V C MH( A) to <S(£) has degree — (r + 1).

113 REFERENCES [1] Donaldson, S.K. and Kronheimer, P.B., The Geometry of Four-Manifolds, Oxford Mathematical Monographs, Oxford Science Publications, 1990. [2] Donaldson, S.K., Polynomial invariants for smooth four-manifolds, 257-315.

Topology 29 N o . 3 (1986),

[3] Fogarty, J and Mumford, D., Geometric Invariant Theory, Springer-Verlag, 1982. [4] Freed, D.S. and Uhlenbeck, K.K., Insianions Berlin Heidelberg Tokyo, 1984.

and Four-Manifolds,

Springer-Verlag New York

[5] Gieseker, D. and Li, J., Irreducibility of moduli of rank two vector bundles, J. Diff. Geom. 40 (1994), 23-104. [6] Gieseker, D. and Li, J., Moduli of vector bundles over surfaces, J. Amer. Soc. (to appear). [7] Li, J., Algebraic geometric interpretation of Donaldson's polynomial invariants, J. of Diff. Geom­ etry 37 (1993), 417-466. [8] Li, J., Kodaira dimension of moduli space of vector bundles on surfaces, Invent. Math. 115 (1994), 1-40. [9] Morgan, J., Comparison analogues, preprint.

of the Donaldson polynomial invariants

with their

algebro-geometric

[10] O'Grady, K., Moduli of vector bundles on projective surfaces: some basic results, preprint.

114

Analysis on Singular Spaces Fang Hua Lin Courant Institute of Mathetical Sciences New York, New York 10012 and The University of Chicago

Dedicated to Professor Gu, Chao Hao on the occasion of His Seventieth Birthday Abstract Here we give a survey with some detailed proofs on some basic analysis on Alexandrov spaces. In particular, we sketched the proof of Sobolev and Poincare inqualities on Alexandrov spaces with curvature bounded below. We also give a proof of Holder estimate for harmonic maps and functions defined on such spaces. Some related mathematical results are also presented. Among the many fields of mathematics in which Professor Gu, Chaohao was a leading figure, one was the nonlinear field theory that includes, in particular, the studies of Yang-Mills-Higgs model (the Yang-Mills theory) and that of nonlinear umodel (the harmonic mapping theory). His joint works with Professor C.N. Yang on Gauge field theories in mid seventieth, and his works on wave maps (i.e., harmonic maps from the Minkowski spaces) in early eightieth have profound impact to the later developments. These works have set up the questions and problems that still today remain fascinating and challenging. The present paper gives a survey of some recent results obtained by the author on harmonic maps defined on a singular space. The main results here have not yet been published though they have been presented at several conferences including the workshop in " P D E method in Geometry" held in May 1994 at the University of Minnesota and, the sixtieth anniversary of Chinese Math. Society meeting held in May 1995. Thus we shall present most results with proofs in details.

1

Introduction

Motivated by many questions in the geometry and the algebra, the study of harmonic maps from a smooth domain to a negatively curved Alexandrov-space has attracted

115

the attention of many researchers recently. Many beautiful and remarkable results have been obtained by Gromov-Schoen7, Korevaar-Schoen10, Jost 9 and others. Here we are interested in harmonic maps (and, in particular, harmonic functions) from a singular space into either a smooth Riemannian manifold or negatively curved Alexandrov space. There are many reasons to study such problems beside its own interests. We shall describe below three basic examples. The first one is related to the study of nematic liquid crystal droplets. The problem also involves a free interface, we shall discuss the behavior of free boundaries in our forthcoming paper12. Here we consider simply the map that describes the configuration of optical director field. The second one is an optimal design problem which involves a vector-valued function which prefer to take value in certain set. It can also be viewed as certain phase-transition prob­ lem. The third example is motivated by the well-known work of Eells and Sampson3 on harmonic maps into negatively curved Riemannian manifold, and the well-known gradient estimate of S.T. Yau for harmonic functions on a complete Riemannian man­ ifold with nonnegative Ricci curvature. The problem is to understand polynomial growth harmonic functions on such manifolds, and, in general, Liouville-type theorem for harmonic maps from such manifolds into nonpositively curved Alexandrov spaces, cf.11. The main questions we will address in this paper are classical existence and reg­ ularity. The latter is related also to the uniqueness question. Due to the domain of these maps is singular, even the Sobolev spaces of maps on such domains have to be defined properly. There are a few subtle points. The known theory of Schoen-Uhlenbeck and also by Giaquinta-Giusti (that in­ volves blow-up of independent variables) uses very crucially the smoothness of both domain and target manifolds, cf. 17,5 . It, therefore, can't be applied to these problems mentioned above. The method in8 involves blow-up of target manifolds is somewhat better. It applies to the case domain may be singular (but requires the target to be smooth), see a recent article16. The method in 7,10 requires also the domain to be smooth. During the AMS summer institute 1992 at the Park-City, Utah, the author found a proof of Holder regularity of harmonic maps from certain singular spaces to nonpositively curved Alexandrov-space (see section 3 below). It is motivated by L. Caffarelli's proof of C2'a estimate due first to C. Evans on certain fully nonlinear elliptic equations of Monge-Ampere type. A very similar result was shown indepen­ dently by Jost 9 in a recent preprint. We shall also mention an interesting work done by J.Y. Chen2 on a related question. The present paper is written as follows. In section 2 below, we describe three basic examples that motivate our study and the statements of some results, we present a simple version of Holder regularity theorem for harmonic maps from a singular domain to a singular target in the section 3. Various applications are also stated, in particular, the results concerning the first two basic examples. In the final section we look at harmonic maps from nonnegatively curved Alexandrov spaces to either smooth Riemannian manifolds or nonpositively curved Alexandrov spaces.

116

2

Basic Problems and Main Results

Here we describe a few basic problems that motivate us to look at mapping problems from a singular domain to a singular or smooth target. E x a m p l e 1. (Liquid crystal d r o p l e t s ) The study of polymer dispersed liquid crystals that used in many display devices leads one to investigate the basic behavior of liquid crystal droplets. A simple mathe­ matical problem can be formulated as follows. Let ft C R 3 be a region occupied by the material which is a mixture of nematic liquid crystal drops and polymer like matrix. Let A be the region consisting of these uniaxial nematics. We want to find a solution, (Q, A), of the variational problem:

min / .f{Q,VQ)dx + aPa{A)

(2.1)

Ln such that Q\an = Qo,

Voi{A) = 0 Vol(Q),

0 < /3 < 1,

(2.2)

where a > 0 is a constant, Pn{A) denotes the perimeter of A inside A (i.e., the Hausdorff 2-dimensional measure of the boundary dA inside ft). Here, according to the theory of De Genne and Landau, Q(x) is a 3 x 3 symmetric, traceless matrix, for each x 6 ft, which represents the order parameter of the liquid crystal. It is easy to see that all such matrices form a linear space of dimension five. It can be also viewed as, by polar decomposition of matrices, double cone over S3/V. Here D is an eight element group which corresponding to all reflections with respect to the coordinate-planes in R 3 . Since A represents a part of regions occupied by the uniaxial nematic drops, the values of Q on A have natural constraints. We let E be the subset of all such 3 x 3 traceless, symmetric matrices that they have at least two equal eigenvalues. An easy exercise shows that E is a three dimensional algebraic variety. In fact, it can be identified as a cone over R P 2 . In particular, E is a three dimensional area-minimizing submanifold in the space of all such matrices ( ~ R 6 ). A typical choice of the integrand f{Q, V Q ) is given by /(Q,VQ)

=

| | V Q | 2 + lnQafuQm + L2Qa(l,yQcr,J>

+ ^{Q2)

(2.3) ~ g M Q 3 ) + ^tr(Q4).

Thus, in general, the problem Eq. 2.1-Eq. 2.3 involves a mapping problem to a singular algebraic variety with a free boundary. The singularity of base domain is coming from the possible singularity of the free boundary. We note, however E is nonpositively curved. The problem Eq. 2.1-Eq. 2.3 is thoroughly investigated in 12 ,

117 and hence we shall not discuss any further here. Instead, we simply state the following result proved in 1 2 . T h e o r e m A . Let (Q,A) be a solution of Eq. 2.1-Eq. 2.3. continuous in ft. The boundary, dA, of A inside ft is smooth.

Then Q is Lipschitz

E x a m p l e 2. ( A n O p t i m a l D e s i g n P r o b l e m I n v o l v i n g M a p s ) We let ft C R n be a bounded, smooth domain, and A is an unknown subset of ft with dA = f U E . Here T C ft, and E C 3ft which is also given. One considers

min

/ {l ( + XA)\Vu\2dx

+

aPn(A

(2.4)

Ln si where a > 0 is a constant, \A denotes the characteristic function of the set A, and u is a map from ft into a compact, Riemannian manifold M. Here u satisfies the boundary condition u|gn = iio, for a smooth map uo- <9ft —> M. The problem Eq. 2.4 for the unknown map u can be viewed as a mapping problem from a singular domain to a smooth target. The singularity of the domain metric is due to the discontinuity of the function 1 + \A >n ft- The discontinuity can be taken place on an unknown free boundary T, which, a priori, can be very singular. Here we shall show, in section 3 below, the following. T h e o r e m B . Let (A,u) be a solution of Eq. 2.4 subject to the Dirichlet boundary condition described above. Then u is C 1 ' 2 near V = dA n ft. We shall refer to 1 2 for the further regularity of u and the regularity of F. We shall also note that the existence of minimal solutions of the variational problem Eq. 2.4, and also the problem Eq. 2.1-Eq. 2.3 is obvious. E x a m p l e 3 . ( M a p s f r o m A l e x a n d r o v s p a c e w i t h curvature b o u n d e d f r o m below) Let us recall the definition and some basic properties of Alexandrov spaces with curvature bounded from below. For more detailed discussions we refer t o 1 , 1 4 and 1 3 . Let X be a complete locally compact length space, i.e., a complete locally compact metric space such that any two points p, q g A' is joined by a minimal segment whose length is equal to the distance |p<j| between p and q, where the length of a continuous

curve c: [a, b] —> X is defined to be sup < £] l c (*t) c (^«+i)l : a = to < t% < ■ ■ ■ < tm = b> For p, q £ X we denote by pq a minimal segment joining p and q. We now fix a number k 6 R. For simplicity, we call a complete simply connected surface of constant curvature k a fc-plane. For any triple points p,q,r in A', we denote Apqr a triangle Apqr in a fc-plane such that \pq\ = \pq\, \qf\ = \qr\, and \fp\ = \rp\. Denote by 0. By X is of curvature > k if the following axiom holds. T h e A l e x a n d r o v c o n v e x i t y : For any minimal segments px and py emanating from a common point p, the angle <(x(s)py(t) is monotone nonincreasing in s,( > 0,

118 where x(s) (resp. y(t)) denotes the point on px (resp. py) whose distance from p is s (resp. t). This statement is equivalent to the following. Take any triangle Apqr in X and any points x £ pq and y 6 pr, where pq and pr are arbitrary fixed. Then there exists a triangle Apqr = Apqf in the fc-plane such that if we take the two points x £ pq and y € pr with \px\ = \px\ and \py\ = \py\, then we have \xy\ > |sy|. Let X be of curvature > fc and fix two minimal segments px and py. The Alexandrov convexity implies the existence of the limit of <x(s)py(t) as s,t —> 0, which is called the angle < xpy. We have < xpy > <xpy, which is an analogue of Toponogov's comparison theorem for Riemannian manifold and which is called Toponogov's convexity. One of very fundamental analytic facts concerning these metric spaces is the socalled (n,<5)-burst points, p 6 X is called an (n,«5)-burst point if these are n pairs a,, bi, i = 1 , 2 , . . . , « of points (different from p) such that TT — 5,


> n/2 — 5,

(i ^

j),

and TT/2 - 5,

- - S,

(i ^ j ) .

Let X„:,5 be the set of (n, (J)-burst points in X , then it was shown in 1 that the Hausdorff dimension of X is equal to the maximal number of n such that Xnjs ^ <j> for sufficiently small 5 > 0, and that „Yn,j is open, dense in X, and is an n-dimensional topological manifold where n = dimjy X (the Hausdorff dimension of X) and J > 0 is small enough. Moreover, if 8 < ~ , then there is a bi-Lipschitz map from a neighborhood p, Up, m X onto a unit ball in R". Following 1 , we say a point p 6 X is nonsingular if it is an (n,5)-burst point for any S > 0. The set of singular points is denoted by SxIt was shown in 1 3 that: if X is an n-dimensional Alexandrov space, then the set Sx of singular points in X is of Hausdorff dimension < n — 1. Moreover, there is a C°-Riemannian structure on X — Sx satisfying the following: (1) There exists an A'o C X — Sx such that X — X0 is of n-dimensional Hausdorff measure zero and the Riemannian structure is C 1 / , 2 -continuous on A'o C X. (2) The metric structure on X — Sx induced from the Riemannian structure co­ incides with the original metric on X. We also note that (cf. 1 ) X — A \ j is of topological dimension < fc — 1 for any 0 < fc < n. Also the set of singular nonboundary points is of topological dimension (and also Hausdorff dimension) < n — 2. Let M(n, fc, D, V0) denote the metric space (with Hausdorff metric) consisting all those Alexandrov spaces (these length spaces described above) with curvature > fc, diameter < D and dimension < n. T h e n the space M(n, fc, D, V0) is compact with respect to the Hausdorff metric.

119

Another fundamental analytic property of an Alexandrov space X G M(n,k, D, V0) is that at each point p € X, there is an unique tangent cone Cp at p (cf.1). To be precise, for a point p £ X, we let EJ, be the set of equivalent classes of minimal segments emanating from p, where px and py are called equivalent if < xpy = 0, i.e., one of px and py is contained in the other. The set T,'p has the distance function naturally induced from the angle between minimal segments from p. We call the com­ pletion of E p the space of directions E p . It is shown in1 that E p is compact. If X is nonnegatively curved, then curvature of E p > 1. Cp is the cone over E p . Moreover, dimCp = n = dimX, and Cp is simply the limit (in the Hausdorff metric) of the family of rescaled space (Xp, Xg), X —y oo, centered at p. A remarkable result proved in14 is that Cp is homeomorphic to a neighborhood Up of p in X. Our main results in section 4 concerning energy minimizing maps from a domain in an Alexandrov space of curvature bounded from below to various targets can be summarized as follows. Theorem C. Let 0 be a open domain in an Alexandrov space X G M(n,k, D, V0), and let W be a smooth, compact Riemannian manifold. Suppose u is an energy minimizing map, then u is Holder continuous in fi away from a relatively closed subset of Hausdorff dimension < n — 3. Theorem D. Let u, X, and 0 be as in Theorem C. Suppose TV is a complete, locally compact Alexandrov space of nonpositive curvature. Then u is Holder continuous in 0. We conjecture that the Holder continuity can be improved to the Lipschitz conti­ nuity in both Theorem C and Theorem D.

3

Holder Continuity of Minimizing Maps

Let us first consider a harmonic map u from a unit ball B\ in R" into a smooth, complete locally compact manifold A^ with nonpositive Riemannian curvature. Such u is automatically energy-minimizing in its homotopy class (cf.10). Theorem 3.1 Let u be as above, then \\U\\aa{Ihl2)
120 Note that if d i a m ( f / ( S i ) ) = 2DU p0 = U(0). We let Ni be the manifold obtained from N by a simple scaling around p0 £ N by a, factor D\, the scaled map U\: B\ —► A] is a harmonic map with diam U\{B\) = 2. Also we note that if U: B\ C R " —)■ A" is a harmonic map, then [/ r : B\ -^ N \s also a harmonic map. Here Ur(x) = U(rx). Theorem 3.1 will follow from Lemma 3.2 by the two natural scalings. We also need the following well-known fact: (cf. 6 ). in B\ C R n ,

L e m m a 3.3 Let f > 0 be a super harmonic function

inf f>C(n) 1/2

I

then

fdx.

R

Since N is negatively curved, for any p € N, consider gp(x) = dist;v(p, u(x)), a harmonic map u: B\ —> N, then A5p(i) > 0

in

(cf. 10 ).

Bx

for

(3.4)

We now prove the main Lemma 3.2. For this we need the following: L e m m a 3.5 Under the hypothesis of Theorem 3.1, assume that 1 < diam U{B{) < 2 and that U(Bi) is covered by M balls Bl,. . . , BM of radius e in N and e < £o (here EQ > 0 will be chosen later). Then U(Bi/2) is covered by M — 1 balls among B\...,BM. Proof. For i = 1 , . . . , M , we take as,- £ B\ such that B1 C B2s(pi), pi = Ufa). Hence by taking e 0 such that 2e < lea < j ^ , we have that Bi/is(p,), i = 1 , . . . ,M cover U(Bi). Since U(Bi) has diameter at most 2, every p, belongs to a closed ball B of radius 2 in N. Let M ' be the maximal number of points in the ball B such that the distance between any two of them is at least ~ . Note that M' < C(N) by the locally compactness of N. Thus we may assume that Si/s(pi), i = 1, ■ ■ ■ , M' cover [ / ( S i ) . It follows that there is one p;, say p\, such that

|[/- I {B 1 / 8 (p 1 )}nB 1 / 2 |>r ? (yv)>o. We consider gn(x) Set fPl{x)

= d\stN(pl,U(x)),

= K - gpi(x)

and let K = sup gpi(x)

> 0 in Bi and Afpi(x)

inf fPl(x)

"1/2

> c(n)

< 0 in flj. Therefore, by Lemma 3.3,

I f„(x)dx B

< d i a m ( f / ( B i ) ) < 2.

> c(n) > 0.

J l/2

The last inequality is true because dimU(B}) /Pl(x)>ion[/-]{fi1/8(p1)}nB1/2.

> 1, which implies K > - and hence

121 Suppose U(Bi/2)

n B2e(pi) ± 4> for all i, since {B 2£ (p;)}£li covers U{Bi),

we have

inf //»„((**))<< & 2e.. It will be a contradiction if we choose e 0 so small that 2e0 < c(r/). Lemma 3.5.

This proves

P r o o f of L e m m a 3 . 2 . Since diam U(Bi) = 2, we can cover U{B{] by M0 balls (M 0 < C(N)) of radius e 0 in A^ Lemma 3.5 says that U(BIH) can be covered by N — 1 balls of radius e 0 . Suppose dia,m(U(Bi/2)) > 1, we repeat above arguments for V(x) = Ul^xj, we conclude that U(Bi/4) is covered by N — 2 balls, etc. It follows that there is a number k < M0 such that diam U(B2-k) < 1. In particular, diam U(B2-M0) < 1. We thus can take S0 = 2~M° is Lemma 3.2. (Q.E.D.) R e m a r k 3.6 We note that the proof given above works whenever one has the estimate in Lemma 3.3, the fact Eq. 3.4 and the local compactness of N. We also note that, by Eq. 3-4, one has 1

The later can be used to relax the assumption that U(B\) has bounded diameter to something much weak (say u £ H1(Bi, N) when the space fP(BuN) is properly defined). R e m a r k 3.8 By Remark 3.6 above, we conclude that if N is a complete, locally compact Alexandrov space of nonpositive curvature, the conclusion of the Theorem 3.1 valids whenever the domain is one of the followmgs: (i) A ball B\ C K" with metric g{x) — gij(x)dx% ® dx', such that

A|{I2 < &(*)&

< jiff,

for a constant X £ (0,1), x € B\, and f £ R"; (ii) A ball B\ in an area-minimizing

hypersurface

in R";

(Hi) A ball B\ in an complete Riemannian manifold M with Ricci > 0 and VOI(BR) > c0Rn. Here V/? > 0, BR denotes geodesic balls of radius R in M; (iv) A ball B\ in a Hesseinberg group, or more general situation

as in ;

122 The reason is that in all these cases, the domain has both Poincare and Sobolev inequalities. The conclusion of Lemma 3.3 is true by 6 . Next we consider an energy-minimizing map u: B\ —¥ N. Here B\ C R n endowed with a metric as in Remark 3.8, (i), and N is a smooth compact Riemannian manifold. By 1 6 one has T h e o r e m 3.9 Let u be as above, and suppose

that

,-,/ % r a, w - du du , E(u) -dx<e B ( «=) j-9^x) /* {*)^^< * ,0, B,

for a positive constant e0 = e0{n, X, N) G (0,1). Then ||u||c»(B, /2 ) < C(a,n,X,N), for some positive constant C and a G (0,1). As a consequence of Theorem S.9, one has also that Hn~2(sing u) = 0. Here Hn~2 denotes (n — 2)-dimensional Hausdorff measure and, sing(u) denotes those x £ B] that u is not Holder continuous in neigh­ borhood of x. Suppose TV is also simply connected, then it follows from 8 that J/

C{X,n,N) ] C ) gi(x)u ' (x)u (x)u udx< ax <^ — j J — ! jg' -T, Ximi xr X]■dxux^"'* ut-u

r,

(3.10)

B Brio) r (0)

for r 6 (0,1) and for an energy-minimizing map it. Let T = {E: E C sing(u) n S i , and E is relatively compact in S i } . Then, it follows from Eq. 3.10 that J- is compact with respect to the Hausdorff metric. Moreover, Hn~2{E) = 0 for E € T. We also note that if x0 £ Bu 0 < p < 1 - | s 0 | , then the set EXOiP = " fl S i G T whenever E G F. From these properties one can show the following (cf. 18 ). C o r o l l a r y 3 . 1 1 Let u, N be as above, and suppose N is simply there is 5{N) > 0 such that H"-2-s(sing u) = 0.

connected.

Then

Finally we want to sketch the proof of Theorem B stated in section 2. Let (A, u) be a solution of Eq. 2.4. Then u is a minimizing map from B\ into M. The energy functional u minimizes is / (1 + xA)\Vu\2{x)dx. Thus Theorem 3.9 and Corollary 3.11 apply to the present situation. However, it does not yield the desired conclusion in Theorem B. For this we using the minimizing property of (A,u). First of all, if x € F, and BR(X0) C fi, then

(a)

jJ

2

n

^|v«| (x)(fx + ff"-'(rHnn-lB crn-\-\ XA\Vu\\x)dx (TC\B (x0< ))
B,-(i 0 )

for a l l O < r < R, C = \dBx\.

123

The statement (a) follows by a simple comparison that keeping u and changing A to A = A ~ Br(x0). Next, we have, by12, that mm{Ln{Ar\Br(x0)),

(b)

Ln(Ac <~\ Br(x0)} > corn}

for some constant Co > 0. Here L"-denotes the Lebesgue measure. We obtain also, by a standard argument, that

, / (i + xx)|Vu|Vx + //"-1(rnsr(3;o)) d Br(xo)

(c)

rn~2

dr

s and that

^/* w

(d)

J/

(i+xx)

,fcr re(o,R)!

i^r "-

2

2 (1 + XA)\Vu\ X/OlVul (1 dx^ < A

Br(x0) Br(*o)

+^

2

+ XXA )\Vu\2 dx (l + A)|V«|

jJ B 2 r (l (i 0 )

/

|\u-u\ u - u2dx, |2^,

for

A 6 (0,1),

B S2r(l0> 2r (io)

and r G (0, .ft/2). Here u = -fB2r(x0)u(x)^xIt is rather easy to deduce from (a), (b), (c) and (d) that the small energy hypoth­ esis of Theorem 3.9 satisfied at each point x0 6 T. Thus u is Holder continuous near T. The C'^-property of u follows from (a), (b) and Morrey's Lemma. We refer to 12 for related discussions.

4

Maps from Alexandrov Spaces

Let X E M(n, k, D, V0), then for any 5 6 (0, ~A, Xn,s is an open dense subset of X with the Hausdorff dimension of X — Xn>s < n — 2. Moreover, for each p £ Xnj, there is an open neighborhood Uv around p in X such that Up is bi-Lipschitz to an unit ball Bi in R™. Since there is a natural metric defined on X — Sx, (cf. section 2), we see Up can be identified, under bi-Lipschitz map, with B\ C R" endowed with a Riemannian metric g[x) = gtJ(x)dxi ® dxj. Moreover (gij(x)) satisfies A / < (fl«(x)) (0«(x)) < j / ,

Vx€fii,

for some A £ (0,1) (note A may depend on p). Thus for a map u from Xnj to any smooth compact Riemannian manifold N or an Alexandrov space of nonpositive curvature, its energy is well-defined (cf. also10 and 9 ).

124 On the other hand, since the HausdorfF dimension of Sx is not larger than n — 1, for a Lipschitz map from X into A', its energy density and its total energy can be easily defined. Let n be an open set in X, we define / / ' ( f t , N) to be the completion of Lipschitz maps from ft into N with respect to the energy norm:

ll/IU'taw) Wf\\mn,N)||/||ff'(tVO = Wfhun.N) ll/lk2(fi,A0 + + ||V/||i,2(n,N)Here V is with respect to the metric which is well-defined on X — Sx- Since / is Lipschitz, thus |[V/]|x^(o,/v) will be defined. In order to show Hx{Vl,N) has the usual properties as for Sobolev - maps. We need the fact that Cp (the tangent cone of X at p) is homeomorphic to Up, an open neighborhood of p in Ar, see 14 . Indeed, it follows from the Bishop-Gromov's vol­ ume comparison theorem (see 1 ), the compactness of the span M(k,n, D, V0) and the work 1 5 . L e m m a 4.1 ( P o i n c a r e ' s i n e q u a l i t y ) Suppose f BR C ft, one has

6 H1(il,N),

then, for any ball

n-2 n-2

i ^ j \J\!{x)-f [j^ m - f RR\^dv\ \ ^ d v

\

1I

2


j\Vf\ dv\

As a consequence, //'(ft,./V) is compactly embedded in L• (il,N), volume form on X,fn is the average of f over BR.

.

Here dv is the

One has also the Sobolev's inequality: L e m m a 4.2 Suppose f 6 / / ' ( f t , R) with f = 0 on 3ft, then ||/||£W»-»(n) 11/llLW-j(n) 0, such that - ^ r / \Vu\2{x)dv < e 0 (p), and r 2 ~" / | V u | 2 ( x ) A ; -> 0 as 0

S.O(P)

Br(p)

r -)• 0 + . If p, in addition, belongs to Xn,s for a suitable small S, then if r0 small enough, one can have e 0 (p) so small that Theorem 3.9 applied to u defined on Br (p).

125 Thus u is C ° , ( 5 r o / 2 ( p ) ) . Since d\mH{X - Xn,s) < n - 2, we see u is Holder continuous away from a set of the Hausdorff dimension < n — 2. To complete the proof of Theorem C, we need the fact that at each p e X, then in an unique tangent cone, Cp, of X at p. Moreover, by Lemma 4.1, Lemma 4.2, one can easily adapt the dimension-reduction argument as in 1 7 , to show that u is Holder continuous in fi\(sing u). Here sing(u) is a relatively closed subset of fi with dim f f (sing u) < n - 3. (Q.E.D.) The proof of Theorem D is more or less contained in the proof of Theorem 3.1, see Remark 3.8.

References [1] Y. Burago, M. Gromov and G. Perelman, A.D. Alexandrov bounded below, Russian Math. Survey 4 7 (1992), 1-58.

spaces with

curvature

[2] J.Y. Chen, Doctoral Thesis at the Stanford University (1991). [3] J. Eells and Sampson, Harmonic Math. 8 6 (1964), 109-160.

mappings

of Riemannian

manifolds,

Amer. J.

[4] N. Garofalo and D. M. Nieu, Isoperimetric and Sobolev inequalities for Caratheodory spaces and the existence of minimal surfaces, preprint. [5] M. Giaquinta and E. Giusti, The singular set of minima of certain functionals, Ann. Scoula Norm. Sup. Pisa, 11 (1984), 45-55.

Carnot-

quadratic

[6] D. Gilbarg and N. Trudinger, Elliptic equations of second order, Springer-Verlag, Berlin (1987). [7] M. Gromov and R. Schoen, Harmonic maps into singular spaces andp-adic superrigidity for lattices in groups of rank one, Publ. Math. IHES, 76 (1992), 165-246. [8] R. Hardt and F.H. Lin, Mapping minimizing 4 0 (1987), 555-588. [9] J. Jost, Generalized harmonic

the Lv norm of the gradient,

maps between metric spaces, preprint (1995).

[10] N. Korevaar and R. Schoen, Sobolev spaces and harmonic targets, Comm. Anal. Geom., 1 (1993), 511-569. [11] F.H. Lin, Asymptotically preprint (1994).

CPAM,

conic elliptic operators

[12] F.H. Lin, Variational problems with free interfaces

maps for metric

and Liouville-type

II, to exist.

space

theorems,

126 [13] Y. Otsu and T. Shioya, The Riemannian structure of Alexandrov spaces, J. DifF. Geom., 39 (1994), 629-658. [14] G. Perelman, Alexandrov's spaces with curvature bounded from below II, preprint. [15] S. Semmes, Find curves on general metric spaces through quantitative topology, with Applications for Sobolev and Poincare inequalities, preprint (1995). [16] Y. Shi, Regularity of harmonic maps from a singular domain, to appear in Comm. in Anal, and Geometry. [17] R. Schoen and K. Uhlenbeck, A regularity theory for harmonic maps, J. Diff. Geom., 17 (1982), 307-336. [18] B. White, Geometry measure theory and the calculus of variations, Proceedings of Symposium Pure Math., Vol. 44, AMS (1986).

127

SUPERCLOSE AND SUPERCONVERGENCE FOR TRIANGULAR ELEMENTS

Qun Lin Ningning Yan and Junming Zhou Institute of Systems Science, Academia Sinica, Beijing, 100080 , China ABSTRACT In this paper, the finite element solution and the interpolation of the solution for the Laplace equation on an isosceles right triangular mesh are discussed. By means of the element analysis, the superclose and the superconvergence property are given, and the better error order of the gradient can be achieved as the results.

§ 1 Introduction The purpose of finite element methods is nothing but to solve partial dif­ ferential equations, which requires the methods to be accurate and economic and easy to program. This paper concentrates on the error order of the gra­ dient for the Laplace equation. Let Q be a rectangular domain, u the exact solution defined on n, u£ the finite element solution of degree—p defined on an isosceles right triangulation T h over Q, and uj, the finite element interpolation of degree-p defined on Th. We have the superclose property as follows: 0(h p + )||u|| h

u —u'

=• 0 ( h

P+

)||u||

2, +2

p=l m>

(Lagrange element);

p= 2

(Lagrange element);

1i

l

r^,i P + TNII n l ■...1 _ a O(h * ) l l uII| | p + 2 1 ( , p = 3-. (Hermite element); where || • || is the standard norm of the Soblev space. By an interpolation postprocessing, we get 00(h ( h p + )||u|| a ,, p = l (Lagrange element); ,+ 1 h p+ llrr' n + u _ u II (Lagrange element); element); | |M j ="■° (0h( Ph+ ' )'!)||u| u| ||p| + 2+ 2 ^ a, , p p==2 2 (Lagrange I| |I L 2h ' p p

l

where n 2 h

•■0(h O(h PP + 2)||u|| + 2 ma , p = 33 (Hermite element); means certain interpolation explained in § 4.

The result for p = 3 is new. In a paper of B.Li (1990) , he gave a counter — example for the normal cubic interpolation(including Hermite type) which is not superclose to the finite element solution. We here change the inte-

128

rpolation condition on the center of gravity and so we get a new interpolation superclose to the finite element solution.

§2

Integral identity (for p = 1,2) 1) P = 1 Let S be the piecewise linear triangular element space, SQ = S ( | H0(£2),

u eS be the piecewise linear interpolation function, then, we can prove the integral identity as follows. Lemma 2.1 V ( u - u 1 ) V v = 0(h 2 )||u|| 3 ||v|| 1 , Y v e s ' .

(2.1)

Preof. Let w = u —u , e, e, are ajacent elements in Cl as in Fig.l, with Z, = (0,0),Z2 = (h,0), Z 3 = (0,h) and Zy = (h, - h). Then *

r w v = v ( w + xx

x

v

r w)

x

x'

r r r c = v ( wdy — wdy + wdy — wdy) ( ',

',

h

'.

Let F(S) = ^ ( ( S - S 0 ) 2 - h 2 ) , where SQ is the midpoint of the segment 1 and 2h is the length of the 1, Then w v = x

x

(-F(x)u

y

x

'

+VTF(y)u try

w

'

)V )v

Fig.l

yyyT' yt' x

-oth 1 )IWI lilUii IMI uUlk . The similar result can be proved for the term j

w y vy .

•U»,

Noting that v = 0 on the boundary s£2, 2.1 follows. 2) p = 2 Lemma 2.2 Let S be the quadratic triangular element space, u eS be the interpolation of u satisfying u I (Z.) = u(Z.) and

i=l,2,3 ,

129

u ds =

1=1,2,3 ,

uds,

where Z. and 1. are the node points and the edges of the element, then J V ( u - u I ) V v = O(h 3 )||u|| 4 i 0 o ||v|| 1 , V v e S ' .

(2.2)

0

Proof

Let Iv be the piecewise linear interpolation of v, then, for all

V6S\ 3

v = Iv + v - I v = I v --£ Ihh2 v' vu < p . + 3J ,, B?

(2.3)

ii -- li

where 2h. is the length of 1., v u is the derivative of order 2 on the tangent direction, and cp. + 3 is the basic function on the midpoint of 1.. Bv the definition of the interpolation, it is easv to see that r r p r (2.4) V ( u - u ' ) V ( I v ) = V w V ( I v ) = w ^ ^ d s - wA(Iv) = 0 . e

So

e

e

On e and e, (see Fig 1), noting that q>6 = 0, on 1 , 12 and
w. +

w h.v
x 3 xx^6i

=

-

h

3

V

i 3

Jw.»,

xx(j

W

xx

<

P6+

a

r

n

Wx^').

where 4 4
J J

xx
W W

J J

i x «e=A V=4 **

J J

2 w WxxijxX((xx--hh)) 2dd xx

u My x(x + y - h ) 2 - —

+—

+ — u „ y ( x - h ) ( x + y) 2 h J c.

wn(x-h)x2dx

130

2 r

2 r

= — w x(x - h)( - h)dx - — j - u h J 3h J '. '. - — r 2-

u

3h J " "

x(x + y - h ) 3 + — T -2

u

3 y

3h J " y

x ( x - h ) dx

( x - h ) x dx

h

- —r

u

2

(x — h)(x + y)

xxyyv

•"

J

5ti

= - |

w[x(x-h)] ffl dx + ^

u„yx(x-h)(2x-h)dx

', - A 3h

'. 5

u^xfr + y - h ) J

- - ^ u „ y y ( x - h ) ( x + y) 5 3n J

= °—k u - y W x - h )] 2dx -rrr u„„x(x + y- h ) 3 -311 J

in

J

h u

-—T

3h 2 J

w

Note that h } = w h,v x

3

( x - h ) ( x + y)

,

(p, + xx

x

w h v
6x

h

x

2,

=—

u

19

.

3

xx

. . 2

6x

.

1

x (x — h) v dx + 7 x**y

'

xx

g

,.3

.

u xxyy

x(x + y — h) v v

J

'

xx

'i

1 f +7 u g

3 ( x - h ) ( x + y) v

xxyyv

' *

J

'

xx

= 0(h4)||u||4J|v||2MJe| =0(h3)||u||4iJ|v||^Ue[ . (2.5) Similar results can be proved for other terms in (2.3). Hence (2.2) follows from (2.3) —(2.5) by noting that v = 0 on the boundary sQ.

§ 3 Integral identity for Hermite cubic triangular element

131

Let S

be the Hermite cubic triangular element space, that is

S*={vec(fi); are continuous on Let u eS be u(Z.) = u I (Z i ) and

v\teP3(e), V e e r * , vx and vy the vertices}. the interpolation of u satisfying , ux{Z.) = u\(Z.) , u y (Z,) = uJ(Z.) 1 = 1,2,3 , V(u-uI)V(A1A2/l3) = 0

(3.1)

I

where Z. , i = 1,2,3, are vertices of element e, X., i = 1,2,3, are area c o o r d i n a t e s of t r i a n g u l a r e l e m e n t e. In t h e e l e m e n t e, any v eS can be written into 3

3

3

v = v0 fl»a , i-1

3V

i-1

(3.2)

i-1

3V

where v. = v(Z ) , v. = — (Z.) , v. = — (Z ), cp.„, a., (p.. are their basis functions, v 0 is the value of v at the center of gravity of e,
V(u-u')Vv= Q

VwVv , Q

where w = u —u , veS , we prove some lemmas as follows. Lemma 3.1 Let e, ej are elements in Q as in Fig.l, then I,=

w(2x + y - h ) +

w(2x + y - h ) = O(h 8 )||u|| 5 0 0 ;

I2=

w(x + 2 y - h ) +

w(x + 2y) = 0(h 8 )||u|| 5 o o ;

(3.3)

(3.4)

I 3 = | w ( x + y)+

w(x + y - h ) = 0 ( h 8 ) | | u | | J m .

Proof. Let E(x,y) = - x(x + y - h), then E ^ = 6 E x = 3 ( 2 x + yy--hh)) .

(3.5)

132 2

2

Note E

and E i are zero on \} and 1 2 , integral by part

rf w(2x +y~h) *

1r»

1 f 2 wE w(2x +y~h) = = -wExxx xxx a

= xx (I — (I — — )E )Eu wdy wdy — — --

33 J i,

=g = 7 ((

rc

m E w E 3j w «" 3j

J » i,

rr

--

.,

2 )) w w (( y y -- h h )) 2 d dy y -- — —

rr

i,

2

2

w mM xX 2(( xX++ yy -- hh )) 2 .. W

(3.6)

•

In the same way w(2x + y - h)

=g g((

-

)w(y + h ) 2 d y - —

W w jM ( X+ + y)2(x-h)2 . J M(x

Let F(y) = | ( ( y - y , ) 2 - h 2 ) , where y, is the midpoint of 1, and 2h, is the length of 1. Noting w(Z.) = w i (Z.) = w y (Z.) = 0, integral by part, w(y + h ) 2 d y = '.

w(y-y,+-)dy »J

=

w f C y - y ^ ' + h C y - y ^ + h^dy

==

wlI4[- 5r^r FFyyyy + + (v(577 + + ll)h, - 77 F Fyyyy + + hh9-^-0-FFwy»ldy Jldy J w ') hi, - g l 45 yyyy v 5 ' 1 6 yyyy 90 yyyyyJ J

1 3 3

h r

= = i_

'.

. 1 „3

1

2 h "l

2

2 ,

. 1 „ l F 2)dy , u F3 + " —uyyyy(77 (77 F )dy -45 F + -— 5 yyyy 45 5

1

2

h r

1

3

„3,

h , . —uyyyyy „F F3dy dy —u . 90 90 yyyyy 1

'■

In the same way. P P h2 2 P p r h p 22 33 2)dy + ~ w ( y h ) d y = u ( 7 F + rr -- F u F33 dy . yyyy ( - 7 F w ( y h ) d y = u + F ) d y + ^ r J " J yyyy 4 5 5 9 0 uyyyyy F dy ■y* J " J 45 5 90 nrn

(3.7)

133 Hence,

\W(y~

\w(y-h)2dy-

\w(y + hfdy

'.

'.

=

(TF^3+~rF2)u 45 5

J

= 0(A 8 )ll«ll 5 i C O • In the same way, * C

w ( y - h ) 22 d y -

',

dxdy-£r( 90

yyyy

<

+\)u I

F*dy

yyyyy

(3.8) w(y + h 22 )dy = 0(h 88 )||u|| 55 oo oo .

(3.9)

i.

Note that x + y —h = 0, on 1, and x + y = 0, on l 4 , Integral by part, 2

w xxx

e e

2

2

x (x + y -—hh) ) + \

i

J 2

1

= — T-

u

O 1

xxxy xxxy

w

jmj \

e, e,

/

j t

.J

1

t

3*i

, .4

1

x (x + y — h) — \

J

2

(x + y) (x — h) \

>

,

u

XJUty

. .2

.3,

(x + y) (x — h)

jumy \

j i

\

>

h x 22 ( x - h ) 22 w rr aa d x

++

* u

1

=— ]2

2,

xxxyy

'

4F 2 (x)F'(x)u

-7 f.

I

c

x N(x + Jy — h) + —

\ /

\ /

u

]2

o

1.

.4,

Ji

, .2

\

i

( A F ' W M -§h2F(x))w

dx + ^ h jjjy

.

(x + y) (x - h)

xxxyyv

v /

MC

xx

^

I

v

''

dx xxx

1,

=— 12

u xxxyyv(x + yJ i) "\ ( x - h )t + — 12 xxxyyv

12

2,

~3

.

+-h Fu Q

t

4 ,

12

1 1

„3

45

u ^"yy x 2 (x + y - h ) * ^"yy

.

dx + —rh —r h F u xxxxy

8

Jr \

dx xxxxx

(3.10) = 0(h )||u||5m . Hence (3.3) can be proved from (3.6) —(3.10). (3.4) can be proved by the similar way. For (3.5), Noting the interpolation condition 3.1,integral by part,

134

0=

VwV(A A A ) =

= -V(

h

J

-

J

w— (A A2A ) d s -

) w [ x y ( h - x - y ) ] dy + ^ - (

h

J

wA(A,A 2 A 3 )

-

J

)w[xy(h - x - y)] dx

- \ (-2x-2y)w. h J Hence w(x + y) = - (

-

)w[xy(h-x-y)]idy >i

h

+ ~(

-

)w[xy(h-x-y)]ydx .

',

(3.11)

>i

In the same way, *

c

w(x + y - h ) = - (

r

i>

+ \(

)w[y(x-h)(x + y)] i dy '<

-

)w[y(x-h)(x + y)]ydx .

(3.12)

w[y(x-h)(x + y)]ydx = 0 ;

(3.13)

'.

>J

Note that w[xy(h-x-y)]ydx + h

h w[y(x — h)(x + y)] % dy =

W

= 45 77

wy(y + h)dy = 4 wF(y)dy

( F 3BJI - 3 6 h 16 ^ F 2w* )dy

>> =

2

3

4

h!!

u yyyy (-=rF — 15 ~ F 45

In the same way ,

2

)dy ' J .

135 w[xy(h — x - y)] I dy= 1

wy(y — h)dy j.

4h

f 2 3 1 yyyyv45

f 2 15 '

J

i

Hence w[y(x — h)(x + y)] i dy +

w[xy(h-x-y)]xdy

i

f

r2 F 3 - ^ h f F 2 )

yyyy' v ^ g

= 0(hS)||u||,aj .

(3.14)

•U«, In the same way, w[xy(h — x — y)] x dy + 1

w[xy(h - x - y)] y dx 1,

+ wtyfc-fcXx + y ^ d y H- w[y(x-h)(x + y)]ydx = 0(h 8 )||u|| 5 > 0 Then (3.5) can be proved from (3.11)-(3.15). Lemma 3.2

(3.15)

w

p 1 i +
(3.16)

> u +
(3.17)

f w y ( V l l + « » „ ) , =O(h 6 )||u|| J i 0 0 ;

(3.18)

z

w

I(

<

eU«,

*

«U

wy(*12+«>„), "OOi^llull^ . •u«, Proof. In e, 2 <£11 = -V[x(h-- x - y) - xy(h - x - y)] , h

(3.19)

136 1

2

"Pn

P 2 i = ^ [ x C h — x ) ( h - 2x - y)] , h 1

2

2

(
(<"» +

and in e , (P,=\[x(h11 h

x) 2 + 2y(x + y ) ( h - x)] ,

» a = - T W x + y)( h - x) - (x + y) 2 (h - x)] , h

So,


+ (

',

J

)w( < p 1 1 + ( p 2 1 ) i dy + ( 'i

'.

wC<55n +

.

'.

M#n+»„),dy '<

wh(h-y)dy

"i

=

w[— F L

'« =— 12

12

u

TOT

*Ht»

——F 6h

yyyw

F dv + — 6h

'.

ldy

uyyyyy F dv . i,

In the same way, i f 2 i f 3 w(

'. so,

|.

'.

(3.20)

137

(

JI

-

)w(«p„ + « . „ ) dy = ^r u F2-^-( + 12 I yvyy* 6h JI

J

].

L

*l I.

1_

)uyyyyy F 3 dy t

= 0(h6)||u||5m . In the same way,

(3.21)

(j - J ) w ( 9^,nn++<« p 2 1 ) iI d yy = 0(h66)||u|| 5oo (I 5oo .

(3.22)

\ M
= 4 ( v 1 - v 3 ) y ( y - h ) ] ;; ~~T t[wwxx( (v iv 1--vv22))xx ( x - h ) + w yy(v, h *.

(3.24)

e

j VwV(v](] (pp10 +v22

4 =— [ w i, ( v 1, - v 2 ) x ( x - h ) + w y ( v 3 , - v 2 ) y ( y - h ) ] . h JJ

(3.25)

c

Proof. In e, V

1 * 1 « ,

+ V

2 * » .

+ V

3 « ' M I

= 4 ( v 1, - v 2 ) x ( x - h ) - 2 7 ( 1 3 v 1 + 7 v 2 + 7 v 3 ) a i A 2 y i h and V

l « ',1100,,-+, -VV22( (iP' 22 0 , -+,V" V33«(PP30y 30y

= ) ( l A, ;^2^3) . 2A]) = 44 (( vv .1 -- vv 3! ))yy( (yy- h- h) -) 2- 27 7( (1133vv 1. ++77 vv 22+ 7+ v7 v3 3)( y • h Hence

,

138

VwV(vl(p10+v2(p20+v3(p30)

=

~ t w x( v i - v 2 ) x ( x - h ) + w y ( v i - v 3 ) y ( y - h ) l h J

-27(13v1+7v2+7v3)

VwV(ljA 2 Jl 3 ) .

Then, (3.24) follows by notice that the interpolation condition (3.2). And (3.25) can be proved similarly. Lemma 3.4 4 h

| w x ( v 1 - v 2 ) x ( x - h ) = 0(h6)||u||5oo||v||lMe . J

(3.26)

«U«,

P*oof. Integral by part, w x(x —h) =

w(y —h)ydy—

w(2x — h) ,

(3.27)

i,

and wxx(x-h)=-

wy(y + h ) d y -

■i

w(2x-h).

(3.28)

'.

By lemma 3.1 , w(2x-h) = | ( 3 I 1 - I 2 - I 3 ) = 0(h8)||u||5m . Note that w(Z.) = w t (Z.) = 0, where Z. is the end point of 1 , integral by part, W

J

w(y - h)ydy =

= -T=

J0

I

—j= T(T - V 2 h)dr 2V2

wF(i)dx

v2 J 1,

1

=

f , l c ! w( F

h

l

1_J , . F )dT

7lJ % ""-T'6 ""

(3.29)

139

', where h, = — h. In the same way, ^ wy(y + h)dy = ^ J u m T ( ^ F 3 ( T ) - | Q F 2 ( T ) ) d t . w(y - h)ydy i,

wy(y + h)dy i4

1

f

= -;L VT J

h2

1

(-^F 3 (x)-^-F 2 (x))dxdy U "™*v90 60

•U.,

= 0(h8)||u||^ . Hence (3.26) follows from (3.27)-(3.30) and V

(3.30)

— V I

2

- II

II

Remark : similar merger as in lemma 3.4 can be completed by combination of e and its left element e2(see Fig. 2), for the term w

v( v i - v 3 ) y ( y - h )

"\

T\ \

«

in (3.24) .

\]

Lemma 3.5

Fig.2

( •

w , ( 9 > u - * « ) , - \"M„

-ViX

=0(h 6 )|| U || 5 ^ ;

(3.31)

=i

™J
o

\

(3-32)

;

(3.33)

* , ( * « - Vn), -Jw,(

a ), = 0(h*)| |u||Sm ;

(3.34)

c3

140

where e 3 is a adjacent element showed in Fig. 3 . Proof It is easy to see that w

«(^ii — ^2i>x =

w

I(«'1i-

.

<

D"uH(x-x0,y-y0),

. f^4 ' ' P2i)x= D uH(x-x0,y-y0),

(t'oM

(*o,lto)

Fig.3 where D 4 u is derivative of order 4 of u, H(x,y) is a funciton satisfying

H(x,y) = 0(h 3 )

, if 0
By translation of coordinate system,

=

P 4 u(x,y) - D 4 u(x - h,y)]G(x - x 0 ,y - y 0 )

= 0(h)||u|| 5 m

|G(x-x0,y-y0)|=O(h6)||u||w

which proved (3.31). (3.32) — (3.34) can be proved similarly. By the aid of above lemmas, we can get our principal integral identity as follows. Lemma 3.6 | v ( u - u I ) V v = 0 ( h 4 ) | | u | | 5 c o ( | | v | | l f l + llvll,^) Q

= 0 ( h 3 S ) | | u | | ^ | | v | | i n , VveS*. Proof. By (3.2), for all veS , in element e 3

3

3

v = v0 fl»H I - 1

1 - 1

3

1 - 1

I = v0
+
V

^

J , K » „ -
+ (V2I-V3x)('P21-
+ (v3x - V ^ i - 9u) + (vly + v2y)((p12 + ^ ) + (vly-v2y)(^12-«)!.22) + (v2j+v3y)(922+
141

+ (v 3 y -v l j )(< i P 3 2 -

a

+ I llu-u'lLjIvll,^ = oa/)iiuii 5 . M Eiiv|| Uie + o(h 4 )iiuii 4a , i I M I = 0(h4)||u||^(||v||lfl + | | v | | 1 3 0 ) . Hence, theorem 3.1 follows by inverse inequality.

(3.35)

§ 4 Superclose and superconvergence In the section 2 and the section 3, it has been proved that j O r t u i y i v l l , , V v e S * " , p = l (Lagrange), j V { u - ^ ) V v = | o ( h 3 ) | | u | | < e o | | v | | j J V v e S * " , p = 2 (Lagrange),

(4.1)

^ ( h ' ^ l l u l l ^ l l v l l , , V v e S h - p , p = 3 (Hermite), where S is the finite element space of order p, u eS is the interpolation of u. Then it is easy to get the results of superclose as follows. Theorem 4.1 Let u , u eS '", p < 3 (when p = l , 2 , u , u eS 0 ' P ), ,P

then,' up and up is superclose, that is r

|Ofh 2 )||u|| 3 , p = l (Lagrange), | U p - u P || 1 J B = O(h 3 )||u|| 4 o o , p = 2 (Lagrange), 3i

Proof

^O(h )||u|| 5 o o ^, p = 3 (Hermite) . Because u h - u ' e S ^ S / " , when p = l , 2 ) , then by (4.1) p

p

U

h I h II h Ml2 ^r -upI ) < C V(uP -u P)V(u ' P II u P — up l l v

1-n

x

ii

-cj*V(u

-u^VCu*-^)

Q

- ■

'ortuiUK-Upll,^ P-i. O^INI^I^-u'JI^, p = 2, ^(h'- 5 )llu|| 3 , M , u —u UP

PII 1>Q

, p = 3,

(4.2)

142

which proved theorem 4.1. In order to get the global superconvergence, we are going to construct the interpolation operator n ^ eC(£2)->Sj h as follows. Let adjacent four elements to be a large element (see Fig.4), then, on any large element E as in Fig,4, p+i P + I . „ n2h uep (E), and for p = 1, n ^ u ( Z . ) = u(Z.) , i = l , - , 6 ; for p = 2, n ^ u ( Z . ) = u(Z.) , i = l , - , 6 , n2huds= i

uds , i = l , — , 4 ; i,

for p = 3 (Hermite type), n 4 2h u(Z.) = u(Z i ) , i = 1,2,3 , — - ^ - ( Z . ) = -(Z.), 3X

'

3X

a(n^u)

Fig.4

i=l,-,6 , ''

3U

5—(Z.) = —(Z.), i = 1,—,6 . ay ■ ay ' It is easy to see that l l n ^ u - u H ^ c h ' +'llull^ ,

(4.3)

and ||n;+,v||


(4.4)

Note that _p+l n 2h

U

h p -

U

TT P + 1 h TTP + 1 . T-tP+1 U + n 2b U p " n 2 h 2h T-.P+ 1 . h I . , _P+1 U = n2h ( U p - U p ) + n 2 h " =

n

U

U

U

~ '

,AC\ (4-5)

By (4.2) —(4.5), we can get the global superconvergence as follows. Theorem 4.2 fO(h )||u|| 3 , p = l(Lagrange), ||nL+'up-u||1=

O(h 3 )||u|| 4 o o , p = 2(Lagrange),

Lo(h 3 5 )||ui| S o o , p = 3(Hermite)„ Remark : When p = 3, we can only get the superconvergence of order p + - (not p + 1 as for p = 1,2). But if we note (3.35) and take v in (3.35)

143

to be 3 G z , which is the Galerkin project of derivative of Green's function satisfying (Vv,V(3G^)) = 3v(z), V v e S h , then", in the inner node Z(which is far from the boundery aft), IUGMI
'

ll 3 G h ||" 11

z

i,i,o


and hence, |3U3(Z)-3u(Z)|=0(h4|lnh|)||u||5^ .

That is on the inner node, the order of superconvergence is almost p+1 for p = 3 , which is similar to p = l , 2 .

§5 1

References

B.Li, superconvergence problem for triangular finite element of high order, Numer. Math. (Chinese), (1990). 2 Q.Zhu, Q.Lin, superconvergence Theory of finite Element, Hunan Science Press, (1989).

144 A G E N E R I C R E S U L T OF A P P R O X I M A T E C O N T R O L L A B I L I T Y

J.L. Lions College de France 3, rue d'Ulm 75231 PARIS Cedex 05 dedicated to Professor GU

ABSTRACT. We consider the Stokes equations in a bounded il of R 3 . We suppose that we can act on the system, through the control which is applied only on the 3 d component of the state equations. This control is concentrated on an arbitrarily small subdomain O of fi, and depends only on the time variable. We show that under these conditions the system is generically approximately controllable; more precisely, given 0 and 0 , and given a time horizon T, the system is approximately controllable at time T after possibly modifying the sets Q and O in an arbitrarily "small fashion" (in the C 2 topology). The first result along these lines, but for a different situation, has been obtained by E. ZUAZUA and the A. 1.

Synopsis. 1. Introduction. 2. Proof of the generic result. 3. Dual formulation. References.

145 1. I n t r o d u c t i o n . Let Q be a bounded open set of R 3 with smooth boundary T. We consider the Stokes equations in fi. If y = y(x,t)

=

{yi(x,t),y2(x,t),y3(x,t)}

denotes the velocity of the flow in O, and if p denotes the pressure, the state equation is given by (1.1)

^-nAy

= -Vp + v(t)g,

(1.2)

divy = 0

(1.3)

y = 0 o n T x (0,T),

(1.4)

y(x,0)=0

infi.

In (1.1) v(t) denotes a scalar control function, where 5 = {0,0, xo} (1.5)

Xo characteristic function of an open set O C O . We assume that veL2(0,T).

(1.6)

Then (1.1)...(1.4) admits a unique solution, in the following sense. We define V = {
1

, div

H£(fl) = {ip\4>€H (n),

(1.8)

V = 0onT}

H = closure of V in L 2 (fi) 3 .

Then the solution y = y(x, t\ v) satisfies (1.9)

y€L2(0,T;V)nL°°(0,T-H).

Moreover (1.10)

t —> y(., t; u)is weakly continuous from[0, T] —> H

146 so that one can define (1.11)

y(.,T;v)eH.

We say that the problem is approximately controllable if y(., T; v) spans a dense subset of H when (1.12)

v describes L 2 (0,T).

We are going to show that property (1.12) holds true genencally with respect to Q and to O. More precisely : given Q and given O (arbitrarily small), the time horizon T being given, (1.13)

one can always find fl and O arbitrarily close to O and to O (in the C topology for instance), such that (1.12) holds true for the couple Cl, O .

Before we proceed with the proof of this result, a few remarks are in order. R e m a r k 1.1. The first generic result of this type has been given in a paper of E. ZUAZUA and the A. 1. In this paper, the control is given by (1.14)

V{x,t)g

where (1.15)

ii6l2(Ox(0,T)).

But O is fixed once for all. In 1. it is proven for a special geometry of fi and of O, that one has property (1.12) genencally with respect to Cl. In the present situation, we have less flexibility on the control function (since it does not depend on x), but we are allowed to change (by an arbitrary small "amount") the set O where the control is applied (and this makes the proof given here much simpler than the one given in 1.).

□

147 R e m a r k 1.2. It is extremely likely that a similar result holds true for boundary control (as studied in 2. and in 3.), when the control is apphed on a part OofV

and where we can change

(by arbitrarily small "amounts" both Cl and O). But not all technical points are settled at the time of writing the present paper.

a Remark 1.3. One can obtain results similar to (1.13) when

g = {0,0,

6(x-b)},

(1.16) S(x — b) = Dirac measure at point 6 G H and by allowing b to move slightly. In this situation, the solution y is weaker, and, in particular, y(.,T; v) does not belong to H for any v in L2(0,T).

One can then have density in a larger space than H. One can

also introduce the subspace of those v's in L2(0, T) which are such that j/(., T\ v) € H. [For spaces of this sort, one can consult 4. and the appendix 5.].

□ Remark 1.4. We have stated in 1989-1990 (cf. 6., 7.) a conjecture about the approximate control­ lability of Navier Stokes equations, i.e. when the left hand side of (1.1) is replaced by (1-17)

^-ryVy-fiAy,

other things being unchanged. We think it is quite probable that analogous results hold true in the non linear situation of Navier Stokes system. But nothing is proven along this line, i.e. generic approximate controllability results for non linear "turbulent"

systems.

a Remark 1.5. The "genericity with respect to Q." in the result (1.13) is used in a very simple manner. Indeed we modify Q so that the spectrum of the spectral problem — Auij = Xj Wj — V7T, ,

(1.18)

div uij = 0 m £ l , uj, = 0 on T ,

148 is simple. And it is known 8. (cf. also 9. 10.) that this is indeed possible to achieve by an arbitrarily small change of fi. Therefore, in the proof to follow, we shall assume that (1.19)

the spectrum of (1.18) is simple

and fi will be fixed. We shall prove (1.13) by making only changes on O . D Remark 1.6. For given Q and O the approximate controllability may be false (cf. Remark 2.1 below). Therefore the result (1.13) cannot be (significantly) improved.

□ Remark 1.7. Slightly modifying the set where the control is applied does not always suffice to insure approximate controllability! For instance let us consider the state equation

•jj| - Ay + y3 = v(x, t) Xo in fix (0,T), (1.20)

y(z,t) = 0 onT x (0,T), 2/(z»0) = OinH .

Then one has never approximate controllability (11., 12.), and slight (even large!) modifications of O (provided O is strictly contained in fi) do not change the situation.

□ 2. Proof of the generic result. Step 1. As we said in Remark 1.5., the spectrum of (1.18) is simple. In the first step, we use Hahn Banach theorem. We consider / e H such that (2.1)

(y(.,T;v),f)

= Q

V^eL2(0,T),

where ( , ) denotes the scalar product in H. We want to prove that, generically in O (2.1) implies that / = 0.

149 We introduce the adjoint state tp given by dtp liAtp = — VCT dt div ip = 0 i n f i x (0, T)

(2.2)

on]?x(0,T),


inn.

Applying Green's formula, one verifies that (2.3)

(j/(.,r; » ) , / ) = / Jo

v(t)(
Therefore (2.1) is equivalent to (V(t), g) = 0

(2.4)

on(0,T)

/
on (0,T)

We have to conclude that, generically in 0 , (2.4) implies that tp = 0 , hence / = 0 . Step 2. Of course one can reverse time, just for convenience. Thus ip is now given by dip ■ nAip = — V<7 dt with the other conditions unchanged. Using (1.18), the solution

tp - ] T (/, wj) expi-Xj

t) Wj.

3

But ip is analytic in t with values say in H, so that (2.4) holds true V t > 0, and we arrive at (2.6)

Y,V,u>j)exp{-\jt) i

( u>n dx = 0 V t > 0 , JO

Therefore (2.7)

w} 3da: = 0 (/, Wj)) / ty,Jo

V.

150

It remains to show that one can always assume, by a possible arbitrarily small change of O , going from O to 0 , that (2.8)

/

wpdx^O

Vj.

Jo Then (2.7) implies that (/, uij) = 0 V j and the proof is completed. Step 3. Proof of (2.8). We embed 0 in a family 0(C) of open sets, depending in a real analytic fashion of (,|(|

<£

, e > 0 given such that O(0) = 0 , or O(0) very close to O (so that no

regularity in O is needed), distance {0(C), O) < t . We assume that I ) (0(C))

(2.9)

contains strictly O . We define

0j(C) = /

wl3 dx .

JO(0

The functions 9j are real analytic in | C [ < e • Therefore one can always find a C* in | £ | < e such that (2.10)

MC*)#0

Vj.

Choosing

6 = 0(0 gives (2.8). D Remark 2.1. If O is such that, for some j , J0 Wj3 dx = 0 , then one has not approximate controllability, which proves Remark 1.6. D Remark 2.2. The method of proof does not extend to the non linear situation.

□

151 R e m a r k 2.3. One can consider similar questions for rotating fluids, i.e. with equation (1.1) replaced by (2.11)

-£-nAy at

+ kxy

= -Vp + v(t)g

Other conditions being unchanged. We do not know if similar results still hold. This is due to the non symmetric operator k x y. If one considers (as in 13.)

(2.12)

tt = Qx(0,L),

6cR2

O = UJX(O,L)

LJeg

and (2.13)

^ - n A y + -ekxy

= -\>p + v{t)g

where fc = {0,0,1}, then in the limit f - > 0 , the analogous generic result is not true.

Remark 2.4. If we assume that approximate controllability holds true, one can consider the following problem : find (2.14)

when v G L2(0,T)

inf- / v 2 J0 is subject to y(.,T;v)€yT

(2.15)

v2dt,

+ l3B

y T given in H, B = unit ball of H, 0 > 0 arbitrariry small.

We consider in the next section the dual formulation of (2.14) (2.15), in order to see what becomes the "genericity" result.

152

3. Dual formulation. We use the duality theorem of Fenchel-Rockafellar 14. in convex analysis. We define the proper convex functional rT

(3.1)

Fi{v)

= \jQ

(3.2)

F2(f) =

onL 2 (0,T),

v dt

0

if

feyT

+ (3B, 0B,

+ oo elsewhere on H.

and the linear map L (3.3)

Lv = y(.,T;v),

C(L £{L2(0,T);H).

L £e

Then problem (2.14) (2.15) can be formulated by (3.4)

inf (F^v) + F2(Lv)) . «et2(o,T)

vELHO.T)

Using 14., this inf equals -in]l(F{(L*f)+FZ(-f)) - i fen nf (F{(L'f)+FS(-f))

(3.5)

where F* is the conjugate function of Fi and where L* is the adjoint of L. One finds that

ni») = \j "2*. F;(f) = P\\f\\+(yTJ)

(I) | | = norm in H)

and using (2.3) L* f = (tp(t),g)

,

ip solution of (2.2).

Therefore Problem (3.5) reads (3-6)

- taf [ \ j \ J

vs(x,t)dx)'

dt + 0\\f\\ -

(f,yT)].

It follows from (1.13) that (3.7)

generically in U and in O, problem (3.6) admits a unique solution, for any /3 > 0 arbitrarily small.

153 Remark 3.1. A direct proof of (3.7) does not seem obvious.

□ References. 1. J.L. LIONS and E. ZUAZUA, A generic uniqueness result for the Stokes system and its control theoretical consequences. Dedicated to C. PUCCI. 1996. 2. G. LEFRANC and B. STOUFFLET, Dassault aviation Technical Report. 1994. 3. E. FERNANDEZ-CARA and M. GONZALEZ-BURGOS, A result concerning con­ trollability for the Navier Stokes equations. SIAM J. Control and Optimization, 33, 4, July 1995. 4. J.L. LIONS, Some methods in the Mathematical Analysis of systems and their control. Science Press, Beijing (in Chinese), Gordon and Breach, Sc. Pub., 1981. 5. LI TA TSIEN, Properties of the function space U. Appendix 1, Chapter 4 of [Li4]. 6. J.L. LIONS, Are there connections between turbulence and controllability ? Springer Verlag Lecture Notes 144 (1990), A. Bensoussan and J.L. Lions, (ed). 7. J.L. LIONS, Exact Controllability for distributed systems. Some trends and some problems, in Applied and Industrial Mathematics, R. SPIGLER (ed.), Kluwer, 1991, p. 59-84. 8. A.M. MICHELETTI, Perturbazione dello spettro dell operatore di Laplace in relazione ad una variazione del campo, Ann. Scuola Norm. Sup. Pisa 26 (3), (1972), p. 151-169. 9. J.H. ALBERT, Genericity of simple eigenvalues for elliptic PDE's. Proc. A.M.S. 48 (2), (1975), p.413-418.

154 10. K. UHLENBECK, Generic properties of eigenfunctions. Amer. J. Math. 98 (4), 1976, p. 1059-1078. 11. A. BAMBERGER, Result reported in [HE]. 12. J. HENRY, Thesis. University of Paris, 1976. 13. J.L. LIONS, Some results and some open problems in the active control of distributed systems. ICIAM, Hamburg, July 1995. 14. T.R. ROCKAFFELAR, Duality and stability in extremum problems involving convex functions. Pac. J. Math. 21 (1967), p. 167-187.

155

Plate Elements With High Accuracy Zhong-Ci Shi* National Lab of Scientific Computing Chinese Academy of Sciences Beijing, China

Shao-Chun Chen Department of Mathematics, Zhengzhou University Zhengzhou, China

and Hong-Ci Huang* Department of Mathematics, Hong Kong Baptist University Hong Kong

Dedicated to 70-th birthday of Prof. Chao-hao Gu. Abstract A class of 12 degree-of-freedom triangular plate elements with 0(h2) con­ vergence rate is constructed by the use of the double set parameter method.

1

Introduction

It is known that for solving the plate bending problem, a conforming element has to be of C 1 continuity that stimulates the development of nonconforming elements. The practical nonconforming plate elements in engineering applications are 3-node triangles with 9 d.o.f. being the nodal function values and two first derivatives. There appear many devices based on different considerations, such as the classical Zienkiewicz cubic element [1] and newly developed TRUNC [2], quasi-conforming [3], generalized conforming [4] and energy-orthogonal [5] elements, etc. All those elements *The work of the author was partially supported by the Research Grants Council of Hong Kong 'The work of the author was supported by the Research Grants Council of Hong Kong

156

have the same asymptotical rate of convergence 0(h) in the energy norm, where h is the mesh size of elements, though the practical performances are not the same. It is shown in [6] that both the quasi-conforming and generalized conforming elements have their consistency error of order 0(h2), one order higher than standard nonconforming ones. However, since the shape function spaces of all 9 d.o.f. elements contain only a full quadratic polynomial, but not a cubic one, the interpolation error then permits at most of order O(h). Therefore, the total rate of convergence is of order 0(h). In this paper, we give a description of how to construct a class of 12 d.o.f. plate elements with 0(h2) order of convergence. The nodal parameters of these elements, besides function values and two first derivatives at vertices, are the normal deriva­ tives at the mid-points of edges. We will use our previous results on the double set parameter method [7],

2

Consistency error estimate

Consider the plate bending problem of finding u 6 HQ(Q) such that a(u,v) = (f:v)yv€H20(V),

(1)

where d2u d'v d2v d2u d2v d2u d2v., i , , , . r ,. , ,, d'ucfv crud w „ a'-u a(u,v) = ^ [AuA, + (1 - - ) ( 2 ^ ^ " ^ W 2 ~ W2^)]d*dy, (/» = /

fvdxdy,

O is a convex polygon in R2, f e L2(Q), Poisson ratio 0 < <J < | . Dividing Q, into a regular family of triangles K with diameter hK < h, and defining on each triangle K a shape function, we obtain the finite element space V),. The finite element approximation of (1) is to find uh € Vh such that a.h(uh,vh) = (f,vh),Vvh

e Vh

(2)

using /

N

W

fA

A

M « ) = E }K \ ^

L/1

vo

&U

d2ud2y

^

+ (1 - ^B^yO^Ty

d2V,d2V^

- dTleyl ~ ^

dV2^'

Suppose that u and uh are the solutions to problems (1) and (2) respectively. According to Strang Lemma [8], we have the following error estimate |u - ^ k *
%eH

\U - vh\2th + sup wkevh

|E, (

! "'W,l)h , \w -',/.

(3)

157 where the discrete semi-norm \vk\zfi is defined by \vh i\2,h. \2,K> M * = J2 M1,K> :

K K

and the consistency error functional is Ehh(u,w) (u,w) = (u,w) + 1(u,w) + 3(u,w) =E Ei(u,w) + E22(u,w) + EE3(u,w)

(4)

with

^H£/jAu-(l-,)g)§^

^H£/jA^l-,)g)§^, £ 2 M = W (i-,)*J*V yy JdK iaA-

anas as one's as

*<«.») = x£ JdK / - an ^»*. The first term on the right-hand side of (3) is the interpolation error and the second one is the consistency error due to the nonconformity of shape functions across interelement sides. Now we give conditions for obtaining a high order consistency error. Lemma 1. Let K be an element and K be its reference. For every K, there exists an affine mapping from K to K. Assume that the following three conditions are fulfilled. C l . Function Vh 6 V/, is continuous at vertices of K and vanishing at vertices lying on 80,. C2. For every polynomialp(s) € Pm-i(F), the integral fFp{s)vhds is continuous over each interelement side F and vanishing when F C dQ, where PT{F) is a polynomial of degree r on the side F. C3. For every p(s) 6 Pm(F), the integral fFp(s)^-ds is continuous over each interelement side F and vanishing when F C <9fi. Then, if the solution u £ Hm+3(Q) n H$(Q), we have \Eh{u,wh)\ „ , m+1 , 1 +1, , M ^ M < Chmm+ \u\ sup I——| \u\ m+u m+umm > 1.

wk
\W\2lh

(5)

Proof. Let a, b be the two endpoints of F. For every p(s) € Pm{F) and Vh € 14, an integration by parts yields Vh s hds. (s)v -J P'( J/ , PP(s)^ds^~ds^ds = =p^P(s)vh\" ^ ~a~JP K ds !

JF

Since p'(s) £ Pm-\(F),

the condition C l and C2 imply the condition

158

C4. For every p(s) € Pm{F), the integral JFp{s)^fcds terelement side F and vanishing when F C dil-

is continuous over each in-

From C3 and C4, using the same technique as in [6], we can conclude that \Ei{u,wh)\ < Chm+l \u\m+3\w^,h,i = 1,2. \Ei{u,wh)\ < Chm+1\u\m+3\wh\%h,i = 1,2. Similarly, from C l and C2, we have m+1 \E | £ 33(u,w ( w ,h^)\) l <<< C/i ? / l m + 1 ||w| w | m + 3 K| K k2 /,,. ..

(6)

(7)

Then, (6) and (7) give the estimate (5).

3

Elements with 0(h2) accuracy

Given a triangle K with vertices p* = (xi,j/j),i = 1,2,3, we denote by Aj,Fj and nt the area coordinates of K, the side pj+ipj_i (mod 3) and the unit outward normal on F{, respectively. Let 6. = 2/i+i Vi+i ~Vi-i,Ci -Vi-uCi == i i i--i i --Xi+i,Ti l i + i , n == (6i+i6i_i (bi+i&i-i ++Cj+iCi_i)/A,( Ci+iCj_i)/A, ( mod mod 3) 3) 2 2 U = \Fi\ /A = (b [b] + c?)/A,e y = n/tj, r , / ^ 1 < i,j t,j < < 3, A = area of K. It can be easily checked that 3

13

3 bi =

t=l

_ ( -[

Ui

S c» = °> r>+i + r ' - i

= _i

«> A = biCi+i - bi+ic,,

i=l

bj_ __Ci_\ d\__ bj_ d>n_ \Fi\' \Ft\)> dx~2A'

c,_ ._ dy~2A'l-l'Z'6-

Now we use the double set parameter method [7] to construct a class of 12 d.o.f. elements with 0(h2) accuracy. Suppose Pz{K) is a full cubic polynomial space on K that may be represented in the form P3(K) = span {Ai,A2,A3,A1A2,A2A3,A3A1,AjA2 - AiA§, AiA|, A2A3 — A2A3, A3Ai — A3A!, AiA2A3}. We choose the initial shape function space as follows PFK* = /OU{VI,<M, = PP33((tf)U{V>i,iM,

(8)

where ipi and ip2 consist of combinations of certain mono-nomials of degree 4 that are 2

2

2

2 2 i>i==fluAjAjAs Oa A?A2A+3 +ei2eAiA^A rpi 0l3^AiA 0i4o^x A2A2 + + eibe\\\\ 3 AIA 3 + 2A2 A 3 + l2\1\ 2\3 + + ibx

2 x\+ +0i6eAi62xA2xl, i =l,2r 2 ,i =

159

with the coefficients 0*., to be determined. Every function v £ PK can be written in the form v =/8lA1 + /32A2 + /J3A3 + AA1A2 + ftA2A3 + /JeAaAi + /J7(A?A2 - AXA|) + ft(A|A, - A2A|) + A(AJA, - A3A?) + /JwAxAjA, + /?m/>i + M2 =Ab,

(9)

where A =(Ai, A2, A3, A1A2, A2A3, A3A1, AXA2 — AiA2, A2A3 — A2A3, A3A1 - \3\\,

\i\2Xi,tpuA),

b =(A> &,•■•, A 2 ) r We take 12 d.o.f. as follows T>{v)=(v1,v2,v3,T=|Fi| M -6/

f

OV

7F 3

vds

>T^l I vds
f

OV

f

OV

[ — ds, - 6 / — (is, JFidn JF2on f

OV

T

- - d s , - 6 / A 2 — d s , - 6 / A 3 —ds, - 6 / Ai—ds)' an JFi on JF2 on JF3 on

(10)

Substituting (9) into (10), after a manipulation we obtain B(v) = Cb,

(11)

where the coefficient matrix C is " 1 0 0 0 3 3 3*!

0 1 0 3 0 3

0 0

I 3 3 0

0 0 0 0 0 1

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

3/

ft

-tl

0

k

tl

-h k

0

3r 3

3r 2

1*1

3r 3

3i 2

3n

ft

3r 2

3)"!

3t 3

-1*3

1^3 1^3 1*2

h tl In 1

If* In

n ^

r

2 l

1*3

1*2 - 1 * 2 1*3 1*3 2

5i 2

<2

2

«1 2

*3

0

-h

AM/5

0 0 0 025/5 026/5 024/5

ft ft

an

"21

Q12

£*22

1*3

Ol3

»23

*11

<5 2 1

<5l2

622

<5l3

<5 2 3

"ft -ft ft ft ft - 1 * 2 - 1 * 2 ft - 1 * 3 ft - ! * 3 ft

where aii = ^(0:2 + 0i3-20i 5 )ti, ai2 = ^(0ii+0i3-20 i 6 )<2,

0 0 0 015/5 ^16/5

160

Ci3 — T ( # I 1 + #i2 - 26,4)<3i

Sa = ~[M* + 29l3 + (2e21 -

4)9a]ti,

6.2 = ^[20a + 30a + (2e32 - 4)#l6]t2, 6.3 = ^[3#« + 2#l2 + (2e13 - 4)0u]t3. It can be verified that the determinant of C takes the value 2

detC = - i ( ^ )

detC*,

(12)

where C* is as follows /

2 # u — 8i2 — #13

—

2^14 + 4 # J 5 — 2 # i 6

V - # 1 1 + 2 # i 2 - #13 " 2 # 1 4 - 2 # 1 5 + 4 # i 6

2#21

—

#22 ~~ #23 ~~ 2#24 + 4#25

—

2# 2 6

- # 2 l + 2 # 2 2 - #23 - 2#24 - 2# 2 5 + 4#26

We have then Lemma 2. The d.o.f. (10) can uniquely determine an element of the shape function space (8) if and only if the matrix C* in (12) is nonsinguler. Then we have b = CT1D(i>).

(13)

Examples. 1. Taking 0U — #22 = 1 and #y = 0 else, we have det C* = 3 / 0. The last two basis functions of Pk in (8) are l(>l - A1A2A3,

"02 = A1A2A3.

Since A1A2A3 + A1A2A3 + A1A2A3 = A1A2A3,

then span {A1A2A3, A1A2A3, A1A2A3} is equivalent to span {AiA2A3, A1A2A3, A1A2A3}, which means that the initial shape function space PK = span {Ai, A2, A3, A1A2, A2A3, A3A1, A^2 — AiA2, A2A3 — A2Aj, A3A1 — A3A1; X^^X^, AiA2A3, A1A2A3}.

(14)

161

2. Taking 014 = 62b = 1 and % = 0 else, we have

detC* = 12^0. 12^0. Then

^ = A?A^ 2 = A ^ . 3. Similarly, by taking tp\ = \\\\ have in both cases

and ip2 = A2A3 or ip\ = AJAJ and ip2 = A3 A?, we det C* ^ 0.

4. We can select other basis functions ipi and ip2 that make the matrix C* nonsingular. Now we introduce the second set of parameters, the nodal parameters as follows: Q(v) Q (V) = = (Vi,Vl (vuvixx,V,vly,V y,v ,v233x,n,,v3y,v31tn)T,V3,V3x,Viy,V3hn)T, 12in,V2, 2x,V 2y,,v3V 1y,v 12tn,v2,v2x,v2V 23in

(15)

where «y,n means the normal derivative of Vh at the mid-point of the side ptpj. According to the double set parameter method, we use certain quadrature schemes to discretize the d.o.f. (10) through linear combinations of the nodal parameters (15) that result: v, =viti = 1,2,3, —— / vds =6{v2 + 1/3) + ci(v2x - v3x) + bi(-v2y + v3y) +

0(h3\v\i
— - / vds =6{v3 + Vi) + c2(v3x - vlx) + b2{-v3y + vly) +

0{h3\v\iiK),

—— / vds =6(i>i + v2) + c3(vlx - v2x) + b3{-viy + v2y) + \F3\ JF3

0{h3\v\itK),

\i*2\ JF2

- 6 / — d s =bi{v2x + v3x) + ci{v2y + v3y) - 4|Fi|023,n, JFi

on

- 6 / 7— ds =b2(v3x + vix) + c2(v3y + vly) - 4|F2|t)31>n, JF2 on

-6 /

JF3

—ds =b3(vlx + v2x) + c3(vly + v2y) - 4|F3|i)12,n, on

- 6 / \2-^-ds=b1v2x JFx

on

+ c1v2y-2\Fl\v23tn

+ 0{h3\vUiK),

=b2v3x + c2v3y - 2|F2|w3i,n +

0{h3\v\iiK),

- 6 / Ai^-rfs =b3vXx + c3vly - 2\F3\v12n + JF3 on

0{h3\v\iiK).

- 6 / \3^-ds JF2 on

The above formulas can be written in the matrix form as D(») = GQ(ti)+ £(«),

(16)

162

where the matrix

cG

"

'1 0 0 0 0 0 0 0 6 -02 6 c3 o 0 0 b2 0 h 0 0 0 0 0 63

0 0 0 0

h -h 0 c2 c3 0 0 C3

0 0 0 1 0 0 0 0 0 0 6 Cl 0 0 0 0 6 -c3 0 0 bi 0 0 0 -4|F,| 0 h 0 b, 0 0 0 0 0 0

0 0 0 ~6i 0

~m\

h C\

0 c3 Cl

0 0

0 0 1 6 6 0 -4|*\ 1 0 0 0 0 0 -2|*i 1 0 0 0 0 0 0 0 0 0 0 0

0 0 0

0 0 0

-Cl

61

c2 0

-b2 0

61

Cl

b2 0 0 b2 0

c2 0 0 c2 0

0 0 0 0 0 0 0 -4|F2 0 0 -2|F 2 0

and e{v) = (0,0,0,84,£ 5 , e8, 0,0,0,e 10 , e u ,e 1 2 ) T ,e, = 0{h3\v\tiK),

i = 4,5,6,10,11,12. (17) Neglecting the remainder terms et and substituting (16) into (13), we obtain b = C- 1 GQ(u)

(18)

That is the interpolation formula determining the final shape function space from the initial one through the nodal parameters (15). The d.o.f.(10) are a set of auxiliary parameters which ensure the convergence but do not enter into the final formulation of the finite element space. If the rank of matrix G is g, then the final shape function space PK associated with the interpolation formulation (18) is a j-dimensional subspace of the initial space PK. It may be shown that the matrix G is singular. Moreover, by deleting the fifth and the sixth rows and the eighth and the twelfth columns, we obtain a nonsingular matrix G* with detG' = -2|F3|2A2#0. Therefore, 10 < Rank(G) < 11.

(19)

It is important to note that from the quadrature formulas (16), it is seen that all remainder terms tt are vanishing for every cubic polynomial v. Hence, from [7] it follows P3(K) C PK. In summary we have Lemma 3. If the matrix C in (11) is nonsingular, then the shape function space PK determined by the interpolation formula (18) has the properties: P3(K)

CPKCPKC

P4(K),

163

and 10 < dim(PK) < 11. Now we are in a position to prove the following convergence theorem for the class of 12 d.o.f. plate elements described above. Theorem. Suppose the domain Q C R2 is a polygon and the triangulation is regular. The 12 nodal parameters are taken as (15). If the basis functions ipi and ip2 in (8) are chosen such that the matrix C in (11) is nonsingular, then the 12 d.o.f. plate elements determined by the interpolation formula (18) have the convergence rate \u-uh\2,h
(20)

where u and u h are the solution of (1) and the finite element solution of (2), respec­ tively. Proof. We have to estimate two kinds of errors in (3). 1. Interpolation error. Let II/i be the interpolation operator associated with the shape function space PK and UK = Hh\K- Prom Lemma 3, PK contains a full cubic polynomial. It follows from the interpolation theory, the affine mapping technique [8] and the property of the remainder terms (17), that \u-UKu\2tKK

(21)

and so inf \u - vh\2,h <\u-

Y\h\2,h < Ch2\u\A.

(22)

2. Consistency error. The nodal parameters (15) are continuous at nodes on interelement sides and vanishing at nodes on dQ. Further, the quadrature formulas (16) expressing the d.o.f. (10) through the nodal parameters (15) only use these nodal values that are related to the same edge of K, and certain geometrical configuration of that edge. Therefore, the d.o.f. (10) are also continuous over interelement sides and vanishing on sides belong to dd. Since 1 and A;+1 are two basis functions of the linear polynomial space Pi(Fi), hence three conditions of Lemma 1 are satisfied with m = 1. We have \Bh(u,wh)\ 2 sup J—r~ < Ch \u\i. WK£Vh

(23)

\U}\2,h

Combining (22),(23) and (3) gives the theorem. Remark 1. Using an undetermined coefficient technique, [9] has constructed another class of 12 d.o.f. plate elements. Since it does not satisfy the condition C3 of Lemma 1 with m = 1, the convergence rate of these elements is only of order 0(h).

164

Remark 2. More specifically, using the nodal parameters (15) and the shape function space (14), a standard nonconforming element can be derived. However, this element has a convergence rate 0(h) too. Remark 3. It is interest to mention that the last three fourth order terms in the shape function (14) take zero values on each element side. It means that alone each side the shape function (14) is a cubic polynomial. Therefore, the three remainder terms, namely e4, £5,66 in the quadrature schemes (17) are vanishing, that is €4 = «5 = £6 = 0 .

References 1. O.C. Zienkiewicz,27ie Finite Element Method (3rd ed, McGraw-Hill, Lon­ don, 1977). 2. Z.C. Shi, Convergence of the TRUNC plate element, Comput. Meths. Appl. Mech. Eng. 62 (1987) 71-88. 3. L.M. Tang et al., Quasi-conforming elements in finite element analysis, J. Dalian Inst. Technology 19 (1980) 19-35. 4. Y.Q. Long and K.Q. Xin, Generalized conforming elements, J. Civil Engi­ neering 1 (1987) 1-14. 5. C.A. Felippa and P.G. Bergan, A triangular bending element based on energy-orthogonal free formulation, Comput. meths. Appl. Mech. Eng. 61 (1987) 129-160. 6. Z.C. Shi, On the accuracy of the quasi-conforming and generalized conform­ ing finite elements, Chin. Ann. Math. 11:B (1990) 148-155. 7. S.C. Chen and Z.C. Shi, Double set parameter method of constructing stiff­ ness matrices, Numer. Math. Sinica 13 (1991) 286-296. 8. P.G. Ciarlet, The Finite Element Method for Elliptic Problems (North- Hol­ land, Amsterdam, New York, 1978). 9. S.C. Chen and T.S. Qi, On 12 d.o.f. plate elements, Computational Mathe­ matics, a Journal of Chinese universities 12 (1994) 154-160.

165 A S Y M P T O T I C ANALYSIS OF T H E L I N E A R I Z E D N A V I E R - S T O K E S EQUATION IN A 2D C H A N N E L AT HIGH REYNOLDS NUMBER ROGER T E M A M 1 2 AND XIAOMING

WANG 2

Dedicated to Professor Gu Chaohao ABSTRACT. We briefly survey recent results on the asymptotic behavior of the linearized Navier-Stokes equations in a two dimensional channel at high Reynolds number presented in [17] and [16]. We also present a new result which treats the case when initial layers exist and which generalizes our previous results in [17] and

[16].

I N T R O D U C T I O N AND M A I N

RESULT

The behavior of wall bounded flows at large Reynolds number is one of the most intriguing and largely open problems, both in fluid mechanics and in mathematical analysis. Two important phenomenons occur: on the one hand, inside the flow, the cascading of energy from large eddies to small eddies and then its dissipation by viscosity; on the other hand, at the boundary and in the boundary layer which may be turbulent vortices are generated which drive the flow. Boundary layer theory is very i m p o r t a n t from the physical point of view and has been extensively studied in the fluid mechanics literature; see e.g. Germain [4], Lagerstrom [7], Lamb [8], Landau and Lifschitz [9], Moffat [11], Prandtl [13], Schlichting [14] among many others. From the mathematical point of view these problems are studied in the long article of Vishik and Lyusternik [18] and the book of Lions [10]; different aspects are considered by Eckhaus [2], Fife [3] and Oleinik [12]. Nevertheless the problem is still largely open in terms of the behavior of the solutions of the Navier-Stokes equations at high Reynolds number. From the mathematical point of view, the difficulty lies in two parts: the nonlinear term and the presence of the pressure. There are several indications of the difficulties brought in by the presence of the pressure: the difficulty in studying the Stokes problem in comparison to that of the Poisson equation; and the long standing open problem of the regularity problem of 3D Navier-Stokes equations in comparison to the global existence of strong solutions to the 3D Burger equations ut — i/Au + (u ■ V ) u = 0 . As a very preliminary step towards the understanding of the behavior of the solu­ tions to the Navier-Stokes equations in a bounded region at high Reynolds number, 'Laboratoire d'Analyse Numerique, Universite Paris-Sud, Batiment 425, 91405 Orsay, France The Institute for Scientific Computing and Applied Mathematics and the Department of Math­ ematics, Indiana University, Bloomington, IN 47405 2

166 we initiated in [17] and [16] the study of the asymptotic behavior of the NavierStokes equations linearized around a constant flow ((/«,,0). These equations in space dimension two, in a channel read

(1)

^--eAu'-rU^Dm' c

(2)

u € V

(3)

+

Vp'^f,

for t > 0,

u' = u0 at t = 0,

whereas the corresponding inviscid equations (linearized Euler equations) read (4)

+ UooD1u0 + Vp° = f,

^-

uc e H,

(5) (6)

for t > 0,

u° = u0 at t = 0.

Here (7)

V = {v € (/y, I oc (n <X) )) 2 ,divi; = 0 , u | a n „ = 0,v periodic in x with period 2ir}, H =closure of V in ( L L ( n ~ ) ) 2

(8)

={v = (vuv2)

6 (Lic(aoo))\dWv=

0,v3\an„

= 0,

v periodic in x with period 27r}, (9)

noo = R1 x ( 0 , l ) ,

n = (0,27r) x (0,1),

and D\ denotes the derivative in the horizontal (x) direction. T h e well posedness of (l)-(3) and (4)-(6) are easy and the convergence of uc to u° in some weak sense (say L 2 (0, T\ L2(ii)2) is classical (see e.g. Ladyzhenskaya [6] or Lions [10]). In [17] and [16] we have improved this convergence result as follows: P r o p o s i t i o n 1. Suppose u0, f 6 H are smooth enough. (10) where K is a generic constant

)\u'-

n°\\L^0,T.M)

<

depending on u0,f

Then

K^\ and T but independent

of £.

In fact if we follow closely the proofs in [17] and [16] we are able to obtain the following two auxiliary results: (H)

h'Wmojy)

(12)

\\Au<\\L,{0,T;H)

<

K£-U\

< K£-3/\

for any q € [1,4/3), where A is the Stokes operator associated with the boundary conditions prescribed in (7) and the generic constant K is of the same kind as in (10) except that it may depend on q in (12) as well. However the convergence in a stronger topology (say L2(0, T; H'(Q)2) or L°°((0, T)x Q) 2 ) is not obvious. In fact this convergence is not true due to the disparity of the boundary conditions between uc and u° or the so called boundary layer problem. Usually this difficulty is overcome by introducing a corrector (boundary layer t y p e

167

function) 9" and we obtain the convergence of u< - (u° + 6') valid up to the boundary in the stronger topology (see e.g. [10] and [18]). In [17] and [16] we have explicitly constructed a corrector 0': we first obtained 6' in an abstract form, i.e., as the solution of an evolution equation, and then in an explicit form using the heat kernel (see [5]), periodic expansion and suitable asymptotic expansions. The main result of [17] and [16] is Theorem 1. Suppose u0 € V and f 6 H are sufficiently smooth. Then u' has the following asymptotic expansion (13) u*(t, •) - u°(t, ■) - P(t, •) = 0{e*l8-«l2) in L2(0, T- IT (St)') for q e [0,1]. Here ^ , x ,

y )

= / ' Jo

(

l - 2 e r f ( ^ E = ) ) { ^ \/2s(t - s) <■ + U^Dm0^;

; J

- ^ - ^ ^ at

x - UUt - s), 1)1 ds

(14) + /'(l-2erf( Jo

,

I

n l W v - U - l t - M

y/2e[t-s)

l

dt

+ U„Dxu°{a; x - Um(t - «),0)} ds, where erf is the standard error function defined as (15)

ert{y) =

-j=j\->2"dz.

Moreover, we have the following estimate on the pressure (16) W - P°\\wo,T,HHa)) < ™l/2 where K is a generic constant as in (10). The theorem was first proved by choosing 8C to be the solution of (17)

-j£-eUr

+ Ua>DxP = Qt

£

(18)

0 = Oatr=O,

(19)

9C = - u ° a t y = 0 and 1,

then proving the convergence of ue — u° - ffc in L2(0, T; //'(ft) 2 ) with explicit rate depending on e, and finally applying an asymptotic formula for 6C. The primary difficulty in carrying out the proof was in the presence of the pressure term which depends globally on the behavior of the velocity field at the boundary and this makes localization efforts hard. To overcome this difficulty we departed from several points of view from classical studies in boundary layers in fluid mechanics. In particular we used a functional analysis global treatment of the pressure term; also we considered a corrector which is not divergence free and which acts only on the tangential velocity. We can produce a divergence free corrector as usually proposed in fluid mechanics but this appears to have no advantage. Finally another aspect of

168 our work is the utilization of L2 type Sobolev norms instead of uniform (L°°) norms, which is useful since a corrector like 9* is small in the L2 norm but not in the L°° norm. It is inferred from (14) that the boundary layer developed after time t has a thick­ ness of order \fe~i which agrees with heuristic physical arguments (see e.g. Batchelor [1]). It is also observed that the transport term UooD\uc takes the effect of mix­ ing the boundary layers which gives us the flavor of possible difficulties that might be encountered in the nonlinear (full Navier-Stokes) case. There is also a result in [16] indicating the dependence of the long time behavior of the boundary layers on whether the driving force has or not a vanishing average over a period in the tangential direction. In this article we want to improve Theorem 1 in the following way. Theorem 1 assumes that u0 € V while the initial value problem (l)-(3) is known to be well-posed if u0 £ H. Hence concerning asymptotics we have avoided the initial layer problem. In this article we shall resolve this difficulty by proving the following result. T h e o r e m 2. Suppose u0, f € H are sufficiently expansion (13) with (14) replaced by

smooth.

Then uc has the

asymptotic

=u0(x - /7oo<,l)(l c U , l ) ( l - 2erf(^=J)) rF{i,x,y) ^ , x , y ) =uo(* Vlet v2er + u« o0 ( x - t / o0 o0 i«, ,00))((ll- 2 e r f ((--7Jj = L) ) v\/2et 2e< + ' ( l - 2 e r f ( // -- "y + /Al-2erf( - ^j t- -' sMU )) ) ^} e( u^ ^; -' U (20)

7o Jo

\/2e(ty/W^s) s) "IX

dt &

+ U^DrU^s; U^D.u^s-x- x - U^ Uodt - s),-s),l)\ds 1)} ds Jo Jo

K

y/2e(t-s)"\ y/2e(t-s)"\

dt dt

+ Uozoz,D ,Diu°{s; Uoo{t- - s),0)1 s),0)\ds, ds, iu°{s;xx - Uoo{t and with (16) replaced by (21)

||p P%mT;H>m « £ ,7/ 2 l|pE -" P°IU«(o,T;W(n)) < ^e'

for any q € [1,4/3) and K a generic constant depending on u0, / , T, q but ofe.

independent

In order to prove Theorem 2, we follow the general approach of [17] and [16]. However we need to modify the proof of the main result by using a slightly different approach. More precisely we first estimate the difference between pc and p° i.e. prove (21) first, and then apply it to prove the rest of the results. Our choice of corrector will remain the same, i.e. the solution of (17)-(19) and with asymptotic expansion we obtain (14).

169 PROOF OF THEOREM 2

Before we move on to the proof of the main theorem, we recall some results from

[16]. L e m m a 1. Under the assumptions

of Theorem 2 we have

||(/ - P n i L ~ ( 0 , T ; L W ) < « 3 / 4

(22)

where P is the Leray-Hopf projector from (L2(Q.))2 onto H, 8e is the solution (n)-(19), and K. is a generic constant independent of e.

of

Proof. T h e result is the same as the first half of Lemma 3.1 (i.e. (3.25). 1) of [16] except that we do not have the assumption UQ € V. Through a careful examination of the proof of [16] we observe that the assumption u0 6 V is not necessary for the first part of Lemma 3.1 of [16]. Thus (22) is valid under the current assumption. L e m m a 2. The operators A and Dt commute, where A, as before, is the Stokes operator.

the operators P and Dt

commute,

Proof. This is Lemma A.2 of [16] and its direct consequences. Proof of T h e o r e m 2. We now turn to the proof of Theorem 2. We consider the equation satisfied by u€ — u°: (23)

du

'~

"

- e A t i ' + P m f l , ( u ' - u°) + V(p f - p°) = 0,

(24)

u* - u° = 0 at t = 0.

After applying the divergence operator to (23) we find (25) (26) (27)

A(f dip* - p°) V Q

ay

- p°) = 0,

= T e A « 4 , at y = 0 , l ,

/ ( p e - p ° ) = o. Jn

We apply the classical Agmon-Douglis-Nirenberg theory and obtain 11/ - P°IU«(o,Tiff'(n)) < re ll v ( ? £ - P°)\\LHo,T;L^ny
< ( T h a n k s to (12))
for qe [1,4/3).

However this inequality is not strong enough to obtain our main result. In order to get t h e strong convergence, we need a refined estimate on pe - p°.

170 Observing that Auc2 =D\u\ '

'

+

D\u\

=D\u'2 - D2D,u\

(divergence free),

and

(30)

for a11

IMIw^nfl) < «lsl^o)llffllJiHn)

S € ff'
we deduce ||P£ - P°||t«(o,T;W'(fi)) ^ K£ ll^ u 2llL'(0,T;//-'/ 2 (3nocnn)) <«£||At4||i«(o,T;t»(Sa»nft))
+ «H^ii,2«i||^0iT!Z,,(n),|piI'2«!lli^0ir,flr»(a)) ,u + «£ || D i « ' ll£,,(o,T;«'(fi)2) II ^

|| i ,, ( o,r ; H'(n) 2 ) U

' HL'(0,T;H2(f2)2)"

Observe further that by differentiating (1) in the x direction we obtain an for Diu* and D\uc which takes the same form as (1) (thanks to Lemma 2), same boundary condition but with / replaced by Dtf or D2f, u0 replaced or D2u0. Thus the same estimates (11) and (12) hold with uc replaced by D2ue. This implies, in particular, when combined with (31), (32)

\\p'-

< «1/2

P°\\mo,T;H'm)

for

equation with the by DiUQ D\uc or

q G [1,4/3),

which is the same as (21). Next we consider the equation satisfied by wc = u' — u° — Qc. We have (33)

—

- eAu; 1 + U^Dna'

+ V(p E - p°) =

eAu°,

c

(34)

w = 0 at t = 0,

(35)

w' = 0 at y = Oor 1.

We multiply (33) by we and integrate over il, noticing that \\(V(p< - P%W')\\LHO,T)

=ll(V(Pf -

P°),

(/ - P ) ^ | | L , ( 0 , T )

I

= ll(V(p -p°),(/-P)r|| L 1 ( 0 , r ) 36

( )

^f

" P°l|L.,0,r;//.,n))ll(/ -

PWh~{0,T;mW)

< ( T h a n k s to (32) and Lemma 1) <«£5/4;

we deduce (37)

jt\w'\2

+

e\Vw'\2
171 and thus (38)

ll"£-u0-*1U~(o.r;LW)<^5/8

(39)

ll"£-"°-01U>(o,r;"W)<^I/8

T h e main result (13) follows from interpolation between (38) and (39), and (14) follows in the same way as it was derived in [16J. This completes the proof of the main theorem. Remark 1. It is observed that the first two terms in (20) correspond to initial layers introduced by the assumption u0 £ V. In order to derive a finer explicit expression of the boundary layer function 9*', we need to derive better asymptotic expansions of 6e at t = 0 and at the boundary of dilao. This will be done elsewhere. Remark 2. When / is zero, vorticity is transported by the inviscid equation. However, as we can observe by applying the curl operator to (14), vorticities are generated near the boundary and are transported along the tangential direction of the boundary by the viscous equations. Remark 3. T h e same kind of technique applies to the case for Navier-Stokes equations linearized around a constant state (Ut,Ui), i.e. (40)

^--eAu'

+ ^D^+

U2D2ue+

Vpc = f.

In this case the thickness of the boundary layer is proportional to t provided U2 ^ 0. ACKNOWLEDGEMENTS

This work was supported in part by the National Science Foundation under Grant NSF-DMS-9024769 and NSF-DMS-9400615, by ONR under grant NAVY-N00014-91J-1140 and by the Research Fund of Indiana University. REFERENCES

[1] G. K. Batchelor. An Introduction to Fluid Dynamics . Cambridge University Press, Cambridge, 1967. [2] W. Eckhaus. Asymptotic Analysis of Singular Perturbations. North-Holland, 1979 [3] P. Fife. Considerations Regarding the Mathematical Basis for Prandtl's Boundary Layer Theory. Arch. Rational Mech. Anal, 38:184-216, 1967. [4] P. Germain. Fluid dynamics. In R. Balian and J. L. Peube, editors, Cours de I'Ecole d'Ete de Physique Theorique, Les Houches, New-York, 1977. Gordon and Breach Science Publishers. [5] C. Gu. Equations of Mathematical Physics. Shanghai Education Press, Shanghai, 2nd edition, 1978. [6] O. A. Ladyzhenskaya. The mathematical theory of viscous incompressible flows. Gordon and Breach, New York, 2nd edition, 1969. [7] P. Lagerstrom. Matched Asymptotics Expansion, Ideas and Techniques. Springer-Verlag, New York, 1988. [8] H. Lamb. Hydrodynamics, reprinted by Dover, New-York, 1945. [9] L. Landau and E. Lifcchitz. Fluid Mechanics. Addison-Wesley, New York, 1953. [10] J. L. Lions. Perturbations singulieres dans les problemes aux limites et en controle optimal, volume 323 of Lecture Notes in Math. Springer-Verlag, New York, 1973.

172 [11] H. K. Moffatt. Six lectures on general fluid dynamics and two on hydromagnetic dynamo theory. In R Balian and J L. Peube, editors, Cours de I'Ecole d'Ete de Physique Theonque, Us Houches, New-York, 1977. Gordon and Breach Science Publishers. [12] O. Oleinik. The Prandtl system of equations in boundary layer theory. Doki. Akad. Nauk S.S.S.R. 150, 4(3):583-586, 1963. [13] L. Prandtl. Veber Fliissigkeiten bei sehr kleiner Reibung. In Verh. Ill Intern. Math Kongr. Heidelberg, 484-491, Leibzig, 1905. Teuber. [14] H. Schlichting. Boundary Layer Theory. McGraw-Hill Book Company, New-York, 1979. [15] R. Temam. Behavior at time t = 0 of the solutions of semilinear evolution equations. J. Diff. Equations, 17:73-92, 1982. [16] R. Temam and X. Wang. Asymptotic Analysis of Oseen Equations in a Channel. To appear. [17] R. Temam and X. Wang. Asymptotic Analysis of the Linearized Navier-Stokes Equations in a Channel. Differential & Integral Equations, 8:1591-1618, 1995. [18] M. 1. Vishik and L. A. Lyusternik. Regular degeneration and boundary layer for linear differ­ ential equations with small parameter. Uspekki Mat. Nauk, 12:3-122, 1957.

173 A N o t e on the Distribution of Critical Points of Eigenfunctions Shing Tung Yau dedicated to Professor Gu for his seventieth birthday

This author has always been curious about the distributions of critical points of eigen­ functions when the eigenvalue A grows to infinity. Perhaps its growth is at least as large as vA. In order to gain some information, we study the two dimensional case where we prove that for certain eigenvalue A related to the curvature of the surface, there is always a critical point of the eigenfunction with some control on the value of this eigenfunction at this point. Let

If

dM ^ (j>, we assume dM is concave and we take either Dirichlet or Neumann conditions. Let —A be the eigenvalue. We define u = \V
(1)

where c is a constant to be chosen. The function u achieves its infimum at some point in the interior of M. (This followsp from the sharp maximum principle and the boundary conditions.) At this point, 0 < Au = 2 £ y$ + 2 £

Vi( A ¥>). + 2A'|V^r

+ 2c^(A¥>) + 2c|Vy 3 | 2

(2) 2A V 2

= 2j2tfj + 2 £ VW* " M* + 'I ^ + 2ap(Vip - \tf>) + 2c\Vip\2.

If Vtp ^ 0 at this point, we can assume ip\ j^ 0 and if 2 = 0. Then ui = u-i = 0 implies that
Supported by NSF Grant No. DMS-9206938

= 0

(3)

174 >Pi2 = 0.

(4)

Hence V22 = Ay? ^
(5) »(V-Ai+c)v». Putting these equations into (2), we obtain

0 < c V + (V - A + c) V + (V - A)| Vyf +VJ2 fiVi + K\V9\2 + cVip2 - c\(j>2 + c\V
2

(6)

ip^viVi

< [(2c - A)(c - A) + V2 + (3c - A)V + ^ - ] v > 2 2 [A' + V +

C

-A +

^

W

l

If we can find a constant c so that (2c - A)(c - A) + V2 + (3c - X)V + 1 ^ - < 0

(7)

and

K+V+c-\+ J^L

< 0.

(8)

Then the sharp minimum principle will give rise to a contradiction. Hence the as­ sumption that Vy? ^ 0 is violated. In other words, if (7) and (8) hold, the minimum of the function |V(/>|2 + c 0, (7) and (8) hold. Then there is a critical point XQ of

\i(x)+cV2(x)>cvt(x0) for all x G M. By integrating (9), we obtain the following

(9)

175 THEOREM 2. Iftp is normalized so that fM (p2 = 1, then ip has a critical point x0 where 2 / 1
A cArea(Af)

By following the shortest path from xo to the nodal line of ip, we obtain the following THEOREM 3. Let

surface

with eigenvalue —A. Let c be a constant so that (7) and (8) hold. Then there is a critical point XQ of ip so that the distance from xo to the nodal line of(p is not greater than ~^h-NOTE: If V = 0 and K < 0, we can choose c = A. If V = 0 and K < 1, we can choose c = | and (7) and (8) are equivalent to K < | . It can be seen that the estimate is optimal for the sphere.

Reference R. Schoen, S.T. Yau, Lectures on differential geometry. International Press, 1995.