Differentiable Functions on Bad Domains

(2)

lim IlMhu - ullp,Q = 0-

(3)

and if p < oo, then h ++0

Inequality (2) is a consequence of the Young inequality (1.1.2/5). To verify (3), we observe that

(Mhu)(x) - u(x) = f K(y)(u(x - hy) - u(x))dy.

(4)

An application of Minkowski's inequality yields IIMhu - ullp,n

IlKII1 sup

tEBh

f Iu(x + t) - u(x)Ipdx

\ 1/p

So

The last supremum tends to zero as h -4 0 by continuity in mean of functions

in Lp(1), p < 00.

1

Remark. The above argument implies that if u E Lp,lo,(Sl), p < oo, and

GCC1,then Mhu->uinLp(G) ash-40.

1

Clearly, equality (3) is not true for p = oo since the limit function u must be continuous in this case. However, the following property holds. 3. Let u E C(S2) and G CC Q. Then Mhu uniformly converges to u on G

ash -4 0. Indeed, (4) implies that I(Mhu)(x) - u(x) I < IlKIklsup{Iu(y) - u(x)l : y E Bh(x)}. We obtain the desired result because the supremum on the right converges to 1 zero uniformly on G. The assertion stated below will be used in Sec. 1.2.2.

Lemma. Let u c L1,10 (1). If

fucr7dx=0

(5)

1.2. Functions with Generalized Derivatives

9

for any cp E C0 00(Q), then u = 0 a.e. on Q.

Proof. First we prove that (5) holds for any bounded measurable function cp having compact support in Q. Let cph = Mhcp denote a mollification of cp with radius h > 0. Since c°h -4 cp in L1(1l) as h -+ 0, there is a sequence {hi}i>1 of positive numbers satisfying hi -+ 0 and cph; (x) -* cp(x) a.e. in 0. One may assume that

sUPPWh, CFC0, i=1,2,..., for a compact set F, and so

i = 1, 2(6) Furthermore, I(ph;(x)I < const IIWII.,si

for all x E SZ and i > 1 (see (2) for p = oo). Now the passage to the limit in (6) is possible by the dominated convergence theorem.

To conclude the proof, we fix an open set G such that G cc 0 and put '(x) = sign u(x) for x E G and '(x) = 0 otherwise. Then (5) implies u = 0 a.e. on G. Since G is arbitrary, u = 0 a.e. on Q. 1.2.2. Generalized Derivatives

Let SZ be a nonempty open set in R", a E Z+ a multi-index and u a locally summable function on Q. Definition. A function v E L1,10 (S2) is called the ath generalized or weak derivative of u if

f uD°rldx = (-1)I°`I

J

vrrdx

(1)

for any 1 E Co (S2). We write v = D°u.

The well-known formula (1) of integration by parts for u c C1(1) (1 = IaI) shows that the notion of weak differentiability generalizes the notion of classical differentiability. By Lemma 1.2.1, the weak derivative is uniquely determined up to a set of measure zero. The following properties are readily apparent from Definition.

If G is an open subset of 11 and v = D°u on Sl, then v = D°u on G. If v = D°u and w = DQv on S1, then w = D°+pu on ft However, the last

1. Basic Properties of Sobolev Spaces

10

property does not mean that the existence of -a derivative Dau implies the existence of derivatives D16u with Q <_ a (i.e., f3 < ai, i = 1, ... , n). Consider the following example. Let SI = {x E R2 : x1, x2 E (0, 2)}. If X denotes the characteristic function of the interval (0,1), then the function Q D X H u(x) = X(x1) + X(x2)

has the derivative 82u/8x18x2 = 0. However, the derivatives 8u/8x1, 8u/8x2 do not exist. Indeed, if we had 8u/8x1 = v E L1,10 (S2), then the equality e(1) = - f

(2)

z

would occur for arbitrary functions C,11 satisfying e E Co (0, 2), 11 E C0 (0, 2),

f?l(t)dt=1.

However, (2) is not true for the function (0, 2) 3 t F-a e(t) = f (k(t - 1)),

where f E Co (-1, 1), f (0) = 1 and k a sufficiently large positive number. I Nevertheless, we shall see later (cf. Corollary 1.2.3) that the existence of the gradient Olu E Lp(Sl) implies the existence of all intermediate derivatives Dau E Lp,10 (SZ), Jal < 1.

1

The assertion stated below suggests another equivalent definition of generalized derivatives.

Theorem. Let u, v E L1,,,,r

) and a E Z. The function v is the ach

derivative of u (v = D°u) if and only if there is a sequence {uk}k>1 C Coo (0) such that Uk -* u and Dank -+ v in L1,10 (S1).

Proof. We have only to show that if v = D' u is the weak derivative, then there exists the sequence of functions mentioned above. Let Mhu be a mollification of u. First we verify the identity

D"Mhu = MhD'u on the set Ph = {x : Bh(x) C S1}.

(3)

1.2. Functions with Generalized Derivatives

11

Put Kh(X) = h-"K(x/h), where K is the mollifier. By carrying out the differentiation under the integral sign and by using (1), we obtain for x E S2h (DaMhu) (x) =

f

Dy Kh(x - y)u(y)dy

(-l)ICkI f Dy Kh(x - y)u(y)dy

a

=

fKh(x - y)Dau(y)dy = (MhDu) (x)

Thus, (3) is true. Let Xh denote the characteristic function of the set S2h, and let uh = Mh(Xhu). Clearly, Uh = Mhu on the set 02h. Furthermore uh E COO (R") (since Xhu E L1,1o,(R")), and D'uh = MhD"u on f22h because of (3). Now

Uh -4 u, D°uh 3 D'u in

L1,10 (S2)

as

h -* +0

by Remark 1.2.1. This completes the proof of Theorem. 1 Combining Remark 1.2.1 and identity (3), we arrive at the following assertion.

Corollary. Let U E LP io,(S2) and Dau E L,10(), p < oo. If G is an open set satisfying G CC 1, then

Mhu -> u, D'Mhu - D"u in

Lp(G)

as

h -* +0.

1

The last theorem enables us to reformulate results of classical differential calculus for weak derivatives by means of approximation. For example, one can check the Leibnitz formula

D°(uv) _ 1: (

aQ

DAuDa-Av,

p
where Dpu E Lp ioc,(Q), DOv E Lp,,1 (1) for all Q < a (i.e., (3; < ai for

1
0

The lemma stated below is another consequence of the above results.

Lemma. Let SZ be a domain in R" i.e., an open connected set, and let u E L1,toc(Sl). If Vju = 0, 1 > 1, then the function u (possibly modified on a set of Lebesgue measure zero) is a polynomial in Pi-1.

1. Basic Properties of Sobolev Spaces

12

Proof. Let Mhu be a mollification of u and fix a subdomain G CC 0. For any sufficiently small h > 0 and any multi-index a, Ial = 1, we have D°Mhu = 0 on G by identity (3). Therefore, the function Ph = MhuIG belongs to PI-1. Applying Fubini's theorem, one can easily obtain the inequality IIPhll1,G <_ IIKIIIIIulli,u',

where K is the mollifier, f? is another subdomain satisfying G CC S2' cc 0 and h sufficiently small. Thus, the set {Ph} is a bounded subset of a finitedimensional space Pt_1i and {Ph;}t>1 is convergent for some sequence {h;}, hi -+ +0. Clearly, limPh; E P1_1. On the other hand, Ph, = Mh;u -4 u in L1(G) (see Remark 1.2.1). Now u is equal to a polynomial a.e. on G. The result follows by the arbitrariness of G. 1.2.3. The Spaces Lp(S2), W, (52), Vp(S2)

Let Q be an open subset of R", 1 < p < oo, I > 1 an integer. The space LIP(Q) consists of those functions in Lp,10 (S2) for which all generalized derivatives of order I are in Lp(S2). We equip Lp(Q) with the seminorm LP' (0) E) u H IlVlullp,n.

The spaces WP(1 ), VP(S2) are introduced as follows

Wp(1) = LI, (1) n L.' (Q), VI (Q) = nk=0LP(Q)

and endowed with the norms IIullW;(n) =

IIullp,c + IlVzullp,o,

Ilullygn) _

Ilokullp,n. k=°

The conventional notation LP = Wp° = Vp = Lp is used in case I = 0.

We remark that if u is an element of one of the spaces LP' (Q), WP'(Q) or VP '(Q), then u is determined only up to a set of Lebesgue measure zero. Let us agree to say that the function u is bounded, smooth, etc. if there exists a representative v which is equal to u a.e. on S2 and v is bounded, smooth, etc.

1.2. Functions with Generalized Derivatives

13

The following theorem suggests an equivalent definition of L!p(S2) in terms of distributions.

Theorem 1. The space LP' (Q) is the space of distributions on Q with derivatives of order l in LP(SZ).

Proof. We have to show that if u is a distribution on S2 with V!u E LP(S2), then u E LP,1OC(1).

Let w and g be nonempty open subsets of S2 such that w cc g cc Q. Let e > 0 be so small that the e-neighborhoods of w and g are contained in g and Q, respectively. We introduce cp E Co (S2) with epl9 = 1 and put v = cpu. Next, let 77 E Co (Be), 77 = 1 in a neighborhood of the origin. It is readily checked that the fundamental solution of the polyharmonic operator At is U(x)

rc

x12!-n,

cIx12!-n

for 21 < n or for odd n < 21, for even n < 21.

log lxi,

That is, for an appropriate choice of the constant c = c(n, 1), the equality 0!U = 6(x) holds. Then At (71U) = 8 + with £ _ A' ((77 - 1)U) E Co (Rn). By using the formula At

l!

_

D2a'

a!

a E Z+,

IoI=1

we arrive at !t

v+ *v=

Da(77U)*Dav,

(1)

Ia1=!

where the star denotes convolution. Note that * v E COO(Rn). Therefore, one has to verify that the sum in (1) is in Lp(w) to obtain v c Lp(w). Since (c) DacpDO-Qu,

D'(cpu) = p
Q

it follows that Dav = Da(cpu) = coDau on g.

Hence for every a E Zn with jal = l Da(r,U) * Dav = Da(,U) * coDau

on w.

1. Basic Properties of Sobolev Spaces

14

We complete the proof by observation that the integral operator with a weak singularity, applied to coDau, is continuous in Lp(w) (cf. Lemma 1.5.1). The assertion stated below is an immediate consequence of Theorem 1.

Corollary. If u E Lp(S2), then all intermediate derivatives Dau, lal < 1, belong to Lp,toc(SZ).

1

We conclude this subsection with the following simple fact.

Theorem 2. The spaces W, (1) and V1 (Q) are Banach spaces. Proof. Let {uj})>1 be a Cauchy sequence in V(SZ). Then {Dau3} is convergent in Lp(SZ) for all multi-indices a, lal < 1. Let

Ilui - ullp,n -- 0,

IIDau.i -

vallp,2 --> 0.

For any 77 E Co (SZ) we have

liz

uDarldx = lim I uiD°r dx

J

cz

_ (-l)' C" lim J r1D°uidx = (-1)IaI J v,,ijdx.

j+ sz

n

Hence va = Dau and the sequence {uj } converges to a function u c VP '(Q). The result has been verified for the space V1(0). The case of W1(SZ) is treated 1 in a similar way. The question of the completeness of the space Lp(SZ) will be considered later in Sec. 1.5.3. 1.2.4. Absolute Continuity of Functions in LP '(Q) In this subsection we show that there are representatives of elements in Lp(SZ) which can be regarded as generalizations of one-dimensional absolutely continuous functions. A function defined on SZ is said to be absolutely continuous on a straight line a if this function is absolutely continuous on any segment of a contained in Q.

Theorem. Let ) be an open subset of R". A function in Lp,10 (SZ), p > 1, belongs to the space LP(Q) if and only if this function (possibly modified on a

1.2. Functions with Generalized Derivatives

15

set of Lebesgue measure zero) is absolutely continuous on almost all straight lines which are parallel to the coordinate axes and its first classical derivatives belong to LP(Q). Furthermore, the classical gradient of the function coincides with the generalized gradient almost everywhere.

Proof. Let u E LP(1l) and let G be a bounded parallelepiped of the form G = {x : xk E (ak, bk), k = 1, ... , n}

such that G C Q. It suffices to show that u agrees a.e. on G with a function absolutely continuous on almost all lines in G parallel to the coordinate axes. For each k = 1,.. . , n, we write G = gk x (ak, bk), where gk denotes the projection of G to the hyperplane orthogonal to the xk-axis. Accordingly, a point x E G is written as x = (xk, xk) with x'k E A and xk E (ak, bk). By Theorem 1.2.2, there is a sequence {ui}i>1 C C°°(1) such that ui -4 u and Vui - Vu in L1,10 (1). In view of Fubini's theorem, {ui} has a subsequence (which we relabel as {ui}) satisfying lim

f

bk

i-+oo ak

(I ui(xk, t) - u(xk, t) I+ I Vui(xk, t) - Vu(xk, t) I )dt = 0

(1)

for each k = 1, ... , n and a.e. xk E 9k Let gk denote the set of all such x'k* Since ui is smooth on G, we have ui(xk, xk) - f kk

at`

(xk, t) dt = u=(xk, ak)

(2)

for all i > 1, Xk E (ak, bk) and xk E gk. Fix any xk E gk. Equation (1) says that the integral in (2) converges to

f k k at (xk, t)dt a /,

uniformly in xk E (ak, bk) as i -+ oo. Because ui(x'k, ) - u(xk, ) in L1(ak, bk), the numerical sequence on the right of (2) is convergent. Appealing again to (2), we obtain that the sequence ui(xk, ) converges uniformly on [ak, bk] to an absolutely continuous function. It remains to observe that, according to (1), this function agrees with u(xk, ) a.e. on (ak, bk).

To conclude the proof of the theorem, we show that if a function u is absolutely continuous on almost all lines, parallel to the coordinate axes, then

1. Basic Properties of Sobolev Spaces

16

its generalized gradient coincides with the classical one. Let vj = au/ax3 be the classical derivative of u. Then for any 71 E Co (fl) integration by parts gives

f rlv dx = - j z

n

a

axi

udx.

Thus, vj is the generalized derivative of u. The proof is complete. 1.2.5. On Removable Singularities for Functions in VP (S2)

Here we describe a class of sets of removable singularities for elements in VP '(Q),

1 C R. The description will be given in terms of the (n - 1)-

dimensional Hausdorff measure. So we begin with the definition of the 8dimensional Hausdorff measure. Let E be a set in R. Consider various coverings of E by countable collections of balls of radii < E. For each s > 0, let

v, (e) = v, inf 1: r:, where r;, is the radius of the ith ball, v8 > 0 and the infimum is taken over all such coverings. By the monotonicity of o,,, there exists the limit (finite or infinite)

H9 (E) = lim a, (E). E-++o

This limit is called the s-dimensional Hausdorff measure of E. Ifs is a positive integer, then v, = mes,(B(9)), otherwise v, is any positive constant. For example, one may put v, = is/2/I'(1+s/2) for any s > 0. In case s is a positive integer, s < n, the Hausdorff measure H. agrees with the s-dimensional area of an s-dimensional smooth manifold in R'. In particular, H,, (E) = mesn(E) for Lebesgue measurable sets E C R" (see e.g. Federer [59, 3.2], Ziemer [221, 1.4.2]).

Theorem. Let u E VP (S2 \ F), where S2 C R" is an open set and F C S2 is a closed in SZ set satisfying

0. Then u E Vp(S2).

Proof. If u E VP ((1 \ F), then u and D'u, jc < 1, are defined almost everywhere on 0 and hence Dau E Lp(S2). It suffices to verify that Dau is the generalized derivative of u on S2, i.e., for all 71 E C0(S2)

fuDai)dx = (-1)H"H f7)DQudx. z

(1)

1.3. Classes of Domains

17

Let

pi (x) = (x1, ... , xi-1, xi+1, ... , xn), 1 < 2 < n,

denote the projection of a point x E Rn on the hyperplane orthogonal to the xi-axis. By assumptions, each set pi(F) has (n - 1)-dimensional Lebesgue measure zero. Thus, almost every straight line which is parallel to the xi-axis is disjoint from F. According to Fubini's theorem, we have

8udx= 77

Ju

axi

rlaudx 1
dx,J L(XI)

S2'

axi

where St' = pi(1), x' = pi(x) and £(x') is the intersection of SZ with the line x' = const. Note that £(x') is in Q \ F for almost all x' E S2'. An application of Theorem 1.2.4 leads to dxi f au f dx' J£(XI) axi ,

/' dx' /'

J'

J£(x')

axi

Jo axi

Equality (1) follows for Ian = 1 = 1. The general case can be concluded by induction on 1. Indeed, let l > 2 and let (1) hold for u E VP-1(Q \ F) with

lal <1-1. If uEVP(1\F), lal=1, and D° =D2D' forsome 1< < i < n, the left part of (1) equals

_

f au Dpidx 1 axi

(2)

since u E VP (Sl \ F). By the induction hypothesis, expression (2) coincides

with the right part of (1) (because au/axi E VP'-'(Q \ F)). This completes the proof.

1.3. Classes of Domains In this section we describe some classes of domains in Rn which are useful for the analysis of properties of Sobolev functions.

1.3.1. Domains of Class C and Domains Having the Segment Property Here we deal with the following two classes of domains.

1. Basic Properties of Sobolev Spaces

18

Definition 1. A bounded domain 52 C R" is said to belong to the class C if each point of 852 has a neighborhood U such that the set U fl 52 can be represented by the inequality x,a < f (xl,... , xii_1) in some Cartesian coordinate system where f E C(G) and G is a domain in R"-1

Definition 2. A bounded domain SZ C R" has the segment property if for each point of 852 there exist a neighborhood W of this point and a nonzero vector b E R" such that x + tb E SZ whenever x E 52 fl W and t E (0,1). Clearly, every domain of class C has the segment property. The following theorem contains the converse assertion.

Theorem. SZ has the segment property if and only if it is in the class C.

Proof. Let us verify that any domain 52 having the segment property is in the class C. For a fixed point a E 852, let W and b be the corresponding neighborhood and the vector from Definition 2. We introduce local Cartesian coordinates y with y = 0 at x = a and with basis unit vector along the axis y to be -b/lbl. One may assume without loss of generality that a = 0. The open line segment

l(y) = {(y', yn - tjbi) : t E (0, 1)}, y' = (y1, ... , Y.-j)

is associated with a point y = (y', y,,). Introduce the closed line segment L={y:y'

=0, -P
where a is chosen to be so small that L C W. The points of L with y < 0 are in 52 since they are in l(0) and 0 E S2. On the other hand, if y E L and

y,a>0,then y¢52because 0E1(y)and0¢Q. We shall construct a neighborhood U of the point a = 0 such that the set U fl SZ can be represented by the inequality y < f (y') with a continuous function f.

Construction of U and f. Let Qd°) (z) denote the open cube in R" with center z E R" and edge of length 2d. For some sufficiently small r > 0, the neighborhood W contains the cylinder U=Qx

(_'0"0)'

Q=

Qr"-1)

(0),

and, moreover,

Q x(-p, -e/2) c 52,

Q x (e/2,,o) c W\ Q.

(1)

1.3. Classes of Domains

19

We define f : Q - R1 by 1 (y') = sup{yn : y = (y', yn) E Sl}.

(2)

Inclusions (1) imply that the set of values yn in (2) is not empty and that If (y') < B/2. Clearly, the supremum in (2) is attained. Furthermore

Un{y:yn< f(y')}CSZ, Un{y:yn> f(y)}CW\Si;

(3)

the first because y E l(y', f (y')), and the second by the maximizing property of f (y'). Thus, the point (y', f (y')) is on 8S2 and is the only such point in U for any fixed y' E Q.

Continuity of the function f : Q -* R'. Given y' E Q and E E (0, o/2), define

y+ _ (yo, PAD + 0, y- _ (yo, PAD - e) By (3) and the estimate If (y') I < e/2, we have

y- E UnSI, Y+ E U\c, and hence Q8(n)

(y-) c u n St, Qan) (y+) C U \ SZ

for some 6 > 0. Let us verify that

If (y') - f(yo)I < e if y' E Qan-1)(y0) Since Qan) (y_) C 0, the inequality f (y') < f (yo) -e contradicts the maximum property of f (y'). If f (y') > f (yo) + e, then l (y', f (y')) n Q(n) (y+)

0,

which is again a contradiction, because (y', f (y')) E n and Q(n) (y+) C U \ SZ. The proof is complete. 1 The proof of the theorem also contains the following assertion.

Corollary. Suppose S1 is a domain in Rn having the segment property. Let a = (a', an) E 8S2 and let W and b be the corresponding neighborhood of a and the vector from Definition 2. If b is parallel to the xn-axis, then W contains a cylindrical neighborhood of a of the form 1 (a') x (a, /3) in which the

20

1. Basic Properties of Sobolev Spaces

part of 852 can be represented by the equation xn = f (xl,... , xi_1) with f E C(QTn-1)(a')) 1.3.2. Domains Starshaped with Respect to a Ball and Domains of Class Co,I

Definition 1. The class C°,1 of domains (as well as the classes Cl and C',I) is described by Definition 1.3.1/1 if we require that the function f from this definition is in C°"1(G) (in C1(G) or in C""' (G)).

Definition 2. A domain 12 C R' is called starshaped with respect to a set G, G C 11, if any ray with origin in G has a unique common point with Oil.

We now study geometric properties of domains starshaped with respect to a ball. The following assertion links the class of such domains to the class Co,1.

Lemma. A bounded domain starshaped with respect to a ball belongs to the class Co,l

Proof. Let SZ be a bounded domain starshaped with respect to a ball Be centered at the origin O. First we show that for all x, y c 851 with cp < 7r/3 the inequality Ix - yI <- 2e-1IxIIyI since

(1)

holds, where cp is the angle between the vectors x and y. We claim that the straight line t, passing through the points x, y, does not intersect the closed ball Be/2. Suppose that z c tf1B,12. Since 1 is starshaped with respect to the point z, this point belongs to the segment [x, y]. Consider the triangles Oxz, Oyz. We have IxI >-e, IyI >-e, IzI <<e/2, whence

IzI <- Iy - zI, IzI < Ix - zI. These inequalities imply

LOxz < LxOz, LOyz < LyOz.

Therefore, the angle cp = LxOy is not less than 7r/3, which contradicts the assumption cp < 7r/3.

1.3. Classes of Domains

21

The distance from the origin 0 to the line a is IxIIyIIx - yl-1 sin o and is greater than o/2 because I fl Be/2 = 0. Hence (1) follows. Local coordinates. To an arbitrary point a E 852, we correspond the Cartesian coordinates x in Rn with origin 0 at the center of the ball Be and with along the vector Oa. So a = (0, ...0, Ial). the

For any fixed x' E Ri-1, Ix'I < p/2, the ray {(x',Xn) : x > 0} intersects at a unique point. Therefore, the part of 52 contained in the cylinder {(x', xn) Ix'I < p/2, x,, > 0} can be represented by the inequality 0 < BE"-1) Xn < f (x'). By Corollary 1.3.1, the function f is continuous on with some e E (0, p/2]. We may choose e E (0, 0/4] to be so small that the angle 13

:

ep = LxOy between the vectors x = (x', f (x')) and y = (y', f (y')) is less than BEi-1). ir/3 whenever x', y' E We shall show that

8p-1DIx' -y'I, x',y' E

if W) - f(y')I

B(n(2)

where D is the diameter of Q. Indeed, inequality (2) implies

If (x') - f (y') 15 (here

2p-1

(Ixl2IyI2 - (x, y)2)1/2

(3)

is the inner product in Rn). Since (Ix12Iy12

(Ix'

- (x,y)2)1/2 12)1/2

I2Iy' 12 - (x/, y/)2 + If (XI)y' - f (y/)x/

< Ix'I ly'l sin(Lx Oy) + If (x')y' - f (y')x'l and

Ix'IIy l

sin(Zx'Oy') < DI! - y'l,

If(x')y'-f(y')x'I <-Dlx'-y'l +elf(x')-f(y')I, it follows that the right hand part in (3) does not exceed 4p-'Dl x' - y'I + I f (x') - f (y')I/2. Thus (2) is valid which completes the proof of Lemma. In fact we have also established the following assertion.

1

Corollary. Let S1 be a bounded domain starshaped with respect to a ball Be centered at the origin of spherical coordinates (r, w). If 852 has a representation r = r(w), then S' ' E) w H r(w) is a uniformly Lipschitz function.

1. Basic Properties of Sobolev Spaces

22

Indeed, if x, y E 8Q, D is the diameter of ) and w2 = x/IxI, inequality (1) yields

Ix - yI < 4e-'D2 sin whenever I wx - wy I < 1.

= 2,o-'D 2Ic.rx - WYI

2

1

Remark. Let S2 be a bounded domain starshaped with respect to a set G C Q. Then [x, y] C S2 if x E G, y c 52, (4)

where [x, y] is the closed line segment with endpoints x, y. Conversely, if G is an open ball, one can easily check that ) is starshaped with respect to G when (4) holds. However, the following example

S2=Bi"1\Ix

x E [1/2,1)}, G={0},

shows that this converse assertion is not true in case G is a point. 1.3.3. Domains Having the Cone Property

Definition. A domain 0 C R" has the cone property if each point of S2 is the vertex of a cone contained in 0 along with its closure, the cone being represented by the inequalities x1 + ... + xr21_1 < ax,2a, 0 < x < b in some Cartesian coordinate system, a, b = const. Remark. One can easily verify that any domain of class C°"1 possesses the cone property. The example of a ball with deleted center shows that the converse assertion is not true. Lemma 1. Let S2 be a bounded domain having the cone property. Then S2 is the union of a finite number of domains starshaped with respect to a ball. We observe that any domain with the cone property is a union of congruent

cones. Thus, it is a union of domains starshaped with respect to balls of a fixed radius. Hence Lemma 1 is a consequence of the following lemma.

Lemma 2. If a bounded domain S2 is the union of an infinite number of domains Ga starshaped with respect to balls BR(xa) C G0 of a fixed radius R, then for each r < R there exists a finite number of domains f2k (1 < k < N) starshaped with respect to balls of radius r satisfying Uk 11 k = ft

Proof. Let G0,1 be a domain in the family {G0,}. We put S21 = UpGp, where the union is taken over all indices 3 such that I xp - xal I < R - r.

1.3. Classes of Domains

23

Clearly, any of the balls BR(xp) contains the open ball B1 = B,(x,,,). Since any Gp is starshaped with respect to B1, it follows that [x, y] c Slj whenever x E B1, y E 521. Hence Q is starshaped with respect to B1. Let G0, be any one of the domains G,, not contained in 521. Repeating the above argument, we construct a domain 522 starshaped with respect to a ball B2 = B, (x02) whose center is situated at a distance greater than R - r from the center of B. In a similar way one can define a domain 523 starshaped with respect to a ball B3 of radius r and center situated at a distance greater than R - r from the centers of the balls B1i B2, etc. Clearly, this process will stop after a finite number of steps because the centers of the balls B1i B2, ... are in a bounded domain and the distance between any two of them is greater than R - r > 0. 1.3.4. Domains of Class C°'1 and Lipschitz Domains Domains of class C°'1 are sometimes called Lipschitz graph domains. We now introduce a wider class of Lipschitz domains. Let SZ and G be domains in R" and let F : Sl -> G be a homeomorphic mapping. We say that F is a quasi-isometric mapping if there exists a constant

M such that lim sup x-'xo

IF(x) - F(xo) I < M, Ix - xol

lim sup y--'yo

I

F-1(y) F-'(yo) I < M y - yoI

-

for any xo E Q, yo c G, and the Jacobian det F' (x) preserves its sign in Q. Definition. A bounded domain SZ is called Lipschitzif each point of its bound-

ary has a neighborhood U C R" such that a quasi-isometric transformation maps U f152 onto a cube.

Clearly, domains of class C°'1 are Lipschitz. The following example shows that the converse is not true. It turns out that a Lipschitz domain may even fail to have the cone property (cf. Remark 1.3.3).

Example. Let S1 C R2 be the union of the rectangles Pk = {x : Ix1 - 2-kI <

2-k-2, 0 < x2 < 2-k-2}, k = 1, 2,... and the square Q = (0,1) x (-1, 0). It is obvious that SZ does not have the cone property (and hence is not in Co,l) Nevertheless, we shall show that SZ can be mapped onto Q by a quasi-isometric mapping. Let F° : x -4 y be defined on SZ as follows: y = x on Q, and

yl = (x1 - 2-k)(1 -2 k X2) + 2 -k, y2 = x2 if x E Pk.

1. Basic Properties of Sobolev Spaces

24

It can be easily verified that F° is a quasi-isometric mapping whose image F°S2

is the union of the square F°Q and the set

{y:yiE(0,1), O5y2
If (s) - f(t) I < 41s- tI, 0 < f (t) < 1/8. X2

1

1/8

1/4

1/2

Q

21

Q

1/8

1/4

1/2

F°S2 Fig. 1

Let 71 be a continuous piecewise linear function on R1, i7(t) = 1 for t > 0 and j(t) = 0 for t < -1. The transformation F1 : y --> z defined by Z1 = y1, x2 = y2 - f (Y1)77(y2)

is uniformly Lipschitz on F°S2 (i.e., each coordinate function is uniformly Lipschitz on F°S2) and F1F°S2=Q. Furthermore, F1 is a quasi-isometric mapping since the Jacobian of F1 is greater than 1/2. Now 52 is mapped onto Q by the quasi-isometric mapping F1F°.

1.4. Density of Smooth Functions in Sobolev Spaces 1.4.1. Approximation of Functions in Sobolev Spaces by Functions in C°° (0) It follows from Theorem 1.2.2 and Corollary 1.2.2 that every element u of one of the spaces L,(1), WP '(Q) and VP(S2) can be approximated by smooth functions on inner subdomains of Q. That is, there exists a sequence {Ilk}k>1 C

1.4. Density of Smooth Functions in Sobolev Spaces

25

C°° (Sl) such that Uk and its derivatives converge to the corresponding derivatives of u in Lp,1oc(S2) if p < oo. Here we show that such approximation can be carried out on all of SZ and not only on inner subdomains of Q. We begin with a standard result on existence of a smooth partition of unity subordinate to a locally finite covering.

Definition. Let {SZk}k>1 be a countable family of open subsets of an open set SZ C R". This family is called a locally finite covering of SZ if

Q = Uk>lnk, 1k cc SZ, k = 1, 2, ... , and any compact subset of SZ intersects only a finite number of sets Al. A smooth partition of unity subordinate to the covering {Ilk} is a set of functions {c'k}k>1 such that

Wk E CO (1k), 0 < 0k !5 1, E Wk (X) = 1, X E k>1

The following lemma says that such partitions of unity exist. Lemma. Let {SZi}i>1 be a locally finite covering of Q. Then there is a smooth partition of unity subordinate to the covering {S2i}.

Proof. First we construct another locally finite covering {Sli}i>1 of the set SZ

such that 1 CC R. Let F1 = 1 \ Ui>25Zi

Then F1 C SZ1 cc SZ and F1 is closed in Q. Hence F1 CC 521. As 1' we take

an open set such that F1 cc Q' cc SZ1. Clearly, the sets SZi, Q2.... form a locally finite covering of Q. Now a set S12 CC SZ2 can be defined in a similar

way, etc. We thus obtain the desired covering {;b>1. Let {iii}i>1 be a family of functions satisfying ili E Co (SZ), 0 < iii < 1, i)i(x) = 1, x E SZL

(the existence of such family easily follows from the properties of mollification mentioned in Sec. 1.2.1). Since

E7]i(x) > 1, i>1

x E SZ

1. Basic Properties of Sobolev Spaces

26

and the sum contains only a finite number of nonzero summands for any fixed x, the required partition of unity can be defined by 7)k(x)

x

xESZ) k>1.

Ei>1 77i (X)

This completes the proof.

1

Remark 1. A locally finite covering of an open set always exists. For example, it can be constructed in the following way. Given SZ, put G_1 = Go = 0, Gk = {x c SZ n Bk : dist(x, 8SZ) > k-1}

,

k = 1, 2,...

.

Then the collection of sets 1 k = Gk \ Gk-2i k = 1, 2, ..., forms a locally finite covering of SZ.

I

Two theorems stated below show the possibility of approximating any element in LP' (Q), W1(12) and 1' (Q) (p < co) by functions in COO (Q).

Theorem 1. Let SZ be an open set in R". If u E Lp(SZ), 1 < p < oo, then there is a sequence

{ui}i>1 c C°°(n) n L,(SZ)

such that ui -* u in Lp,10 (1) and IIVI(ui - u)llp,u -+0.

(1)

Proof. Let {SZk}k>1 be a locally finite covering of SZ and let {cPk}k>1 denote

a partition of unity subordinate to the covering {SZk}. Consider a sequence {ei}i>1 satisfying Ei E (0,1/2),

Ei -> 0.

Suppose u E LP' (Q). For any k, i > 1, we introduce a mollification Vk,i of the function cpku, the radius of the mollification being so small that supp Vk,i C SZk and (2) II(Pku - vk,illp,i + IIVI(Wku - vk,i)Ilp,i < Ek, i, k > 1.

Clearly, the function ui = > k> 1 vk,i is in C°° (SZ) for each i > 1. We have u = Ek>1 Wku, where the sum contains a finite number of nonzero terms on every open set w, w CC Q. Therefore IIu - ui lllp,w + II VI (u - ui)

IIp,-'

< E (Il wku - vk,i llp,u + II VI (Wku - Vk,i) Ilp,w) k>1

< ei(1 - Ei)-1.

1.4. Density of Smooth Functions in Sobolev Spaces

27

By Fatou's lemma Ilu - uillp,,, + IIV1(u - ui)IIP,n < 2Ej, i > 1,

(3)

and hence ui E L,(1). Now (3) implies that {ui} is the required sequence. I The following theorem can be proved in a similar way.

Theorem 2. The space WP(SI) n C' (Q) is dense in W, (Q) and the space VP (SI) n C°° (SI) is dense in Vp (SI) if 1 < p < oo.

Remark 2. If u E C(Q) n L,(1), 1 < p < oo, then there exists a sequence {ui}i>1 C C-(Q) n L,(0) satisfying lIV (ui - u)llp,n +sup{lui(x) - u(x)I : x E SI} -+ 0. Analogously, if u E C(Q) n W, (1), p < oo, then there is a sequence {ui}i>1 C C°° (SI) n W, (SI) such that Ilui - uII WP(n) + sup{lui(x) -u(x) I: x E SI} -> 0.

The space WP(SI) can be replaced by VP(Q) in this last assertion.

Indeed, using the notation of Theorem 1, we can require in addition to (2)

that Il1Pku -

Vk,illo,n < Ek,

i, k > 1.

Then for every compact set F CSI Ilu - uillC(F) <- :IIWku - Vk,illC(F) <- 2Ei. k>1

1.4.2. Approximation by Functions in Cm (SI)

Note that the space CI (Q) cannot be always replaced by C°° (S2) in theorems of the preceding subsection. Consider the following example.

Let (r, 0) with 0 < 0 < 2ir be polar coordinates. The boundary of the domain SI = {(r, 0) : r E (1, 2), 0 E (0, 2-7r) } consists of the circles r = 1, r = 2

and the interval {(r, 0) : r E (1, 2), 0 = 0}. The function u = 0 is summable on SI along with all its derivatives, but it is not absolutely continuous on segments of straight lines x = const > 0 which intersect Q. By Theorem 1.2.4,

1. Basic Properties of Sobolev Spaces

28

the function u does not belong to Lpl(S21), where S21 is the annulus {(r,O) : r E (1, 2), 0 E [0, 2ir)}. Therefore, the derivatives of this function cannot be approximated in mean by functions in C°° (fl). A necessary and sufficient condition for the density of C°°(S2) in Sobolev spaces is unknown. The following two theorems give simple sufficient conditions.

Theorem 1. Let S2 be a domain of class C. Then the space C°° (1) is dense in Lp(1l), WP(Q) and VP '(Q), p E [1, oo).

Proof. We restrict ourselves to the space VP '(Q). By Theorem 1.4.1/2, it suffices to approximate a function u E C°° (Q) n VP (S2) by functions in C°° (Q).

Let {Ut}i' 1 be a finite covering of 8S2 such that U; n 8S2 has an explicit representation in Cartesian coordinates for every i = 1, . . . , N. Let, furthermore, {77j}i' 1 be a smooth partition of unity subordinate to this covering. It will suffice to construct the required approximation for ur7t. We can assume without loss of generality that

0

0<x, < f(x)},

where G C Ri-1, f c C(G) and f (x') > 0 on G. Suppose u has a compact support in S2 U {x : x = Ax'), x' E G}. Clearly, the function u,(x) = u(x', x,, - e) is smooth on S2 for any sufficiently small positive e. It is obvious

that IID°(u. - u)Ilp,n = II(Dau)E - D'ullp,n -> 0 as e -4 +0 for every multi-index a, 0 < I al < I. The result follows.

1

Theorem 2. If S2 is a bounded domain starshaped with respect to a point, then C°°(1) is dense in W1(1l) and Vp(1), p E [1, oo). The same is true for the space L,(1), i.e., for any u E LP(Q) there is a sequence {u;}j>1 c C- (a) such that (1.4.1/1) holds. Proof. We may assume that 0 is starshaped with respect to the origin. Given a function v defined on 0, put

(A1v)(x)=v(XE(1+i-')1, i=1,2..... It is an easy matter to show that Aiv -* v in Lp,1OC(Sl) if v E Lp,1°°(S2), and the convergence holds in Lp(S2) if v E Lp(S2).

1.4. Density of Smooth Functions in Sobolev Spaces

29

Let u E LP(s)). By the definition of generalized derivatives

D°(Atu) = (i+ 1)` Ai(D°u), lal = 1, and therefore Aiu E LP((1 + i-1)Sl). Furthermore, we have IID°(u-Aiu)IIP,n <

(1- (i+1)') IIAi(D°u)IIP,n

+IIAi(Dau) - D'uII P,j2.

The right part tends to zero as i -4 oo. Hence I I V1(u - Aiu) I IP,Q - 0. Since SZ CC (1+i-1)Q and in view of Corollary 1.2.2, there is a mollification ui of Aiu with sufficiently small radius such that ui E Cm(S2), IIui - AiuIIWP(n) < 1/i.

Now {ui}i>1 satisfies (1.4.1/1). The spaces WP (1) and VP (SZ) can be treated in the same way.

I

Remark. The condition SZ E C and the property of 0 to be starshaped with respect to a point are not necessary for the density of C' (Q) in Sobolev spaces. Consider the planar domain

SZ={(x,y)ER2:XE(0,1), 0
First we state the following lemma.

1. Basic Properties of Sobolev Spaces

30

Lemma. Let S2 be a domain in R. If u E LP(S2), p E [1,oo), then the sequence of functions min{u(x), k} u(k)(x)

if u(x) > 0,

max{u(x), -k} if u(x) < 0,

(k = 1,2 ....) converges to u in Lp,1°°(S2) and the sequence of their gradients converges to the gradient of u in Lp(Q). The same is valid for the sequence

u(x) - k-1 if u(x) > k-1, u(k)(x) =

0

if lu(x)I < k-1,

u(x) + k-1

if u(x) <

-k-1.

Proof. By Theorem 1.2.4 u(k) E LP(S2) and Vu(k) = XkVU, where Xk is the characteristic function of the set {x E Q: lu(x) I < k}. Thus, IIV(u - u(k))IIp,n = 11(1 - Xk)VUIIp,1Z.

The convergence to zero of the right side follows from the dominated convergence theorem. The same theorem provides the convergence u(k) -* u in Lp,1°°(a) The proof for the sequence u(k) is similar. 1 Theorem. The set of functions in LP '(0) fl L,,.(S2) f1 C°° (S2) with bounded support is dense in Wp1(S2), 1 < p < oo.

Proof. Let u E Wp (S2). Since u(k) -+ u and u(k) -* u in Wp (S2), the set of bounded functions u E L' (Q) with mes,a(supp u) < oo is dense in WP (S2). Suppose u satisfies these conditions and consider the sequence

uk(x) = q(x/k)u(x), k = 1,2,..., where q E Co (B2), q = 1 on B1, 0 < q < 1. Then we have c IIuk - UIIW (o) <- IIUIIp,S1\Bk + II VUIIp,o\Bk

+ k

-> oo. It remains to approximate every function Uk by functions in

LP (0) f1 Lc,,, (S2) f1 Coo (S2) with bounded supports. Such approximation can be

1.5. Poincare's Inequality and Equivalent Norms in Sobolev Spaces

31

constructed in the same way as in Theorem 1.4.1/1 (with the aid of a partition of unity and mollification). 1 The preceding lemma and theorem imply the following corollary.

Corollary. If SZ is a domain with finite volume, then for any u E Lp(SZ), p E [1, oo), there is a sequence {uk} C LP(SZ) fl Lm(SZ) fl C°° (SZ) with bounded

supports such that Uk -+ u in Lp,1a°(1) and IIV(u - uk)IIP,n - 0.

Remark. The assumption mesa(Q) < oo is essential in the last assertion. For example, the set Co (R') is not dense in L1(R') if p < n. Indeed, by passing to spherical coordinates and by using Hardy's inequality (1.1.2/7), we obtain II IXI-'VIIP,Rn < C IIVVIIP,Rn

for any v E Co (Rn). Hence JIVIIP,Bi <- CIIovilp,R^,

and there is no sequence in Co (Rn) convergent to 1 in Lp,1oc(Rn) such that the sequence of gradients is convergent to zero in LP(Rn).

1.5. Poincare's Inequality and Equivalent Norms in Sobolev Spaces 1.5.1. Sobolev's Integral Representation We begin with the following simple assertion.

Lemma. Let SZ be a bounded domain in Rn with diameter D. Suppose l > 0 and put (Tv) (x)

= fn

I

v(yn-1 dy,

x E Q.

(1)

Then T is a continuous linear operator: LP(SZ) -p LP(SZ), 1 < p < oo, and IITII <- n mesn(Bl)D1/l.

(2)

Proof. Let K(z) = Izj`-nX(z), Z E Rn, where X is the characteristic function of the ball BD. Extending v(y) to be zero outside SZ, we obtain Tv = (K * v) I n. Now (2) is a consequence of the Young inequality (1.1.2/5). 1

1. Basic Properties of Sobolev Spaces

32

The following integral representation is a useful tool for analysis of summability and continuity of elements in Sobolev spaces.

Theorem 1. Let SZ be a bounded domain starshaped with respect to a ball B6i B5 C 0, and let u E LIP (Q). Then for almost all x E S2 x

E

u(x) = d-"

13

f APP (y) u(y)dy

a

6

101
+E IcI=i

f f. (X, o

,B)D'u(y)dy,

(3)

where r = Ix-y1, B= (y-x)/r, cpp E C0 (B1), fc, are infinitely differentiable functions such that ff1 < c (D/b)n-1,

c is the constant independent of 11 and D is the diameter of Q.

Proof. It suffices to put b = 1. Let cp E Co (B1) and jcp(z)dz = 1. B,

First we assume that u E C°O(Sl). If x E S2, z E B1, the line segment [z, x] is contained in S1 and so the following Taylor's formula applies:

u(x) _ E D u(z) (x - z)0 IPI
+ l f'(1

- t)i-1 E Dau(z + t(x - z)) (x - z)adt. ICI=1

u(x)

= E fB, D )3! z) (x

- z)13W(z)dz

101
+1

1

a! 1aj=1

f

dt f (1 - t)1-1Dau(z + t(x - z)) (x - z)',p(z)dz. l

(4)

1.5. Poincare's Inequality and Equivalent Norms in Sobolev Spaces

33

Since

f D0u(z)(x - z)0cp(z)dz l

(-1)101

fj u(z)Dp((x -

101<1,

the former of the last two sums in (4) is a polynomial of degree 1 - 1 in R' which can be written in the form of the first term on the right side of (3). We extend DUu to be zero outside S2 for jal = 1 and perform the change of variable y = z + t(x - z) in the last integral over Bl in (4). Then

x - z = (1 - t)-1(x - y), dz = (1 - t)-"dy, and the general term of the latter sum in (4) is

a.

I

f

dy

f D°u(y)(x - y)°cp

(y - )

1

1 -tt

(1

-dt)n+1.

The last expression can be given as

I

al R n D°u(y) (x - y)°K(x, y)dy

with 1

K(x> y) =

fo

tx) 1-t

dt

(1 - t)n+l = r

n

r

cp(x + ee)en lde.

Note that the function Rn \ {y} D x H K(x, y) is in Cm(Rn \ {y}) for any fixed y E Rn. Moreover, K(x, y) = 0 if the line segment [x, y] is not contained in the "cone" Vx = UZEB1 [x, z] (see Fig. 2).

Fig. 2

1. Basic Properties of Sobolev Spaces

34

Thus, formula (3) holds where

fa(x,r,0)=

(a)llea Jrr

v(x+00)en-'dp.

Let us verify the estimate 1A,1 < cDn-1. We have

f

Iff(x,r,0)I <_

I de.

I

+p8EB1

The last integral is dominated by 2 sup { I cp(z) I : z E Bl }, which gives the required estimate. Representation (3) has been proved for u E C°O(S2). Suppose u E L,(1) and p < oo. By Theorem 1.4.2/2, there is a sequence {u%};>1 such that

u; E C°°(0), ui -> u in Lp,1OC (0),

V1(u: - u)Ilp,o -> 0.

Passing to the limit in (3) for ui and using the continuity of integral operator

(1) in L,(11), we arrive at (3) in the general case. If p = oo, (3) is also valid because L00' (Q) C LP(Q) for any p < oo. This completes the proof of Theorem 1. A simpler integral representation holds for u E Co (Rn).

1

Theorem 2. If u E Co (Rn), then u(x)

(nv) n

a

l

IR"

Du(y) d1

,

where vn = mesh (Bl) and the remaining notation is the same as in Theorem 1.

Proof. For fixed x E Rn and 0 E Sn-1, we have u(x)

f r1-1 art u(x + rO)dr.

Since

at

Ba (Dau) (x + r9),

1111 u(x + rB) Ia1=1

1.5. Poincare's Inequality and Equivalent Norms in Sobolev Spaces

35

it follows that u(x)

r1-1 °

1:

Q

j (Dau) (x + rO)dr. B

IaI_1

Integration over 0 E Ss-1 yields (5). 1.5.2. Generalized Poincare Inequality The following assertion will be frequently used throughout.

Theorem. Let 1 be a bounded domain having the cone property and let w be an arbitrary nonempty open set, i C Q. Put

SZE={xER":x/eESZ}, EE (0,00). Then for any u E Lp(Q,:), p > 1, there is a polynomial PE E P1_1 of the form

PE(x) = E-"

((xl 13 101<1

\E/

Q (Y) \EI

u(y)dy

such that W E Co(w), 101 < 1, and the estimate Ilok(u- PE)IIp,c

G c(n,l,w,Q)E1-k IIolullp,i,

(1)

holds fork=0,...,1-1. Proof. It suffices to consider the case E = 1 and then use a similarity transformation. Clearly, we may also assume that w is a ball. First, suppose that p < oo and u E CI(Q) n Lp(S2). By Lemma 1.3.3/1, SZ is the union of a finite set of subdomains starshaped

with respect to a ball. Let G be any of such subdomains and let B be a corresponding ball. We construct a finite collection of balls {B(`)}!"_0 such

that B(°) = B, B(`) n B(i+1) # 0, B("`) = w.

Note that G is starshaped with respect to a ball contained in B(°) n B(1). An application of representation (1.5.1/3) to any derivative D'ru, IryI = k, yields IID"ullp,G <- IIQIIp,G + c(n, 1, G) E IaI=1-k

IIT(D"+7u)IIp,G.

(2)

1. Basic Properties of Sobolev Spaces

36

Here T is the integral operator with kernel Ix - yI t-k-' defined on Lp(G); the polynomial Q E PI-k-1 may be given by

Q(x) = (-1)171 E x0 fBOflB(1) D7iPP(y)u(y)dy IP1<1-k

and 00 E Co (B (0) fl B(1)). It is readily seen that IIQIIP,G < c(n, 1,

G)IIUIIp,B(o)nB(1)

Combining the last inequality, estimate (2) and Lemma 1.5.1, we arrive at IIVkuIIp,G < C (IJVIUIIP,G + IIuIIp,B(1)nB(o)) , 0 < k < 1,

where c does not depend on p. Also IIuJIp,B(+) S C

IIuIIp,BMnB('+1))

for i = 1, ... , m - 1. Hence IlokuIlp,G < C (IIotuHIp,c + IIuHIP,w)

Summing over all G, we deduce II VkuIlp,c < c(n, 1, 0, w) (II VzuIIP,sc + IIuIIP,.)

(3)

Once (3) has been proved for u E Lp(Q)f1COO(1l), Theorems 1.4.1/1 and 1.2.2

imply that every element in Lp(1k) has the gradient of order k in Lp(1) and that (3) holds for any u E Lp(SZ), p < oo. From (1.5.1/3) for u E Lp(w) and Lemma 1.5.1 follows the inequality IIu - PIIp,w < c(n,1,w)IIo(ullp,, where

P(x) _

xP f cPP(y)u(y)dy, (PP E Co(w).

It remains to replace u by u - P in (3) to establish (1) (with e = 1). Thus, Theorem has been proved for p < oo. In case p = oo the result is verified by the passage to the limit in (1) as p -4 oo. I

1.5. Poincare's Inequality and Equivalent Norms in Sobolev Spaces

37

The assertion stated below immediately follows from the theorem.

Corollary. The spaces LP(Q), Wp(Q) and VI(S2) coincide for bounded domains having the cone property. Remark 1. The above theorem admits a more general formulation for functions on domains having the cone property. This formulation follows from the theorem combined with Sobolev's imbedding theorem (cf. Remark 1.8.2/5).1 The next assertion will be useful later on.

u -a )(u) E R1 be a

Lemma. Let S2 be a domain in R" and let Lp(S2) linear functional such that

Ilu - )(u)Ilp,n < C IIVullp,o, C = const,

(4)

for all u c LI (S2). Then for any 1 = 1, 2.... there exists a linear mapping

LP(c)Du'PEPI-1 satisfying IIVk(u - P)Ilp,O < CCl-l°IIVIUIlp,o

(5)

with 0< k <1-1 and c = c(n,p,l). Proof. Let u E Lp(S2) and a E Z+ a multi-index of length l - 1. Since D°u E LI(1), there exists a number )° such that IID°u - a° Ilp,u < C II V (D°u) IIp,n. Hence IIVI-1U1 IIp,o < cC IlViullp,:,

where

ul(x) = u(x) -

a°x°/a!. I°1=1-1

Continuing this process, we construct a function Uk E Lp k (S2) of the form

uk(x) = uk_1(x) - E a°x°/a!,

1 < k < 1,

I°1=1-k

satisfying IlVz-kUklIp,Q 5 CC IIVI-k+luk-IIIp,n, u0 = U.

1. Basic Properties of Sobolev Spaces

38

Note that the coefficients A,, are linear functionals of u. Thus

P(x) = E A,,,xa/a!. IMP«-1

is the required polynomial.

0

Remark 2. The validity of the inequality inf{Ilok(u - P)IIP,92: P E PI-11 < C IIvzulln,ii, u E LP(SZ),

(6)

for a certain k, 0 < k < 1, does not generally imply (6) for other values of k. A counterexample will be given later in Sec. 2.10. Remark 3. Inequality (4) (and, consequently, (5)-(6) for all k = 0, . . ,1- 1) holds not only for domains having the cone property. For instance, (4) is true for a domain Q C R' that is the union of a finite number of domains of class C (cf. Exercise 1.9 at the end of the chapter). However, for such domains the norm II IIp,n on the left of (6) cannot be generally replaced by the norm IIq,D with some q > p. This is verified using the domain II .

11 = {(x, y) E R2 :

Ij' < exp(-1/x)}

and the function u(x, y) = x2I exp ((px)-1). I The finite sums of domains in the class C along with the domains having the cone property do not exhaust the class of domains supporting the Poincare inequality. To confirm it, consider the following example. Example. Let S1 be the union of the triangle {(x, y) : x E (0, 2), 0 < y < x/2}

and the squares {(x, y) : 2-k < x < 21-k, 0 < y < 2-k}, k = 0, 1, 2.... (see Fig. 3). This domain does not have the cone property. Clearly, SZ is not a finite union of domains of class C. Nevertheless, SZ supports inequality (4). y

2

---------------

Fig. 3

x

1.5. Poincare's Inequality and Equivalent Norms in Sobolev Spaces

39

Indeed, by unifying SZ with its mirror image S1', we obtain a new domain

G with the cone property. Let u E Lp(S2) and put v(x, y) = u(x, y) for (x, y) E SZ and v(x, y) = u(x, -y) for (x, y) E W. Then v E LP (G) and IIVVIIP,G = 21/PjjVujjp,o. By the above theorem, there is a linear functional LP(G) 3 v H £(v) E R1 satisfying

lw - t(v) lip,G < C ilwlIp,G, C = cont. Hence (4) holds with A(u) = f(v).

1.5.3. The Space L,([) and Normings in Lp (Q) Let SZ be a domain in R" and let LP(S2) denote the factor space LP(SZ)/Pi_1 which consists of the classes zi = {u + P}, where u E Lp(S2), P E 'PI-1. We equip LP(1) with the norm JIViuIIP,a.

Lemma. The space L,(SZ) is complete.

Proof. Assume that {9 k}k>1 is a Cauchy sequence in Lp(Q). Then for any Uk in the class uk and any multi-index a, jai = 1, D'uk -+ va in Lp(S2). We shall show that v,, = D°u for some u E LP(SZ).

Let B be an open ball, B C Q. Suppose that {SZj}j>o is a sequence of domains with compact closures and smooth boundaries satisfying

BCSlo,

1z CQj+1,

U,SZI=Q.

Let Pk denote a polynomial in P1_1 which is specified by the set w = B and the function uk in Theorem 1.5.2 (with e = 1). Since Uk is in the class itk, wk = Uk - Pk is in the same class. By Theorem 1.5.2, {Wk} is a Cauchy sequence in Lp(1 ) for any j = 0, 1, .... Thus, the sequence {wk} is convergent in Lp,I0,(SZ).

Let u be the limit of this sequence. Clearly, for any multi-index a, jai = 1, and any 71 E Co (SZ), we have

JI uD"ridx

= lim fn WkD°rldx k-+oo

r r _ (-1)1 k J (D'wk)rldx = (-1)1 J v,,ydx. 0

n

1. Basic Properties of Sobolev Spaces

40

The proof is complete. The proof of this lemma also contains the following assertion.

1

Corollary 1. If {uk} is a sequence in L,(Sl) such that lim IIVI(uk - u)IIp,o =

0

for some u E LP(Q), then there exists a sequence of polynomials Pk E PI-1 satisfying Uk - Pk -* u in Lp,1oc(f2).

I

Let 11 be a domain in R' and w an open nonempty set with w CC Q. We introduce the norm in Lp(5l) by IIuIIL,(l) =

(1)

IIuIIP,. + IIVzuIIp,f2.

A combination of Corollary 1 with Theorem 1.5.2 gives the following result.

Corollary 2. Let Q be a domain in R. The space LP(S2) with norm defined by (1) is a Banach space. Different sets w induce equivalent norms. 1.5.4. Equivalent Norms in Wp(SZ)

We begin with a description of a wide class of equivalent norms in WP (0).

Theorem. Let S2 be a bounded domain satisfying the condition Lp(Q) C Lp(S2) and let F(u) be a continuous seminorm in WP(12) such that F(u) = 0 and u E PI-1 imply u = 0. Then the norm F(u) + IIViullp,ci

(1)

is equivalent to the norm in Wp(1l).

Proof. Let B(S2) be the completion of WP '(Q) with respect to the norm (1). The identity map I : WP(Q) -+ B(Q) is linear, one-to-one and continuous. It follows from Lemma 1.5.3 that B(1l) C Lp(1). Hence, B(Q) = WP(f) and I maps WP(S2) onto itself. By the Banach theorem I is an isomorphism. I Let I be a bounded domain such that Lp(S2) = Vp(S2) (for example, S2 has

the cone property). By II we mean a projector of WI(Q) onto the subspace i.e., a linear continuous mapping lI : Wp(Q) -+ 'PI-1

1.5. Poincare's Inequality and Equivalent Norms in Sobolev Spaces

41

satisfying IIz = H. Then one may take InuIIP,n as F(u). Since II(u IIu) = 0, Theorem implies the equivalence of the seminorms II V uIIP,n and Ilu - lIullvi(sl). In particular, the estimate IIVk(u - flu)IIP,Q < c(l, n,p, Q)IIo1uIIP,11

(2)

holds for k = 0,1, ... , l - 1. Let E be a positive parameter and 0, = ESZ = {Ex : x E SZ}. Putting 4D(x) = Ex, we observe that

lI£(v) = (II(v o D)) o 0-1

(3)

is a projector of WP(SZE) onto P1_1. By inserting the function u = v o 4) (v E WP(SZE)) into (2), we obtain the inequality IIVk(v - IIe(v))IIP,c. <- CE1-kIIVivIIP,c1E, 0 < k < 1,

(4)

with the same constant c as in (2). Note that it is possible to specify the projector II: WP(Q) -+ P1_1 by

1! f (D)()( x - y)°

(IN (x) =

(y)dy,

(5)

IcI<1

where V) E Co (1l), ffV)(x)dx = 1. A combination of (2)-(5) with Theorem 1.5.2 leads to the following assertion.

Corollary. The polynomial P£ E P1-1 in Theorem 1.5.2 can be defined by PE(x)

1.

= E-n IaI<1

a

f

Dau(y)(x -

where b E C0 00(w) and f iidx = 1.

One more example of a projector: WP(1) as follows. Let K E Co (S2) and

f K(x)dx = 1, f K(x)x"dx = 0,

(6)

1

P1-1, Q, = E Q, can be given

1 < Ial < 1 - 1.

1. Basic Properties of Sobolev Spaces

42

Then the mapping v H 1IEv defined by

(Ilex) (x) =

E_n ICkI
(y)e D-v(y)dy, a. f K `

satisfies 112 = II.., and estimate (4) holds provided 52 is bounded and possesses the cone property.

1.6. Extendability of Functions in Sobolev Spaces In this section we discuss space-preserving extensions of functions in L,(0)

and VP (S2) into the exterior of Q.

1.6.1. Extension Across the "Plane" Part of a Boundary Here we describe a well-known procedure of a finite order reflection. Let w be a

domain in Rn-1, x = (x', xn) E Rn, x' _ (x1, ... , xn_,), and let (a, b) C R1 be a finite interval. Consider the cylinder 0 = w x (a, b) and denote by COO (S2) the space of restrictions to 12 of functions in COO (w x Rl).

Lemma. The space LP(S2) fl 0, (P) is dense in L,(S2), p c [1, co).

Proof. We can assume without loss of generality that S2 = w x (-1, 1). According to Theorem 1.4.1/1, it suffices to approximate a function u c L,(S2) n COO (Q) by functions in Lp(11) fl 01(Q). Put

ut(x , xn) = u(x', txn), t E (1/2, 1). Then ut E C' (w x (-t-',t-1)) and, furthermore, IID'(u - ut)IIp,c < IID'u - (D'u)tJIP,n + (1 - t`^)II(Dau)tjjp,st, a = (a', an),

j al = 1.

The right part of the last inequality tends to zero as t -a 1. Also ut -+ u in Lp,,OC(S2). Thus, ut - u in L,(1l). We now introduce the functions

W, X,,) H v(t)(x',xn) = ut(x',xn)rl(xn),

1.6. Extendability of Functions in Sobolev Spaces

43

where 77 E Co (-t-1,t-1), 771(_11) = 1.

Then v(t) E C°°(0) and v(t) -+ u in Lp(SZ) as t -* 1 which completes the proof.

I

Theorem. Let a E (0, oo), Sl+ = w x (0, a) and l > 1 an integer. If SZ = w x (-all, a), then there exists a linear operator E : C°° (c +) - C'-1(Q) with the following properties:

1) Eul

= u, i.e., E is an extension operator;

2) the restriction of E to the set Lk(SZ+) fl C°° (Sl+) can be uniquely extended to a continuous map Lk (SZ+) H Lk(Q) f o r all k = 0, ... , l and p E [1, oo);

3) the constant c in the inequality IIVk(Eu)IIp,lt < c IIVkUIIP,Q+, u E Lk(SZ+), 0 < k < 1,

(1)

can be chosen to depend only on 1.

Proof. Let u E C°°(1+). We define Eu to be the reflection of order 1 - 1 of the function u, that is u(x)

(Eu)(x)

_

cju(x',

if x E SZ+, (2) if x E SZ \ SZ+,

j=1

where the coefficients cj satisfy the system

(- j)kcj = 1, k = 0,1, ... ,1 - 1,

(3)

j=1

(cf. Fig. 4). Note that the system has nonzero determinant (the Vandermonde determinant). Now the inclusion Eu E Cl-1(SZ) is a consequence of the readily verified equalities lim

y-++0

8ku(x', y) 8yk

_

lim y_i_O

ak(Eu)(x', y) , 0 < k < l - 1. ayk

1. Basic Properties of Sobolev Spaces

44

It remains to note that statements 2) and 3) follow from above Lemma and 1 the obvious estimate (1) for u E LP(1 ) n C°°(SZ+), 0 < k < 1. y

-1/3

x

-1

-8/9

Fig. 4: reflections of orders 0,1,2 for the function u(x) = 2x(1 - x)

Remark 1. The operator E defined by (2) is continuous as an operator: Lk (SZ+) - Lk (Q), 0 < k < 1,

and inequality (1) is valid for p = oo and c = c(1). Indeed, let u E Lk00 (SZ+). By Theorem, Eu E LP (W' x (-a/1, a)) for any p E [1, oo) and any bounded subdomain W', W' C w. Thus, the function Eu has all generalized derivatives of order k in Q. Estimate (1) with p = oo is easily deduced from (2).

Remark 2. Let Q C R' be an open parallelepiped with edges parallel to coordinate axes. Applying the reflection described in Theorem along different coordinate directions, we extend a function from Q into a wider parallelepiped. Multiplying this extension by a smooth cut-off function, we obtain an exten-

sion from Q into R. Thus, there is a linear bounded extension operator: VP (Q) --* Vp (R"). The same procedure leads to the following assertion. Let

Q be as above and G a domain in R. Then there is a linear continuous extension operator:

11, whose norm is bounded by a constant independent of G.

Remark 3. Let SZ be a domain in Rn with sufficiently smooth compact boundary. Combining the extension procedure described in Theorem with a partition of unity and a local mapping of SZ onto a cylinder, one can construct (cf. Babich [15], Nikol'ski [170]) a linear continuous extension operator: VP (SZ) -> Vp (R").

1.6. Extendability of Functions in Sobolev Spaces

45

1.6.2. Domains of Class EVP

Definition. Let 1 < p < oo and 1 > 1 an integer. An open set SZ C Rn is said to belong to the class EVP if there exists a linear continuous extension operator E : VP(si) - VP(Rn),

i.e.,Eulufor all uEVP(1). As it follows from the preceding subsection, domains with smooth compact

boundaries are in EVP, 1 < p < oo, 1 = 1, 2, .... Theorem stated below describes a wide class of domains in EVP.

A domain Q C Rn is called a special Lipschitz domain if there is an orthogonal transformation T of Cartesian coordinates such that

TO ={x=(x',xn)ER' : xIERn-1, xn>W(x)}, where W is a uniformly Lipschitz function on

Rn-1

Theorem. Let Q be an open set in Rn and let there be positive numbers r, M, N (N an integer) and a sequence {Ui}i>1 of open sets satisfying the conditions:

1) if x E ac, then Br.(x) C Ui for some i; 2) every point x E Rn is contained in at most N sets Ui; 3) for any i > 1 there is a special Lipschitz domain 52i with function Vi such

that UjnQ=Uiflhl and Iwi(x) - Wi(y')I C MIA - Y' J, x',y' E Rn-1.

Then there exists a linear operator E mapping functions defined on SZ into functions defined on Rn and having the following properties:

a) Eul, = u; b) E is a continuous operator. VP (Q) -> VP (Rn) for all 1 < p < co and

1=0,1,2,...;

-

c) the norm IIEIIvp(n)_+VP,(Rn) is bounded by a constant depending only on

n,p,1,r,M,N. This theorem is due to Stein, and its proof can be found in his book [194], Chap. VI, § 3. In particular, the theorem applies for domains of class C0,1. I

46

1. Basic Properties of Sobolev Spaces

A combination of Lemma 1.3.2, Lemma 1.3.3/1 and above Theorem says

that a bounded domain having the cone property is the union of a finite number of domains in EVp, 1 < p < oo, 1 > 1. However, the converse is generally not true. It can be shown (see the comments to Sec. 1.6) that domain S2 in Fig. 3, described in Example 1.5.2, is in EVp for all p E [1, oo] and

l = 1, 2, .... But this domain does not possess the cone property. Another example of a planar domain in EVp , p > 2, without the cone property will be given in Sec. 2.12.

1.7. Change of Coordinates for Sobolev Functions Let 0 and G be open sets in Rn and let F = (F1,. .. , Fn) be a one-to-one bi-Lipschitzian transformation of 11 onto G. This means that all coordinate sunctions F1, .

.

, Fn are uniformly Lipschitz on SZ and all coordinate functions

.

(F-1)1i ..., (F-1)n are uniformly Lipschitz on G. Note that the map F is differentiable almost everywhere on S2 (see Rademacher [175]) and that the elements of the Jacobi matrix F' are in Lo.(Q). Furthermore, the general transformation formula

f

c

f (y)dy =

f(f

o

holds for all f E L1(G) (see e.g. Federer [59, 3.2.3]).

Theorem. Let F : 92 -4 G be a bi-Lipschitzian map and let the elements of the Jacobi matrix F' be in VV 1(S2), l > 1. If u c VP '(G), then the function F*u = uoF belongs toVp(92) and its derivatives D°F*u, Ial < 1, are expressed by the classical formula

(D°F*u) (x) _

cpp(x) (F*DQu) (x).

(2)

1
Here

n

7 T7 'Pp (x) = E cs 11 11 (D9'Fi) (x), s

i=1

(3)

i

and the summation is taken over all multi-indices s = (sib) such that

Esij = a, is

Isi,l

E(Isi;l -1) = lal i,j

- IQI.

1.8. Summability and Continuity of Functions in Sobolev Spaces

47

Moreover, the operator F* is bounded as an operator: VP '(G) -+ VP '(Q) and has a bounded inverse.

Proof. It is readily shown that the elements of the Jacobi matrix (F-1)' are in V, 1(G). Therefore, only the boundedness of the operator F* : VP '(G) -+ VP (1) should be verified.

Let l = 1, p < oo. If u E C°°(G) fl VP '(G), then F*u is an absolutely continuous function on almost all straight lines parallel to coordinate axes. The derivatives of this function are given by n

(DiF*u) (x) = E (Dku) (F (x)) (DiFk) (x)

(4)

k=1

for almost every x E ft Hence and in view of (1), we have lVvllp,n < const Iloullp,c.

(5)

Thus V E Vp (SZ) by Theorem 1.2.4. Approximating every function u E Vp (G)

by functions in C°°(G) fl VP (G) (see Theorem 1.4.1/2), one obtains (4) and (5) in the general case. We arrive at the same result for p = oo if we take into account that V,(G) C C°'1(G). The proof will be continued by induction on 1. Let l > 1 and let (2) hold for lal = l -1, where cpp are subject to (3). Since DQu E V, (SZ), the functions F*D-ou belong to VP (11) as shown above. Furthermore, it follows from (3)

that W' E V., A- By differentiating the right part of (2) and by using (4) with DAu in place of u, we obtain (2) for lal = 1. Therefore lV1F*nllp,n < const Ilull vv(c)

The result follows.

1.8. Summability and Continuity of Functions in Sobolev Spaces 1.8.1. On Continuity of the Imbedding Operator: WP(Rn) -+ L,(Rn)

Let 1 < p < co, 1 < q < oo. In the present subsection we are concerned with the question whether a constant c = c(n, p, q, l) > 0 exists such that llulIL,(Rn) 5 CIIUIIWD(Rn)

(1)

48

1. Basic Properties of Sobolev Spaces

for all u E Co (Rn). For brevity, the indication on Rn is omitted below. One can easily check (with the aid of a cut-off function and mollification) that Co is dense in W. Therefore, once inequality (1) has been proved for all u E Co, the same inequality holds for all u E WP. In this case WP C Lq. Moreover, the identity map: WP -+ Lq is linear, continuous and one to one, i.e., it is a topological imbedding.

By inserting the function Rn E) X H u(Ax) into (1), we find that (1) is equivalent to a parametric inequality I1U11q < C

Al+n/q-n/PIIV,uHIP)

(An/q-n/PIIuIIP +

,

(2)

where A is an arbitrary positive number. It is clear that the exponents of A in (2) cannot be both positive or negative (otherwise letting A -+ 0 or A -* oo gives a contradiction). Thus, the conditions q > p and l + n/q - n/p > 0 are necessary for (1) to be valid for all u E Co . Minimization of the right part in (2) with respect to A leads to the following results. 1. If lp < n and q = np/(n - lp), then (1) is equivalent to (3)

Ik4llq <- C II VzulIP.

2. If p < q < oo and 1 + n/q - n/p > 0, then (1) is equivalent to IIU Iq C COnst IItIIpHIOltIIP-V

(4)

with v = (l - n/p+n/q)l-1, const = c (1- v)"-lv and c the same as in (1). Let us show that the conclusions just stated are really valid. The case p = 1 = 1, q = n/(n - 1) will be separately considered.

Lemma 1. The inequality n-1)/n

x n/fin-1)dx ) ( (f I4L()I

GC

f IVu(x)Idx

(5)

holds for all u E Co with c = n-1/2. Proof. For the simplicity of presentation,' we restrict ourselves to the case n = 3. We have Iu(x)I <-

fRI 8t

(t, x2,

x3)

dt, x E R3,

° inequality (5) with exact constant will be proved in Sec. 1.9 by a different (and less elementary) method.

1.8. Summability and Continuity of Functions in Sobolev Spaces

49

and, clearly, the last inequality remains true if t occupies the second or the third component of the argument. Therefore 3

lu(x)l3/2

au

(JkfR'

axk

1

1/2

dxkl

(6)

.

Integrating (6) with respect to xl and then applying Holder's inequality gives lu(x)l3/2dx1

fRl

<

(JR'

au

(L

au dx1 ax1

ax1

1/2 f3

On

dx1/

axk au

ffl2 ax2

dxldx2

fR2 ax3

\ 1/2 dxl dxk I 1/2

dxldx3)

.

(7)

By integrating (7) with respect to x2 and by using Holder's inequality, one arrives at fR2

au ax2

I u(x)

l3/2dxldx2

au

dxldx2

fR2 ax1

au dx fR3 ax3

dxldx2

\ 1/2

Finally, integration of the last inequality with respect to x3, combined with Holder's inequality, yields

\ 2/3 (I lu(x)3/2dx /

I

1/3

3

Uf

au

=1 R3 axk

dx

The geometric mean is majorized by the arithmetic mean and hence 1

I nlIL3/2(R3) <

33

3

k_1

au R3 axk

dx <

1V3 R3

IVuldx,

which is the required result (for n = 3).

Let 1 < p < n and put q = np/(n - p). If we replace u in (5) by and use Holder's inequality to bound the right part, then llullq(1-1/n) < n-1/2q(1 - 1/n)II luIq(1-1/p)VuIl1 < n-1/2q(1 - 1/n)lluliq(1-1/p)IIo1IlP.

1 1Iq(1-1/n)

1. Basic Properties of Sobolev Spaces

50

Therefore

Ilullnp/(n-p) < n-1/zp(n - 1)(n - p)-1Iloullp This along with

I V IDi-iul l 5 n1/2 V,-i+lul, j = 1, ... ,1 - 1, implies

p(n - 1) IIV1

_JtIInp/(n-7p) 5

lIVl-i+lullnp/(n-(j-1)p),

n - jp

(8)

where jp < n. Putting j = 1, . . . , l in (8) and then multiplying all inequalities obtained, we arrive at the following assertion.

Corollary 1. If lp < n, p > 1, then (3) holds for all u E Co with q = np/(n - lp) and r(n/p - 1) c = (n - 1)1 r(n/p) Inequality (3) is known as the Sobolev (p > 1)-Gagliardo (p = 1) inequality. We have established it with explicit (but not exact) constant. I Let lp < n and p < q < np/(n - lp). An application of Holder's inequality gives Ilullq

IluliPllullnp/(n-lp)'

where u E Co and v = (1- n/p + n/q)l-1. Combining the last estimate with Corollary 1 yields (4) and hence (1). We now turn to the case lp > n.

1

Lemma 2. If lp > n, p > 1, and u E Co , then sup Jul < c (diam(supp u))1-n/p llOlullp

with c depending only on p, n, 1.

Proof. Theorem 1.5.1/2 says that lu(x)I < c f Iolu(y)I

y Ix

dy l n-1,

c = c(n, l).

Putting D = diam(supp u) and using Holder's inequality, we obtain 1/p'

lu(x) I <- c Ilviullp (f1--yj
(`-n)p'dy)

,

1; = PAP -1),

(9)

1.8. Summability and Continuity of Functions in Sobolev Spaces

51

which gives the desired result. The following assertion is an immediate consequence of Lemma 2.

1

Corollary 2. If lp > n, p > 1, then sup lul

C IIUIIwi

for all u E Co with c = c(n, p, l). We now consider a special case p = 1, 1 = n. Because

z,

u(x) =

dy1 ... f y° 00

anu(y)

a

ay1

"' yn

dyne u E Cp00

e

one obtains the estimate sup Iui < IIVnuiIi.

0

The next assertion complements Corollary 2.

Lemma 3. If p > 1 and A = l - m - n/p E (0,1) for some m = 0, ... , l -

1,

then there is a constant c = c(n,1, p) such that sup IVmu(x) - Vmu(y)I < cIIuIIwn Ix - yla zi4y

(10)

for all uECO. Proof. We may assume m = 0. By Theorem 1.5.1/2 lu(x + h) - u(x) I <

JIK&(x+h_y)_Ko(x_y)iiDou(y)idy, IQ1=1

where x, y, h E R', I Ka (z) I < c I zI1-n and

I K.(z + h) - Ka(z)I < c

Ihiizi1-n-1

for IzI > 31h1.

Therefore, the right part of (11) does not exceed c IhI

I

IIz-yI>_3Ih1 1

+cJ Iz-yI<-4IhI I

dy

Iotu(y)I Ix - y1n+1-1 dy

(IVzu(y)I + IVzu(y+h)I) Ix -

yIn-1

(11)

1. Basic Properties of Sobolev Spaces

52

An application of Holder's inequality to each of the last two integrals yields

Iu(x+h)-u(x)l 1. Here inequality (1) will be established for all q c [p, oo).

Lemma 4. Let 1 < p = n/l < q < oo. Then there is a constant co > 0 depending only on n, 1, such that Ilullq < Co ql-l/P+lI9IIu IW,

for all uEC0

(12)

.

Proof. First we obtain an auxiliary estimate Ilullq < clg8(diam(suppu))n/gllV,UllP

(13)

with cl = cl (n, 1) and s = 1 - 1/p + 1/q. Let D = diam(supp u). It follows from (9) that Iu(x)I< -

IViu(y)IX(0,D)(Ix-yI)

C2 J

Ix - yl n-I

dy, c2 = C2 (n, 1).

Use of Young's inequality (1.1.2/5) gives

Ilullq

C2 (f

zl (I-n)/sdz

Thus Ilullq C C2 (sgmesa(B1))3

18II DIUII

/

P

Dn/gllolUII ,

and (13) is true. To prove (12), we consider a collection of cubes

Qj={xERn:-1<xn-jn<1, n1=1,...,n}, jEZn, and introduce a smooth partition of unity {77j} for Rn subordinate to the covering {Qj }. One may assume that Okrlj (x) I is uniformly bounded with

1.8. Summability and Continuity of Functions in Sobolev Spaces

53

respect to x,j for k = 0,...,1. Note that every point in R" belongs to at

most 2" cubes in the collection {Qj}. Hence Ilullq <

2"c9-1,

i

II7li

1q q,

uECo

,

and the last sum does not exceed

C q (2V )"

IIV(iiu)IIP 9

in view of (13). Since 1/p

,a3>0,q>p, it follows that 1/q 114119 <-

IIV(?7;U)IIP)

with c = 2"n a, c1 Q8. Thus (12) is true. 1 In fact, a stronger result than that stated in Lemma 4 can be established in the case lp = n. The next lemma offers exponential integrability of functions in WP, which implies their q-summability for q > p. Below [p] stands for the integral part of p.

Lemma 5. If lp = n, p > 1, then there is a constant c = c(n, l) > 0 such that for all nEWW

f F (c

lulp IIuIIp

WPI

)dx<1, J

where p' = p/(p - 1) and [p!

F(t) = exp(t) - L tk/k! k-0

(14)

1. Basic Properties of Sobolev Spaces

54

Proof. It is sufficient to assume IIuIIw, = 1. Clearly 00

k

fF(cIuIP')dx=

k!II?IIP,k.

k=[p]+1

We apply (12) with q = p'k to bound the general term of the last sum. Then

J

F(cJuIP')dx < E00ckc0'k(p'k)k+l/k! k=[p]+1

The series on the right is convergent if c p'c e < 1, and its sum can be made not greater than 1 by an appropriate choice of c. Lemma 5 is proved. 1 We summarize above observations in the following theorem.

Theorem. (i) If p > 1, lp < n and p < q < np/(n - lp), then the space W, is continuously imbedded into Lq.

(ii) If lp = n, p > 1, then WP is continuously imbedded into Lq for any q E [p, oo). Moreover, there is a positive constant c depending only on n, l such that estimate (14) holds for all u E WP', u 0 0. (iii) If lp > n for p > 1 or if l = n and p = 1, then WP is continuously imbedded into L00. Furthermore, any function in WP almost everywhere agrees with a bounded continuous function. (iiii) If (l -m)p > n for some m = 0, ... , 1- 1 and A = l -m-n/p E (0,1), then every function in WP almost everywhere agrees with a function in C' n V'; whose gradient of order m is uniformly Holder on R" with exponent A. If u

is the representative in C', then estimate (10) holds. 1.8.2. Sobolev's Theorem Let S2 C R^ be an open set and p a (nonnegative) u-finite Borel measure on Q. By Lq(fl, µ) we mean the space of equivalence classes of qth-power integrable functions with respect to p. If q E [1, co), the norm in this space is 1/q IIUIILq(SZ,µ)

= (\L

ulgdpL)

1.8. Summability and Continuity of Functions in Sobolev Spaces

55

The last notation will be also used for q E (0, 1) when the right side is a pseudonorm.b For q E (0, 1), the space Lq(SZ, p) becomes a complete metric space if the distance between its elements u, v is defined by I I u - V I I L (fl ) In case p = mesa we omit measure in notations of norms and spaces and write I I-

I I L, (n) or simply I I-

I l q,1.

We now formulate the classical Sobolev theorem.

Theorem. Let p E [1, oc) and 1 > 1 an integer. Suppose that Q is a domain in R' which is the union of a finite number of domains in EVP. Let p be the s-dimensional Lebesgue measure on SZflR'. Then for any u E C' (Q) nV1(SZ) the estimate k

E IIVju11LQ(S2,µ)

(1)

C IIu1IVp(n)

j=0

is true, where C is a constant independent of u, and the parameters n, s, p, q, 1, k satisfy the conditions:

1) p>1, 01, n=p(l-k), s
4) p > 1, n < p(l - k);

5) p=1, n
Esupsl lojul <-

(2)

j=o

If SZ belongs to the class EVP, then in the case 4) the theorem can be refined as follows:

6) If.\=l-k-n/pE (0,1), then forallue VP(12)nCm(SZ) sup

x,yccl,x#y

lVku(x) - Vku(y)1 < C ix - y1

(3)

°

bA linear space is called pseudonormed if a functional llxll > 0, defined on its elements,

satisfies the conditions: a) if llxll = 0, then x = 0; b) llaxll = lalllxll for a E R'; c) if llxkll -> 0, llykll -+ 0, then llxk +ykll - 0.

1. Basic Properties of Sobolev Spaces

66

7) In the case (1 - k - 1)p = n inequality (3) is valid for all A E (0, 1) and u E Vp (I) n C°O(S2).

For s = n, this theorem is a consequence of Theorem 1.8.1 which has been proved in the preceding subsection. In the general case the proof of the Sobolev theorem can be found in the paper by Gagliardo [70] and in the book by R. A. Adams [4] (Sec. 5.1-5.19). Here we note that a more general version of Sobolev's theorem is known when p in (1) is an abstract measure on 52 satisfying certain conditions (see Maz'ya [136, 1.4.5] and Comments to Chapter 1 of the present book). Below we make some remarks concerning the Sobolev theorem.

Remark 1. The assumption q > p in 1)-3) can be omitted when SZ is bounded. The relations between p, 1, n, k, A in 4)-7) are sharp. This fact is verified with the aid of the trial functions q(x) I xIO', q(x)xk log I log Ixl I, where

q E CO-(B1), q(x) = 1 for Ixi < 1/2.

Remark 2. It follows from 1)-3) that the restriction operator C' (Q) nVp(S2) E) U H UIReno

can be uniquely extended to a linear continuous operator: VP (q) -> 9c(Rs n S2).

The conditions q < sp/(n - (1- k)p) in 1), 2) and q < oo in 3) are sharp. This can be verified by using the trial functions

x a xiq(e-lx) and

x H q(x)xi log I log IxII,

where q is the same as in Remark 1 and e > 0 sufficiently small. In particular,

if 52 C R" is an arbitrary domain and if p, 1, k, n, s are as in 1), 2) or as in 3), then the linear continuous operator mentioned above cannot exist for q > sp/(n - (1 - k)p) in case (1 - k)p < n and for q = co in case (l - k)p = n,

p>1.

Remark 3. Let S2 be a bounded domain having the cone property.

By

Lemma 1.3.2, Lemma 1.3.3/1 and Theorem 1.6.2, this domain is the sum of

a finite number of domains in E V P for all 1 < p < oo and l = 1, 2, ... . Hence VP (S2) is continuously imbedded into VQ (1) for (l - k)p < n, 1 < q < np/(n - (1- k)p). The same is true for any q E [1, oo) provided (1- k)p = n. In

1.9. Equivalence of Integral and Isoperimetric Inequalities

57

case (l - k)p > n or p = 1, 1 - k > n the space VP (Q) is continuously imbedded into Ck(Sl) n V,(S1).

Remark 4. Suppose 1 E C°'1. Then SZ E EVp for all p > 1 and 1 = 1, 2, .. . (see Theorem 1.6.2). Under conditions 6), 7) in above Theorem, the space VP '(Q) is imbedded into the space obtained by completion of C°° (1) with respect to the norm liulI V k (S1) +

sup a,gES2,x#y

I Vku(x) - Vku(y) I lx - YlA

Remark 5. Let SZ be a bounded domain having the cone property. Combining Sobolev's theorem with Theorem 1.5.2, we can write (1) in the form k

E IlVi(U - P)II Ly(S2,,) ! G IIOIUIIp,n, i=0

where P is the polynomial from Theorem 1.5.2 and p the s-dimensional Lebesgue measure on 1 n Its. This enables us to establish the continuity of the restriction operator:

Lp(SZ) -4 qk(O n R')/PI-1 if conditions 1)-3) hold. Analogously, the continuous imbeddings

L4(c) C

Ck(n)/P1-1,

i4(c) c

Ck'11(a)/Pl-1

can be introduced under conditions 4)-7) and SZ E C°'1

1.9. Equivalence of Integral and Isoperimetric Inequalities The well-known fact that among all bodies of given volume the ball has the least area of the boundary can be stated in the form of the inequality

mesn(G)1-11' < n-1vn 1/ns(aG).

(1)

Here vn = mesn(Bl), G C Rn is an arbitrary open set with compact closure and smooth boundary, and s denotes the (n - 1)-dimensional area. Inequality (1) is called the classical isoperimetric inequality. For its proof see e.g.

1. Basic Properties of Sobolev Spaces

58

Ljusternik [121], Hadwiger [82], Schmidt [181], Burago and Zalgaller [32]. It turns out that this inequality is tightly connected with integral estimate (1.8.1/5).

Theorem. Let SZ C R" be an open set, q c [1, oo), and let u be a Bore] measure on Q. (i) Suppose that sup {µ(G)1/1/s(8G)} = D < oo,

(2)

(GI

where {G} is the collection of open subsets of Il with compact closures in St bounded by C°°-manifolds. Then IIUIIL.(Q,µ)

C IIDUIILl(n)

(3)

for all uECo (1) withC
Lemma 1. If S2 is an open set in R" and u E C' (Q), then the sets {x u(x) = t} are manifolds of class C°° for almost all t E R1. The proof of Morse's theorem can be also found in the book by Maz'ya [136] (Sec. 1.2.2). The second fact involved is a so called coarea formula.

Lemma 2. Let 0 C R" be an open set. Let f be a nonnegative Bore] measurable function on SZ and u E C°'1(I). Then

f f(x)IDu(x)Idx=fdtfSx:lu(

x)l=t}

where Hn_1 is the (n - 1)-dimensional Hausdorff measure. The proof of this formula appears in the book by Federer [59] (see also Ziemer [221, 2.7.1] and Maz'ya [136, 1.2.4]).

Proof of Theorem. (i) By writing Iu(x)I = J o"O X(°,lu(x)I)(t)dt= foo" Xm,(x)dt,

(4)

1.9. Equivalence of Integral and Isoperimetric Inequalities

where Mt = {x c S2 arrive at

59

Iu(x)I > t}, and by using Minkowski's inequality, we

:

II kII Lg(n,µ) 5 f µ(Mt)1/9dt. 0

According to Lemma 1, almost all sets Mt are bounded by smooth manifolds. Hence

D

IIUIHL,(st,µ)

f

O

s(8Mt)dt.

It remains to note that the last integral equals IIVuIIL,(st) by Lemma 2.

(ii) Fix a set G in the collection {G} and consider a mollification uh = XG * Kh of the characteristic function of G with nonnegative mollifier Kh and

sufficiently small radius h > 0. Then uh E Co (SZ) and uh = 1 on the set Gh = {x E G : dist(x, 8G) > h} (cf. Sec. 1.2.1). The Gauss-Green formula yields

Duh(x) _ - f Dy(Kh(x - y))dy G

=

f G

Kh(x - y)v(y)ds(y),

where v(y) is the inner unit normal to 8G at y. Therefore

L IVuh(x)Idx

f

ds(y) frR, K h(x - y)dx = s(8G).

aG

n

By inserting Uh into (3), we obtain

p(Gh)1/9 < Cs(8G). Passage to the limit as h -> 0 gives

µ(G) 1/a < Cs(8G)

and hence D < C. It follows from the theorem just proved and the classical isoperimetric inequality (1) that the estimate IIUIILn1(n_1)

n-1 Vnl/nIIVuIIL

1. Basic Properties of Sobolev Spaces

60

holds for all u E Co (Q) with exact constant (cf. Lemma 1.8.1/1). 1 We refer the reader to the book by Maz'ya [136] (Sec. 1.4.2, Chapters 3 and 4) for a wider treatment of connection between imbedding theorems and isoperimetric inequalities.

1.10. Compactness Theorems In this section we state several assertions showing that the sets bounded in Sobolev spaces can be compact subsets of some other function spaces. We begin with the following lemma which is valid for arbitrary domains. Lemma. If Il is a domain in R" and w a subdomain such that w cc SZ, then every bounded subset of Vp (SI), p E [1, oo), is relatively compact in VP-1(w).

Proof. It suffices to consider the case l = 1. Suppose that U is a bounded set in VP '(Q). Let W E Co (1l) satisfy W = 1, 0 < cp < 1, and let G be a domain such that supp cp C G CC Q. We shall show that the set

W={w=cou:uEU} is relatively compact in Lp(G) (which clearly implies the desired conclusion). By M. Riesz's compactness theorem [179], one should check the boundedness of the set W in Lp (G) and the equality lim sup { Ihl-+0

l JG

I w(x + h) - w(x) I pdx : w E W

1

= 0,

(1)

where h E R. Only (1) needs verifying. We extend every function w E W by zero to the exterior of ft Then w c Vp (R"). Assume for a while that w is smooth. In this case

Iw(x+h) - w(x)I =

dt

w(x + th)dt

r1

< IhI

J0

I (Vw) (x + th) Idt,

and Minkowski's inequality gives

IIw( + h) - w(')IIp,R^ 5 IhI

IIVwIIp,Rn.

1.10. Compactness Theorems

61

This estimate is validated for an arbitrary w E W by using mollification. Now (1) follows from the last estimate and the inequality IIVwIIP,R- <_ Ilwllcl(c) sup {Ilully; (n) : U E U}, W E W.

This concludes the proof.

I

Remark 1. The lemma is also true for p = oo. In this case it is a consequence of the Arzela-Ascoli compactness theorem. The next assertion concerns domains with finite volume.

Theorem 1. Let 0 be a domain in R" with finite volume. Then bounded subsets of L,(1), p c [1, oo), are compact in n-dimensional Lebesgue measure. That is, any sequence bounded in LP(S2) has a subsequence convergent in measure on Q. Proof. Let {uk}k>1 be a sequence bounded in LP(S2). We introduce a collection {Ij}.i>1 of domains with smooth boundaries satisfying Sid CC 52j+1i

Ujc23 = Q.

One may assume that the norm in L,(1l) is given by IUIIL,(sl) =

IIRIIP,91i + IIVzuIIP,c

(cf. Corollary 1.5.3/2). In view of Corollary 1.5.2, a set bounded in LP(c) is bounded in VP (SZj) for every j > 1. Therefore, this set is relatively compact in Lp (f2j) for every j > 1 by the above lemma. This means that there is a family of sequences {uk°)}k>1i {uk1)}k>1,..., each sequence being a subsequence of

the preceding one, uk°) = uk, k > 1, and {ukj)} is convergent in Lp(S2j) for j = 1, 2, .... Put vk = ukk), k = 1, 2, .... Then {vk} is a subsequence of {Uk} and {vk} is convergent in Lp,10 (1). Let v be its limit. We claim that vk -a v in measure. Indeed, let b > 0 and put ek = {x E S2 : Ivk(x) - v(x)I > b}. Clearly

mes(ek) <mes(S2\1)+b-PIIvk-vIIPO., j=1,2,...,

(2)

1. Basic Properties of Sobolev Spaces

62

where mes = mes,,. The first term on the right tends to zero as j -+ oo, while the second term can be made arbitrarily small when j is fixed and k sufficiently large. Hence mes (ek) ---> 0 as k -+ co. This establishes the theorem. I Corollary 1. Bounded subsets of the spaces WP(S2), VP (S2) are compact in measure if S2 C R" is a domain with finite volume. The following result also concerns domains with finite volume. Theorem 2. Let I be a domain in R" with finite volume. If the space Lip(S2) is continuously imbedded into Lq(I ), where p E [1, oo) and q > 0, then the imbedding operator: L,(51) -+ L, (Q) is compact for any r E (0, q). In this assertion Lp(S2) can be replaced by WP '(Q) and Vp (S2).

Proof. We shall show that if {vk} is a bounded sequence in L, (S2) and vk -* V in measure, then vk -> v in Lr(S2) for any r E (0, q). The result then follows by reference to Theorem 1. First we observe that v E Lq(Q). This is a consequence of the continuity of the imbedding LP(S2) C Lq(S2) and Fatou's lemma. Next, for any 8 > 0, one can write

ivk - vlydx=J f2

Ivk - vlydx+ f Ivk - vlydx,

Sd\ek

k

where ek is given by (2). By Holder's inequality, the last integral does not exceed (mes (ek)) 1-r/9llvk - vllq IIvk - vllr,S2 <_ brmes (1) + C[mes (ek)]

1-r/q (3)

with some positive constant C independent of k. Given e > 0, put 8 = (2rmes (a))-l/re and choose an integer N such that the last term in (3) is less than (e/2)r for all k > N. Then IIVk-vllr,o < e fork > N. This completes the proof for the space ' (52). The argument for the spaces WI (Q) and VP '(Q) is the same. One should 1 only refer to Corollary 1 instead of Theorem 1. Theorem 2 implies the following corollary.

Corollary 2. If 1 is a domain in R' with finite volume, then the space VP (0), p > 1, is compactly imbedded into Vy-1 (0) for any q E [l, p).

1

1.10. Compactness Theorems

63

The theorem stated below shows that the restriction and imbedding operators mentioned in Remarks 1.8.2/2-3 (which are continuous by Theorem 1.8.2) turn out to be compact for certain values of p, n, s, k, 1, q.

Theorem 3. Let SZ C Rn be a bounded domain and let it be the union of a finite number of domains in EVP. Suppose that p > 1, 1 > k > 0. We have:

1° Ifs is a positive integer such that (l - k)p < n, n - (1 - k)p < s < n and 1 < q < sp/(n-(l-k)p), then the restriction operator n(Q) E) u H uIR,nn is compact as an operator: VP (c1) -+ Vqk (R8 n c2);

2° If (1 - k)p = n, then the restriction operator mentioned in 1° is compact as qk (Re n Q) for q E [1, oo) and s < n; an operator. VP (cZ) 3° If (1 - k)p > n, then the space V(S1) is compactly imbedded into the space Ck(c2) fl VV (Q) with norm

o sups Ioiul.

The proof of Theorem 3 can be found in the paper by Gagliardo [70] and also in the books by R. A. Adams [4] (Sec. 6.1-6.7) and by Maz'ya [136] (Sec. 1.4.6).

1

Remark 2. If s = n and p, 1, k, q, n are as in 1°, 2°, the conclusion of Theorem 3 follows from Theorem 1.8.1 and Theorem 2.

Remark 3. Let the hypotheses of Theorem 3 hold and let p, 1, k, n, s be as in

1°. If q = sp/(n - (l - k)p), the restriction operator V(Q) 3 u H uIR*nn is continuous as an operator: VP (c2) -4 9k(Re fl 0) by Theorem 1.8.2. However, this operator is not compact. Indeed, suppose without loss of generality that cZ contains the origin. Let 'i E Co (B2), ii = 1 on Bl and define

ui(x) = 24"I gxi?1(2ix), x E Q, i = 1, 2, ... . Then ui E Co (c2) for sufficiently large i, and it is readily checked that {ui}i>1 is bounded in VP(c2). Put Ai = {x E R" : 2-i-1 < Ixi < 2-i}, i > 1. If

j > i + 2, then IIVk(ui - uj)IILq(R°flA;) =

liVkui1lLq(R"f1A;)

c(k, s, q) > 0.

Thus, {u2i} does not have a subsequence convergent in Vgk(R8 fl 1). In par-

ticular, this counterexample shows that for an arbitrary domain cZ C Rn the compactness of the restriction operator mentioned above implies q < sp/(n - (1 - k)p).

1. Basic Properties of Sobolev Spaces

64

1.11. The Maximal Algebra in Wp(1Z) Let A be a subset of a Banach function space. The set A is called an algebra with respect to multiplication if there is a constant c > 0 such that the inclusions u c A, v E A imply uv c A and IIuvJI < c IIuji IIvil

Note that the space WP = Wp(R") is not an algebra when lp < n, p > 1 or l < n, p = 1. Indeed, if W1 were an algebra, the following inequalities would occur IIuNIIp/1

<-

11U1111 'N

cIIUIIWp,

(1)

where u is an arbitrary function in WP and N = 1, 2, .... By letting N -+ 00, we could obtain lI

IIoo <_ c Ilullwi

Clearly, the last inequality is not true for the values p, 1 mentioned above (cf. Remark 1.8.2/2). Thus, W1 is generally not an algebra with respect to pointwise multiplication. However, it is possible to describe the maximal algebra contained in W. Namely, the following assertion holds. Theorem. The subspace WP n L,,. with norm II - Ilwp + II - IIL_ is the maximal algebra contained in W1 . In particular, the space W1 is an algebra if

lp>n,p>1 orifl>n, p=1. We need an auxiliary multiplicative inequality to prove this theorem.

Lemma 1. Let 1 < p < oo and l > 2 be an integer. For any u E W1 n L,,. and any j = 1, ... , l - 1, the estimate 1_

j11

(2)

IIV,UII. -< c Ilullo o'"` IlullwD

holds with c independent of u. We shall deduce Lemma 1 with the aid of another auxiliary inequality.

Lemma 2. If u E Co (R"), p E [1, 00), 1 < q < coo, IIVUIIr < c IIo2uIIp'2IIuIIQ"Z,

2r-1

= p-1 + q-1, then (3)

where c is a positive constant depending only on n.

Proof. It is sufficient to assume p > 1 and q < coo. Then the extreme cases p = 1 or q = oo follow by letting p tend to 1 or q tend to 0o in (3).

1.11. The Maximal Algebra in Wy(91)

65

We remark that (3) is a consequence of the one-dimensional estimate

f Iu

1rdx <

co (f u'lPdx)

zp

24

lulgdx)

U

(4)

with 2r-1 = p-1 + q-1 and co an absolute constant (the integration is taken over R1). Once (4) has been established, (3) is validated by integrating with respect to the other variables and by applying Holder's inequality. Note that for any u E Co (Rl) 2c1 lu Ilr,i <

il1+r-1-P-' 11U 1111P',

lilp-1_r-1-1IIUIlq,i,

+

(5)

where cl is an absolute positive constant, i an interval and Izl its length. Inequality (5) follows from the same inequality for i = (0, 1) by dilation and

translation. In turn for i = (0, 1) this inequality can be obtained from the estimate

u'(x) - (z - y)-1(u(z) - u(y))

f u" (t)I dt, a

where x, y, z E [0, 1], y # z. If y E (0, 1/4), z E (3/4,1), this estimate gives 2-'lam (x)I <- lu(y)I + lu(z)I +f I u"(t)Idt. 1

0

Integrating with respect to y and z, one arrives at

lu'(x)l < 8 f lu(t)ldt+2 fo l lu"(t)ldt, x E [0,1]. 1

0

This yields (5) with cl = 2-4. Inequality (4) is a consequence of the estimate c1llu llr,A <- IIu lIIP/2IIullg12

(6)

where 0 is an arbitrary interval in R1 with finite length. The last is verified as follows.

Fix a positive integer k and introduce the closed interval i of length I0 /k with the same left point as A. Let us consider (5) for this interval i. If the

1. Basic Properties of Sobolev Spaces

66

first summand on the right of (5) is greater than the second, we put i1 = i. In this case Cr J

Q l

1+r-r/P

u Ird2 <

Ul Iu 'IPdxk L

/

P

sit

Suppose the first term on the right of (5) is less than the second. We then increase the interval i leaving the left endpoint fixed until these two terms are equal (clearly, the equality must take place for some i with lil < oo because 1 + r-1 - P-1 > 0). Let i1 denote the resulting interval. Then

r

cJ lu'ydx <

(j1

u"'dx)

\L l uldx)

(8)

Putting the end point of i1 to be the initial point of the next interval, repeat this process with the same k. We stop it when the closed finite intervals i1i i2.... (each of length at least IAI/k) form a covering of the interval A. Note that the covering {i1i i2. ..} contains at most k elements, each ie supporting estimates (7) or (8) (with i1 replaced by i9). Summing these estimates and applying Holder's inequality, one arrives at 1+r

ci

I lu'lydx
(fIu"IPdx)P

\

2,

+ (f lu"IPdx) 2p Y

IulQdx)

Now (6) follows by letting k - oo because 1 + r - r/p > 1. The proof of Lemma 2 is concluded.

I

Proof of Lemma 1. It is sufficient to assume p < oo. First let u E Co (R") If aj = Il V3ullP1/2, inequality (3) implies a < c aj-laj+1 for By induction on 1, we obtain aj < ca0- /1 a1 and (2) holds for smooth functions u with compact support. Let us turn to the general case u E WP n Lo.. Suppose that supp u is bounded and consider a mollification uh of the function u with radius h. Since IIuhlloo < c IIuII. and by inequality (2) applied to uh, the estimate IIVjuhII1P/i
(9)

1.11. The Maximal Algebra in WP(fl)

67

is true. We have limh,o uh = u in Wp, and (2) is established by the passage to the limit in (9) as h -* 0. Let us remove the assumption on the boundedness of supp u. To this end we introduce a cut-off function r) E Co° (B2) such that 0 < ri < 1 and 171 Bl = 1. If we put rlk(x) = r)(x/k), k = 1, 2, ..., and insert the function uik into (2), then IIojuIIPL/7,Bk < C IIU7kIIWD IIuII

It remains to pass to the limit as k -+ oo to establish Lemma 1. We now can give the proof of the theorem stated above.

0

Proof of Theorem. Let A be an algebra contained in W. Inequality (1) implies A C L. n W. To show that the space A = Lm n WP is an algebra, we let u, v c A be arbitrary. Then t

II IVkul . Iot-kvI

IIVi(uv)IIP 5 c

IIP

k=0 c

Ilokull 1f IIo!-kvll, k=0

by Holder's inequality. It follows from Lemma 1 that IIo!(uv)IIP

IIuII00-k,,` IIUIIwp

c

IIvIIk,` IIvIIWpk)/`

k=0

which does not exceed c I I U I I A I I V I I A. Thus, A is an algebra and hence the space WP n Lc,. is the maximal algebra contained in W, .

If lp > n, p > 1 or l > n, p = 1, then W, C Lm by Sobolev's theorem, and the space WP is an algebra for these p and 1. This concludes the proof. I Let S2 be a domain in R". We can ask whether the space W, ) n Lm (Q) is an algebra with respect to pointwise multiplication. Clearly, for 1 = 1 the answer is affirmative. Since Stein's extension operator from a domain SZ E Co,1

is continuous as an operator:

W1 (Q) n Lm(Q) - WP(R") n L,,, (W),

1. Basic Properties of Sobolev Spaces

68

(see Theorem 1.6.2), the above question has the affirmative answer for such domains. Moreover, the answer is affirmative if Q is the finite sum of domains in Co,'. For example, SZ can be a bounded domain having the cone property (cf. Lemma 1.3.2 and Lemma 1.3.3/1). However, it turns out that the space W, (Q) fl L,,. (0) is generally not an algebra. A corresponding counterexample will be given in Sec. 2.8.

1.12. Application to the Neumann Problem for Elliptic Operators of Arbitrary Order In Sec. 1.12.1 we prove the equivalence of the continuity of the imbedding operator: L,(1k) -4 Lq(Q) to a Poincare type inequality. Some applications of Lemma 1.12.1 to the study of the solvability of the Neumann problem for strongly elliptic operators are given in Sec. 1.12.2. 1.12.1. Necessary and Sufficient Condition for the Continuity of the Imbedding LP(Q) C Lq(1)

The following assertion gives the required condition.

Lemma. Let SZ be a domain in R° and suppose PI-1 C Lq(SZ), q > 0. In order that the space Lp(Q), p > 1, be continuously imbedded into Lq(SZ), it is necessary and sufficient that for all u E L,(SZ) the inequality inf{Ilu - QIIq,S1 : Q E PI-11 -< C IIViuIIp,s1

(1)

be valid with some positive constant C independent of u.

Proof. Necessity. Fix an open ball B CC Q. Then IIUIIq,11 < C (IIkIlp,B + Ilolullp,S1)

for all u E L,(SZ) with C independent of u. Thus, the left part of (1) does not exceed

C(inf {IIu - QII p,B : Q E P1-1} + Ilolullp,sl). Reference to Theorem 1.5.2 gives (1).

Sufficiency. Let (1) be valid. Since Pi_1 C Lq(SZ), the set LP(Q) is contained in Lq(SZ). Consider the identity mapping of the Banach space L,(1)

1.12. Application to the Neumann Problem for Elliptic Operators ...

69

into Lq(SZ). This mapping is linear and closed and thus is continuous by the closed graph theorem (see Rudin [180, 2.15]). This completes the proof of the lemma. 1.12.2. Solvability of the Neumann Problem

First we describe the operator of the Neumann problem for a differential operator of order 21, 1 > 1. In this subsection SZ designates a bounded domain in R. Let aap be real functions in Lm (SZ) for a,,3 E Zn, lal = 101 = 1 > 1, such that a,,p = ape. Suppose there is a constant v > 0 satisfying 10

E

aap(x)D' u Dpu)dx > v IIVzullz,si

(1)

Ia1=1131=1

for all u E L2(SZ).

Definition. Let 1 < q < oo. The operator A. of the Neumann problem for the differential operator

Dc (apD13u)

u H (-1)1 1a1=1131=i

is determined by the conditions 1) u E L'2(Q) n Lq(Q), Aqu E Lqs (Q), 1/q + 1/q' = 1; 2) for all v E L2 (Q) n L.(Q) the following identity holds

J

vAqudx =

J

a,,p(x)Dpu D°v)dx.

(

D

I°1=1131=1

It is readily seen that the mapping u N Aqu is closed and that the range Im(Aq) is contained in the set Lq, (SZ) e P,_ of functions in Lql (1) which are orthogonal to the space PI-1.

Lemma 1. If the generalized Poincare type inequality inf {11v - Pllq,cl : P E Pi_11 < C IIV,vll2,ci, C = const > 0,

(2)

is valid for all v E L2(1l), then

Im(Aq) = Lq,(SZ) ePt_1

(3)

1. Basic Properties of Sobolev Spaces

70

Moreover, if Aqu = f , the following estimate holds (4)

II VIuII2,o < CV-1II f Il q',cz,

where v and C are the constants in (1) and (2) respectively (hence u is uniquely

determined up to a polynomial term in PI-1). Proof. Let f E Lq' (52) e Pz_1i v E L2(1 ). Then (2) implies

I

vf dx < C IIfllq',cIIIVzvll2,Il.

(5)

Thus, the functional L2 (S2) 3 v H fn v f dx is continuous on L2 (52) and can be expressed in the form [u, v] with u E L2 '(Q) and

[u, v] = f, ( E aCp(x)D13u D°v) dx.

(6)

1'1=1131=1

By Lemma 1.12.1, the space L2(1) is continuously imbedded into Lq(S2), hence

u c L2(S2) fl Lq(1) and f = Aqu. We now turn to estimate (4). Let Aqu = f . Then [u, v] =

ffvdx Z

for all v E L2 (S2) and therefore [u,

u]1/2 = sup

l

fvdx : v E L2 (S2), [v, v] =1

}.

An application of (1) and (5) yields v1/2IIV1uII2,o

[u,u]1/2 < CV-1/2IIfIlq',Sz

which leads to (4) and concludes the proof of Lemma 1. 1 We continue the study of the Neumann problem Aqu = f with the following assertion. Lemma 2. Let (3) be valid. Then (2) holds for every v c L2(S2) fl Lq(S2).

1.12. Application to the Neumann Problem for Elliptic Operators ...

71

Proof. Let V E L12 (Q) n L. (Q),

IIoivII2,o = 1.

The linear functional

Lq,(SZ) ePl-i E) f H

(f,v) =

f fvdx z

can be expressed in the form F (f) _ [u, v], where [ , ] is defined by (6) and u is an element in L2(SZ). Hence IF"

(f) I < [u, u] 1/2[v,

V]1/2

< C[u, U]1/2' C = const,

and the set {Ft,(f)} is bounded for every f E Lq,(Q) e PI-1. Thus, IIFvII < const. We claim that the following lower bound for IIFvII holds

IIFvII 2inf{IIv-PIIq,c :PEPi_1}.

(9)

Indeed, by the Hahn-Banach theorem, the functional F can be extended to a linear continuous functional on Lq, (Q) with the same norm. That is, there is an element w E Lq(I) satisfying IIwIlq,sa = IIFII and

(f,w) = (f,v) for all f ELq,(1)ePi_1. We now check that v-w c Pj_1. Let {P0}I0I<1 be a basis of Pl_1 orthonormal in L2(SZ). Put

ng = E (g, P0)P0. IaI
Then (IIg, h) _ (g, IIh) for all g E Lq(SZ) and h c Lq, (Q). In particular, if g = v - to, then

(g-ng,h) _

(g,h-IIh) =0

for any h E Lq, (I) because (h - IIh)1Pl_1. Hence g = ng E Pi-1. Therefore IIFvII = IIv - PIIq,sz

for some P E PI-1 and (9) follows. Finally

inf{IIv - Pliq,si : P E Pi-1} < const

1. Basic Properties of Sobolev Spaces

72

which completes the proof of Lemma 2.

1

Corollary 1. If the set L2(0) n Lq(1) is dense in L2(1), then the continuity of the imbedding operator. L12(1l) -4 Lq(SZ) is equivalent to (3).

Proof. According to Lemmas 1, 2 and Lemma 1.12.1, we should verify the validity of (2) for all v E L2(Q) provided (2) is true for all v E L2(Q) n Lq(SZ). Let W2,q(SZ) = L2(Q) n Lq(SZ). The factor-space W2,q(SZ) = WZ q(1l)/Pi_1

is Banach when equipped with the norm IIVIIW,.9(n) = inf {IIv - Pliq,c : P E Pi-1} + IIV1VH2,o, V E W2,q(1),

where v = {v - P : P E P1_1}. Furthermore, we introduce the Banach space D2(Q) = L2(SZ)/Pl_1 with norm (cf. Lemma 1.5.3) IIvIIL2(n) = IIoivI12,n.

Suppose (2) holds for every v E W2,q(S2). Then IIvIIL2(cz)

Ilvll

(1 + C)IIvIIL2(sz)

(10)

for every v E W2 ',,(Q) with the same C as in (2). Since the space WW,q(sl) is dense in L2' (Q), it follows that W2,q(1) is dense in L2(Q). Hence and from (10) we obtain D2(Q) = WZ,q(SZ). Now the second inequality (10) is valid for all v E L2(c), and therefore (2) is valid for all v E L2(1). I The set LZ(I) n Lq(SZ) is dense in L2 (1l) by Corollary 1.4.3, and we arrive at the following assertion.

Corollary 2. Let Aq be the operator or the Neumann problem for the second order strongly elliptic operator. Then Im(Aq) = Lq, (SZ) e 1 if and only if the space L2(1) is continuously imbedded into Lq(I ). 1 Let a E L,,.(SZ), a > const > 0 a.e. on Q. Then the operator

u H Bqu = Aqu + au

(11)

has the same domain as Aq. Consider the Neumann problem Bqu = f with f E Lql (SZ). If 1 < q < 2, then its solvability is a trivial consequence of

Exercises for Chapter 1

73

the continuity of the functional v H fn f vdx on. the space WZ (Q) with inner product

f ( > aap(x)DAv D'u + a(x)uv) dx. 1-1=101=1

In case q > 2 the argument similar to that in Lemmas 1, 2 and Corollary 1 leads to the following result.

Theorem. If the set L2 (T) n L. (Q) is dense in W2(1), then the continuity of the imbedding operator. W2(Q) -4 L,(SZ) is necessary and sufficient for the equation Bqu = f to be uniquely solvable for all f E Lq, (1). I The question of the discreteness of the spectrum of the operator B2 is reduced to the study of the compactness of the imbedding W2(P) C L2(SZ). Namely, B2 has a discrete spectrum if and only if the imbedding just mentioned is compact, see e.g. Birman and Solomyak [24], Theorem 10.2.5.

Exercises for Chapter 1

1.1. Letx=(y,z)ER", yER"-e, z ER', 0<s
(i)Leteither 1
(ii) Let p = n - s > 1 and u(y, z) = 0 for z E R8, Iii = 1. Then

f to

IU(x)IP

dx < c f IVu(x)Ipdx.

(IyU I log IyI I)p

Hint. Use spherical coordinates y = (p, 0) and apply (1.1.2/7) with respect 0, z). Make the substitution I log eI = r in (ii).

1.2. Let u, Vu E L1,10 (1). Verify the chain rule V(f o u) = f'(u)Vu, where f E C°"1(R') (see Ziemer [221, 2.1.11]). 1.3. Let Q C R" be an open set. For h > 0, let P h = {x : Bh (x) C Q}. Show that a function u E Lp,io.(SZ), p E (1, oo), is in Li (SZ) if and only if

Ilu( + t) - ullpsIitl = O(ItI)

as

Itl -4 0, t E R",

1. Basic Properties of Sobolev Spaces

74

(see Gilbarg and Trudinger [74, 7.11], Ziemer [221, 2.1.6]). Check that this is not the case for p = 1.

1.4. Construct a bounded domain starshaped with respect to a point which is not a domain of class C. Hint. Consider the domain SI U 0' in Fig. 3 (Sec. 1.5.2). 1.5. Prove the assertion converse to Corollary 1.3.2. Namely, let Q be a domain

with compact closure whose boundary admits an explicit representation r = r(w) in spherical coordinates. Show that if Sn-1 E) w H r(w) is a uniformly Lipschitz function, then S2 is starshaped with respect to a ball centered at the origin of spherical coordinates. 1.6.

Let x = (y,z) E Rn, y E Rn-e, z E Re, where 0 < s < n, n > 2.

Suppose u E WP (Rn), p E [1, oo). In addition, assume that s < n - 2 if p = n - s and that u(0, z) = 0 a.e. z E Re if p > n - s. Prove that there is a sequence {uk} C Co ({x E Rn : y # 0}) with uk -* u in WP(Rn). Hint. Let p = n - s. For small e > 0, introduce a cut-off function hE by 10

h,(Q) _

for o E (0,e),

- 2log(Q/e)/loge for o c 1

[e,e1/2],

for p > e1/2

If p # n - s, then he is continuous piecewise linear on [0, oo), h.. (p) = 0 for p < E, h, (p) = 1 for p > 2E, he is linear on [E, 2e]. Put u, (x) = h, (I y1)u(x), and, using Exercise 1, show that uE -+ u in WP (Rn) as e -p 0 in case supp u is bounded.

1.7. Let SZ C Rn be a domain with finite volume and let 1 < p < oo. Show that the inequality inf{llu - Allp,n : A E R1} < const IIVuiIP,si

holds for all u E Lp(S2) if and only if the ordinary Poincare inequality Ilu - ull p,S2 < const lloullp,Q, u =

1

mes(Q)

fo

udx,

is valid for all u E LP(S2).

1.8. Consider a domain

0 ={(2,xn)ERn: 12l
Exercises for Chapter 1

75

with f E C(B,.). For all u E LP (1) check the estimate Ilullp,n <- const (IIVullp,a + Ilullp

B;n-1)x(° a)),

a = min f (x')/2.

Hint. Apply the one-dimensional inequality IIvIIp,(°,b) ,5 (b/a)1"pllvllp,(°,a) + bllv'IIp,(°,b),

v E C'([O, b]), a E (0, b).

1.9. Prove (1.5.2/4) when domain 0 is the finite sum of domains of class C. Hint. Use the preceding exercise. 1.10. Verify the validity of inequality (1.5.2/4) for bounded domains starshaped with respect to a point (see Deny and Lions [51]).

1.11.

Show that the set Co (R") is dense in Lp(R") if n < p < oo (cf.

Remark 1.4.3).

1.12. Let SZ E Co,'. Show that the boundary trace operator C°° (St) i) u H u180 can be uniquely extended to a linear continuous operator T : Wp(Q) Lp(ac ), 1 < p < oo. Hint. Consider Q defined in Exercise 1.8 with f E C°'1(Br) and apply the inequality Iv(b)I <- (b - a)-1IpIIvIIp,(a,b) + (b - a)1-1/pllv'Ilp,(a,b),

v E C' ([a, b]),

where v = u(x', ), b = f (x'), a = f (x') - min f (x'). See Necas [162, p. 15]. 1.13. Let 1 be the same as in Exercise 1.8 with f E C°"1(Br). Let T be the operator introduced in the preceding exercise. Prove that for a.e. x' E B,. (Tu) (x', f (x')) -> u(x', x")

as x" -* f (x').

Hint. Assume that u = 0 in the vicinity of x" = 0 and use the absolute continuity of u(x', ) for a.e. x' E B,.. 1.14. Let SZ E C°°1 and let T be the operator introduced in Exercise 1.12.

(i) Show that if u E Wp (f ), p E [1, co), and Tu = 0, then u is in Wp (1), i.e. in the closure of Co (Sl) in Wp (Il). (ii) Show that the factor-space Wp (S2)/4Vp (11) is isomorphic to the space of

the boundary traces If : f = Tu, u c W(1)} endowed with the norm inf {Ilullwp(si) : Tu = f I.

1. Basic Properties of Sobolev Spaces

76

Hint. Put u = 0 on R" \ 0 and employ the same argument as in Theorem 1.4.2/1.

1.15. Let w C R", g C Re be bounded domains having the cone property and put Rn+', e, 6 > 0. wE = ew, g6 = 5g, GE,6 = we x g6 C Let u E LP' (G,,6). Show that there is a constant C > 0 independent of E, 6, u such that the following inequality holds inf

PEPi_1

k=0,...,l-1.

IIVk(u-P)IIp,Ge.6

Hint. Consider the case l = 1 and apply Lemma 1.5.2. See also Stanoyevitch [192].

1.16. Let S2 be the finite sum of domains in the class C. Show that the space V(0) is compactly imbedded into Vl-1(S2). Hint. For an arbitrary small e > 0, verify the inequality C Ilullp,c

e Iloullp,o + IIullp'n., U E

VP (S2),

where S2E C S2 and C > 0 is a constant independent of E, u. First consider a simple domain defined in Exercise 1.8. Prove that for this domain

C

f Iu(x)IPdx < fx
f (x')-e

lu(x',xn)IPdxn +e In, IVuIPdx

with 2E E (0, min f (x')) and u E C1(S2). See also Agmon [5], p. 21, 30.

1.17. Let 1 C R" be an unbounded domain with finite volume. Prove that V(S2) cannot be continuously imbedded into Lq(I) if q > p > 1. Hint. Consider a sequence {rk}k>1 of positive numbers satisfying mes (S2

\ Brk) = 2-kmes (S2),

k = 1, 2, ...

.

Assume VP (S2) C Lq(S2) for some q > p and infer from this imbedding that

rk+1 - rk < const

2-k(p-1-a-1)/l,

k = 1, 2, ... .

Then the series X:k(rk+l - rk) is convergent, which contradicts the unboundedness of Q. See also R. A. Adams [4, 5.30], Maz'ya [136, 4.4.5].

Exercises for Chapter 1

77

1.18. Let SZ be the domain described in Example 1.5.2 (cf. Fig. 3). Show that W, (0) fl L,,, (1) is an algebra with respect to pointwise multiplication.

1.19. Let SZ be a domain in R. Prove that the space W2 (S2) fl L,,. (SZ) is an algebra with respect to pointwise multiplication if and only if W22(S2) n L. (11) is continuously imbedded into L4 (a). Hint. Verify the estimates IIuvll <- C (IIuII IIvII + IIVuIIL4(n)IIVVIIL4(o)),

IIouIIL,(D) <- c (IluV2uhIL,(0) + IIu2II2), where 11

II = II

IIw; (o) + II . IIL-(f2) and c > 0 independent of u, v, Q.

1.20. Let a E [0, 1] and v the volume of the unit ball in R". Show that n-1

(fR n

(nv)

< (n - a)

I u(x)I

II VUIIL1(Rn)

for all u E Co (R") with sharp constant. Hint. Use Theorem 1.9 and the isoperimetric inequality (1.9/1). See also Maz'ya [136, 2.1.4].

Let f be a differentiable function on R" without simple zeros, i.e. f (x) = 0 implies V f (x) = 0 (clearly this is true for differentiable f > 0). 1.21.

Show that for all a E (0, 1] the following inequality holds

Ca+1) \

a

/

If(x)I'

sup

x,yERn,x$y

IVfx) - Vf(y)I Ix - yI.

with exact constant (see Maz'ya and Kufner [139]).

1.22. Construct a domain SZ of class C for which the set flk 0Ck (SZ) is not contained in C°''(1) for any A E (0, 1). Hint. Consider the square {(x, y) E R2 : IxI < 1, IyI < 1} with deleted cusp {(x, y) E R2 : x E [0,1), IyI <_ exp(-1/x2)},

and the function u(x,y) = X(o,l)(x)X(o,l)(y) exp(-1/x). Cf. also Gilbarg and Trudinger [74, 4.1], and Fraenkel [64, Remark A].

1. Basic Properties of Sobolev Spaces

78

1.23. Let 0 be a domain in R". For any x, y E 52 put o(x, y) = inf t('y), where the infimum is taken over all rectifiable curves y of length 1(y) joining x to y, such that y \ ({x} U {y}) C S2 (if the set of such y is empty, we put o(x, y) = oo). Let f be a function on 812 satisfying

If (x) - f (y) I < Mo(x, y) for all x, y E 852, M = const.

Show that there is a function u on n subject to ule, = f and I u(x) - u(y) I <- Mp(x, y),

x, y E S2.

Hint. Consider a function u(x) = sup{ f (y) - Mg(x, y) : y E 812}.

1.24. Let 12 be a domain in R". Check the following "transitivity property" of the imbedding LP(12) C Lq(12). If p E [1, oo), q E (0, oo) and the continuous imbedding just mentioned is valid, then LP', (12) is continuously imbedded into

Lq, (12), where pl E (p, oo), pi 1 - q1 1 = p-1 - q-1. Hint. Fix a ball B Cc 12 and assume that IIUIIq,cl < C II VUIIP,f,

C = const > 0,

for all u E LP(S2), ul a = 0. Put U = Ivlgl/q and apply Holder's inequality.

1.25. Suppose that L11(12) is continuously imbedded into Lq(12), q E [l,oo). Prove that if pl(1 - q-1) < 1, p E [1, oo), then the imbedding Lip (0) C Lq (52)

is continuous, where q* = p/(1 - pl(1 - q-1)) for pl(1 - q-1) < 1 and q* is any positive number for pl(1 - q-1) = 1. (Clearly q* is the Sobolev exponent

np/(n - lp) if q = n/(n - 1) and lp < n). Hint. Proceed by induction on l and employ the preceding exercise.

Comments to Chapter 1 1.2. The spaces L,(12) and Wp(12) were introduced and studied by Sobolev [188-190]; but there are earlier roots, see Levi [116], Fubini [67], Evans [53], Tonelli [201], Friedrichs [66], Rellich [176], Nikodym [169]. The history of the name "Sobolev spaces" was described by Fichera as follows.

"These spaces, at least in the particular case p = 2, were known since the very beginning of this century, to the Italian mathematicians Beppo Levi [116] and Guido Fubini [67] who investigated the Dirichlet minimum principle for elliptic equations. Later on many mathematicians have used these spaces in

Comments to Chapter 1

79

their work. Some Rench mathematicians, at the beginning of fifties, decided to invent a name for such spaces as, very often, French mathematicians like to do. They proposed the name Beppo Levi spaces. Although this name

is not very exciting in the Italian language and it sounds, because of the name "Beppo", somewhat peasant, the outcome in French must be gorgeous since the special French pronunciation of the names makes it to sound very impressive. Unfortunately this choice was deeply disliked by Beppo Levi, who at that time was still alive and - as many elderly people - was strongly against the modern way of viewing mathematics. In a review of a paper of an Italian mathematician, who, imitating the Frenchmen, had written something on "Beppo Levi spaces", he practically said that he did not want to leave his name mixed up with this kind of things. Thus the name had to be changed. A good choice was to name the spaces after S. L. Sobolev. Sobolev did not object and the name Sobolev spaces is nowadays universally accepted", see Fichera [61], p. 146-147.

Theorem 1.2.3/1 was proved by Deny and Lions [51] for l = 1 and by Maz'ya [136, 1.1.2] for 1 > 1. Concerning the contents of Sec. 1.2.4, we remark that the property of absolute continuity on almost all straight lines parallel to coordinate axes was used as a foundation for the definition of function classes similar to LP in papers by Tonelli [201], Nikodym [169], Calkin [41], Morrey [159] and others. Theorem 1.2.5 is well known (see e.g. Vodop'yanov and Gol'dshtein [209], Hedberg [90], Reshetnyak [178], Chap. 1, Theorem 1.5). An analogous result for quasiconformal mappings was proved by Vaisala [211]. In the paper [209] the class of removable singularities for WP (i.e., the class of those sets F for which the conclusion of Theorem 1.2.5 holds with l = 1) was characterized in

certain capacitary terms. It should be noted that there are examples of sets F c R' of removable singularities for functions in Wn with 1 (F) > 0 (see Ahlfors and Beurling [7] for n = 2, and Aseev [13] for n > 2). 1.3. Theorem 1.3.1 is due to Fraenkel [63] who studied various relationships between classes of domains appearing in the theory of Sobolev spaces.

The condition of being starshaped with respect to a ball and the cone property were introduced into the theory of WP-spaces by Sobolev [188-190]. Corollary 1.3.2 can be found in the book by Maz'ya [136]. Lemma 1.3.3/2 was proved by Glushko [76]. Example 1.3.4 is taken from [136, 1.1.9]. See also Morrey [160, p.77].

1.4. Theorem 1.4.1/1 for l = 1 was proved in the work by Deny and Lions [51]. It was also proved by Meyers and Serrin [156]. Theorem 1.4.2/1 is due to Gagliardo [70], and Theorem 1.4.2/2 is found in the book by Smirnov [183].

80

1. Basic Properties of Sobolev Spaces

In his book [160] (Remark, p. 64) Morrey asked whether the elements in VP '(Q) on a domain with sufficiently wild boundary can be approximated by functions in C' (S2). A partial answer was given by Lewis [117] who proved that the set C°° (S2) is dense in WP (0), p c (1, oo), provided S2 is a planar domain bounded by a Jordan curve. A multi-dimensional analog of this theorem is

not known. Also it is not known whether this theorem can be extended to higher derivatives. It should be noted that even the class of bounded functions need not be dense in Sobolev spaces of higher orders on non-Jordan planar domains. A corresponding counterexample will be given in Sec. 2.3 of the present book. A class of planar domains S2 (different from the class C) for which C°° (S2)

is dense in VP '(Q), 1 < p < oo, has been described by Smith, Stanoyevitch and Stegenga in their recent paper [185]. A bounded domain S2 C R" has the interior segment property if to every x c 81 there correspond a number r > 0 and a nonzero vector y E R" such that z + ty E S2 provided 0 < t < 1 and z c S2 n Br (x). (Clearly domains of class C have the interior segment property, cf. Theorem 1.3.1). A domain S2 is said to be weakly starshaped with respect to a point x E 0 if the line segment [x, y] is contained in S2 whenever y c Q.

The following result from the paper by Smith, Stanoyevitch and Stegenga [185] complements Theorems 1.4.2/1-2. Let SZ C R2 be a bounded domain which is either weakly starshaped with respect to a point or has the interior segment property. If mes2(Br(x) n (R2 \ S2)) > 0 for any x E %1 and any r > 0, then C°° (S2) is dense in Vp (S2) for all p E [1, oo) and all l = 1, 2, ... . The converse to this assertion holds in a greater generality. Namely, let S2 be a domain in R2. Suppose that z E 852, that there is a number r > 0 for which mes2 (Br (z) n (R2 \ Q)) = 0 and that z is a limit point of nondegenerate components of 852. Then C°° (S2) is not dense in Vp (S2) for any 1 > 1 and any 1 < p < oo. These results cannot be generally extended to greater dimensions.

Lemma 1.4.3 was proved by Deny and Lions [51], and Theorem 1.4.3 was presented in the book by Maz'ya [136], (Sec. 3.1.2). In connection with approximation of elements in Sobolev spaces by smooth

functions, we mention a deep result due to Hedberg on approximation by smooth compactly supported functions. We say that a closed set F C R" admits (l, p)-spectral synthesis if any u c VP '(RI) that satisfies D°u = 0 on F (up to some "small" subset of F) for 0 < Ial < l - 1 belongs to the closure of Co (R" \ F) in V. According to Hedberg's theorem (see Hedberg [89], Hedberg and Wolff [91]), every closed set F C R" admits (l, p)-spectral synthesis

Comments to Chapter 1

81

if p c (1, oo). Among consequences of this theorem are uniqueness theorems for the Dirichlet problem for elliptic differential equations of arbitrary order (see Hedberg [90]). Later Netrusov [165, 168) gave another proof of this theorem valid for much more general spaces, and also established (1, l)-spectral synthesis of the closed subsets of R'. Recently Belova [17] has extended Hedberg's proof to weighted Sobolev spaces. We refer the reader to the book by Adams and Hedberg [3] (Chap. 10) for a detailed treatment of this subject.

1.5.1. Integral representations (1.5.1/3), (1.5.1/5) were obtained by Sobolev [189, 190] and used in the proof of imbedding theorems. The proof of Theorem 1.5.1/1 follows the argument of the paper by Burenkov [33]. Various generalizations of Sobolev's integral representation are due to Il'in [101], Besov and Il'in [21] (see also the book by Besov, Il'in and Nikol'ski [22]), Calder6n [40], Smith [184], Reshetnyak [177].

Here we mention an integral representation for smooth functions different from (1.5.1/3) and adapted to so-called anisotropic Sobolev spaces. Let li > 1 be integers for i = 1, ... , n. Put .i = It 1 , . = (A1, ... , An), JAI = Al +... + An. If x E Rn, t > 0, we set t-ax = (t-A1x1i ... , t-Anxn). Given r > 0, b > 0 and nonzero numbers a1,. .. , an, consider the horn V of radius r and span b defined by

V = {xER' :xi/ai>0, t<(xi/ai)'' < (1 + b)t, tE(0,r), 1 1. If an open set G C S2 satisfies the condition x + V C S2 wherever x c G, then for almost all x c G the following identity holds u(x) = r-1'\1 J Ko (r-ay) u(x + y)dy

+f, E t-I\Idt f Dj' u(x + y)K1 (t-'y) dy.

(1)

j=1

Here Kj are certain functions of class C°° (Rn) such that for each j = 0, ... , n the support of K3 is contained in the parallelepiped

{xER' :xi/ai>0, 1<(xi/ai)l'<1+b, 1
1. Basic Properties of Sobolev Spaces

82

is described in the paper by Besov and Il'in [21] (see also the book [22], §7). Representation (1) was used to obtain imbedding theorems for anisotropic Sobolev spaces W, (S2), 1 = (ll, ... ,1,) in domains satisfying the so-called horn condition (see [21]). These spaces are characterized by the finiteness of the norm n

IID;`ullp,n + llullp,0, 1 < p < oo.

llullW'(n) = i=1

Let u and D"u be extended to be zero outside 0. Then the right part of (1) (denoted by v) has sense for all x E Rn. It can be shown that IIVIIW'(R.n)

< const llullwp(n), 1

This may be used to construct a bounded extension operator: WP(Q) -+ Wp(Rn) for certain classes of domains. See Besov and Il'in [21].

Note that (1) and Sobolev's representation (1.5.1/3) require some restrictions on 92. It turns out that for functions vanishing on 81 along with its

derivatives up to some order, an integral representation similar to that of Sobolev holds for an arbitrary bounded Q. Let WP (S2) denote the closure of Co (S2) in the norm 11 ll wp (0). The following result is due to Maz'ya [134], [136, 1.6]. Let S2 C Rn be a bounded domain. If l > 1 is an integer such that l < 2m and if u E L' (Q) n Win(11), then for almost all x E 0

D7u(x) _

B

Q

dy

KQ,7(x) y)D u(y) I x - yl i-1

Here ry is any multi-index of length l - 1 and KQ,.y a measurable function on S2 x S2 with IKQ 71 < c(n,1, m). This integral representation implies that if S2 C Rn is an arbitrary bounded domain, then for all u E VP (S2) n Wi `(0) with I < 2m, Sobolev type inequalities (1.8.2/1-2) hold under the same conditions

as in Sobolev's theorem. The assumption l < 2m is essential. Examples of domains were constructed for which Sobolev's inequalities fail in case l > 2m. See Maz'ya [134], [136, 1.6].

1.5.2. Let n be a domain in Rn with finite volume. The Poincare inequality llu - Ullp,n < constllVullp,sj,

u E LP(S2),

(2)

Comments to Chapter 1

83

where u is the mean value of u on 0, was established in Volume 2 of CourantHilbert [47] in case SZ is the finite sum of bounded domains of class C and p = 2. Theorem 1.5.2 can be found in the book by Maz'ya [136], Sec. 1.1.11. The problem of the geometric characterization of domains for which (2) or

its generalizations are valid has recently received a great deal of attention. See e.g. Ziemer [220], Evans and Harris [54-56], Boas and Straube [25], Hurri [96], [97], Martio [122], Smith and Stegenga [187], Stanoyevitch and Stegenga [193], Smith, Stanoyevitch and Stegenga [186]. An important class of domains in this context is the class of so-called A-John domains. We say that a bounded domain S2 is A-John for A > 1 if there is a constant C > 0 and a distinguished point x0 E Sl such that every x E SI can be joined to x0 by a rectifiable arc ry c SZ along which dist(y, acl) > C I'Y(x, y)I',

y E 'Y.

is the length of the portion of ry joining x to y. Clearly the class of bounded A-John domains increases with A. One can easily show that bounded domains having the cone property form a subclass of 1-John domains (1-John domains are also known as John domains named after F. John who considered such domains in his work on elasticity [103]; this terminology was introduced by Martio and Sarvas [123]). It turns out, in particular, that if Il C R" is John, then (2) is valid for all p E [1, co) [122], and if 1 is A-John, then (2) is valid for all p E ((A -1) (n -1), oo), p > 1, [187, Theorem 10]. Hurri-Syrjanen and Staples have shown in their recent paper [99] that the image of a John domain under a quasiconformal map with Jacobian in Lq, q > 1, supports inequality (2) for all p > p0 with some po E (1, n) depending only on the map. We also mention the paper by Koskela and Stanoyevitch [111], where a general class of domains is given for which Poincare inequalities are preserved under Steiner symmetrization. Here 1,y (x, y) I

1.5.3-1.5.4. The completeness of the space Lp(Sl) was proved by Deny and Lions [51] for l = 1. We follow the argument of this paper in the proof of Lemma 1.5.3. Theorem 1.5.4 was established by Sobolev [190].

1.6. The extension by finite order reflection described in Theorem 1.6.1 was used by Hestenes for functions in C' (Q) [95] (see also the paper by Lichtenstein [119]). The same procedure was justified by Nikol'ski [170] and Babich [15] for the space VP '(!Q), where Q E C1.

The fact that domains of class are in EVP (1 < p < oo) was established by Calder6n [40]. His construction of the extension operator was based on an C°,1

1. Basic Properties of Sobolev Spaces

84

integral representation analogous to (1) and on the theorem on the continuity of singular integrals in L. Theorem 1.6.2/1 is due to Stein ([194], Chap. VI, Theorem 5). If SZ has the form Q

= {(x', xn) : x' E Rt-1, xn > W(x')

with uniformly Lipschitz cp, Stein's extension operator u -4 Eu is defined by

Eu(2

,

J

u(x', x,, + t6 (x', xn))0(t)dt, xn < co(x'). 1

Here 8 is a smooth function equivalent to the distance to SZ, and ii a function in C°° ([l, oo)) satisfying O(t)t' -+ 0 as t --> oo for any k = 0, 1, ... , and

f i (t)dt = 1, 1

0,

J

1

s = 1, 2, ...

.

00

The extension operator for the domain of the general form, described in Theorem 1.6.2/1, is constructed with the aid of an appropriate partition of unity. A necessary and sufficient condition for S2 to be in EVP is not known. The case p = 2, 1 = 1, n = 2 is the exception. Vodop'yanov, Gol'dshtein and Latfullin [210] showed that a simply connected planar domain belongs to the class EV2 if and only if its boundary is a quasicircle, i.e., it is the image of a circle under a quasiconformal map of the plane onto itself. By a theorem of Ahlfors [6], this last condition is equivalent to the inequality Ix - zI < c Ix - y1, c = const, where x, y are arbitrary points in 8S1 and z an arbitrary point in the arc of au of minimal length connecting x to y. The Ahlfors condition is sufficient for a bounded planar domain to be in EVP for all p E [1, oo] and l = 1, 2.... (cf. Gol'dshtein, Vodop'yanov [80] for I = 1, Jones [104] for l > 1). Some necessary conditions for S2 to belong to EVP with lp > n were given by Vodop'yanov [207] in terms of a so-called relative metric in Q. In the paper by Jones [104] a class of n-dimensional domains in EVP is introduced. It is wider than the class of domains in C°"1 and coincides with the class of quasidisks for n = 2. Jones' result is as follows.

Theorem. Let 1 be a domain in R. Suppose there exist e E (0, oc) and 8 E (0, oo] such that any two points x, y E SZ, Ix - yI < 8, can be joined by a rectifiable arc y C Q satisfying the inequalities

2('y) < Ix - yI/e, dist(z,8Q) > eIx-zIIy-z1/1x-yk,

Comments to Chapter 1

85

where £(y) is the length of y and z an arbitrary point in -y. Then SZ is in EVp for any p E [1, oo] and l = 1, 2, .... The linear extension operator VP (1) -*

VP (Rn) can be constructed in such a way that its norm is bounded by a constant depending only on n, p, 1, E, 6 and diam (S2).

The proof of this theorem is given in [104]. It should be noted that the Jones extension operator: VP '(Q) -* Vp (Rn) depends on 1, whereas that of Stein (cf. Theorem 1.6.2) is the same for all values 1, p. Domains satisfying the assumptions of Jones' extension theorem are also called (E, 6) -domains [104]. We point out the following result due to Herron and Koskela [92]. Let SZ C R" be the image of an (e, oo)-domain under a quasiconformal map. Then S2 E EVl if and only if SZ is an (El, 61)-domain for some E1, 6 E (0, oo). Let w be a nonnegative measurable function on SZ. By the weighted space VP ,,, (Q) (p > 1, 1 = 1, 2, ...) we mean the space of functions u on SZ having weak derivatives up to the order I and satisfying 1/P

/ r

IIulIvp,W(n) _

I

IoI« J

ID'ulPwdx)

< oo, 1

One says that the Muckenhoupt AP condition is fulfilled for w if either 1

\ 1/p / r (f wdx I J \ QQ Q I

or

f

\ 1-1/p <_ const, 1< p< oo,

1

f wdx < const ess inf {w(x) : x c Q}, p = 1, mesn(Q) Q for all cubes Q C R. S. K. Chua showed [43] that if 1 < p < oo, w is subject to the AP condition and S2 C Rn is an (E, 6)-domain, then there is a linear continuous extension operator: Vp,,,,(S2) -4 VP,,,(Rn). Recently Garofalo and Nhieu [73] have generalized Jones' extension theorem to Sobolev spaces generated by a family of vector fields.

Let 1 E EVp for some I = 1, 2.... and p E [1, oo], and let E : VP '(Q) -* Vp (R') be the corresponding bounded extension operator. Then E can be improved in the following sense. There is another bounded extension operator E1 : Vp(S2) -* VP(R') such that E Coo(Rn \ S2) and 11,_1`1-IDa(Eiu)IIL9(R°\!i) < cons t IIU V, (Q),

1. Basic Properties of Sobolev Spaces

86

where o(x) = dist (x, 8SZ) and kal > 1. The exponent lad - l is generally sharp. This result is due to Burenkov and Popova [36]. Operator El can be defined by AEu, where A is an approximation operator preserving boundary

values, constructed in the paper by Burenkov [34].

1.7. In connection with Sec. 1.7 see also Gol'dshtein and Reshetnyak [78] (Chap. 5, Sec. 4.1), Maz'ya [136, 1.1.7], Maz'ya and Shaposhnikova [154, 6.4.3]. Another and more explicit expression for functions cpp in (1.7/2-3) was given by Fraenkel [62].

1.8. Lemma 1.8.1/1 is due to Gagliardo [70]. The proof of (1.8.1/5) providing the smallest constant was independently and simultaneously proposed by Federer and Fleming [60], and by Maz'ya [124].

Inequality (1.8.1/3) for lp < n, p > 1, and q = np/(n - lp) was proved by Sobolev [189]. The best constant in this inequality for l = 1 was found by Aubin [14] and Talenti [197, 198]. This best constant is 1-1/2n-1/P

C

P- 1

1-1/n

I

F(1 + n/2) r(n)

1/n

r(n/p)r(1+n-n/p)}

Exponential integrability of Sobolev functions in case lp = n (such as stated in Lemma 1.8.1/5) appeared in the works by Pohoiaev [173], Yudovich [219] and Trudinger [202]. Theorem 1.8.2 is the classical Sobolev theorem [188-190] which was refined in the works by Morrey [159], Il'in [100] and Gagliardo [70]. A generalization of Theorem 1.8.2 to abstract measures was given in the book by Maz'ya [136] (Sec. 1.4.5). Namely, let SZ be the same as in Theorem 1.8.2 and let p be a Borel measure on SZ such that sup {r-8µ(n n B,.(x)) : x E R", r > 0} < oo

with s > 0 (in particular, ifs is an integer, p can be the s-dimensional Lebesgue measure on SZ fl R8 as in the classical Sobolev theorem). Then the conclusion of Theorem 1.8.2 remains valid. The proof of the generalized Sobolev theorem

is based on D. R. Adams's theorem on Riesz potentials [1, 2] for p > 1, (1 - k)p < n, and on an estimate for the norm in L. (R', p) by the L1-norm of the lth order gradient due to Maz'ya [133], [136, 1.4.3].

1.9. Theorem 1.9 is due to Maz'ya [130] (see also [136, 1.4.2]). The proof of this theorem given here contains some improvements borrowed from the paper

Comments to Chapter 1

87

by Talenti [198]. In the case SZ = R' the supremum on the left of (1.9/2) is comparable to the same supremum over all balls in Rn [136, Theorem 1.4.2]. We point out a recent paper by Bobkov and Houdre [26] where the connection between Sobolev type estimates and isoperimetric inequalities has been studied in the setting of metric spaces. Generalized isoperimetric inequalities for Markov operators have been studied by Kaimanovich [107]. 1.10. Lemma 1.10 for p = 2 is due to Rellich [176]. Theorem 1.10/1 for l = 1 was proved by Maz'ya [136, 4.8.4]. In fact, Theorem 1.10/2 is a consequence of the following criterion for compactness of sets in Lq(1k, µ), where µ is a finite measure. A set U C Lq(I, µ), q E (0, oo), is compact if and only if U is compact in measure and the norms of the functions in U are absolutely equicontinuous (see Krasnosel'ski et al [112], Lemma 1.1). A version of Theorem 1.10/2 for weighted Sobolev spaces has been proved

in the recent paper by Hajlasz and Koskela [85]. Let a E C(12), a > 0, and let WP o(1), p E [1, oo), be the space of functions u with finite norm 11auII Lp(c) + IIaVuIIL,(c). Suppose p is a finite measure on SZ which is abso-

lutely continuous with respect to the Lebesgue measure. Then the boundedness of the imbedding WA ',(Q) C Lq(SZ, µ), q > 1, implies the compactness of the imbedding WP l,, (Q) C L,. (S2, 1c) for any r E [1, q) (see [85], Theorem 5).

Corollary 1.10/2 for bounded domains can be found in the paper by Fraenkel [63]. Theorem 1.10/3 was proved by Kondrashov [110] for p > 1. In case p = 1 this theorem is due to Gagliardo [70].

1.11. A general form of inequality (1.11/2) is due to Gagliardo [71] and Nirenberg [171] The proof of (1.11/2) follows the paper by Nirenberg [171], where it was also shown that the space WP n L... is an algebra. A description and various properties of the algebra of multipliers in WP(Q), i.e., the function space {y E Lp,1°°(SZ) : ryu E WP(Q) for all u E WP(SZ)} can be found in the book by Maz'ya and Shaposhnikova [154]. 1.12.1. Lemma 1.12.1 for l = 1 was proved by Deny and Lions [51]. A criterion for the validity of the Poincare type inequality 11u - uIjq,c < const IIVujjp,n,

u=

mes(SZ)

f udx,

(3)

for all u E CI (Q) nLp(1Z) is the existence of a relative isoperimetric inequality (if p = 1) or a capacitary isoperimetric inequality (if p > 1) for subsets of SZ, see Maz'ya [136] (Sec. 3.2.3, 4.3-4.4); cf. also Sec. 8.5.2 of the present book.

1. Basic Properties of Sobolev Spaces

88

The last decade much attention has been paid to the analysis of geometric properties of domains 0 C R", for which (3) (or its generalizations) holds for all u E COO (S2)nLP(S2). See e.g. Bojarski [27], Hurri-Syrjanen [98], Chua [44], Buckley and Koskela [29, 30], Hajlasz and Koskela [85].

Let 1 < p < n and put q* = np/(n - p). Bojarski [27] has verified (3) with q = q* provided 0 is a bounded John domain (this class of domains was defined in the comments to Sec. 1.5.2). The same result can be obtained from the earlier works by Besov [19, 20], where the imbeddings of anisotropic Sobolev spaces into Lq(S2) with limit exponents were proved in case Q satisfies the so-called flexible horn condition. In the isotropic case the class of such domains contains the class of John domains. Weighted versions of (3) for John domains have been obtained by HurriSyrjanen [98] and Chua [44]. Weighted Poincare type estimates for bounded A-John domains have been established by Hajlasz and Koskela [85]. Their results in an unweighted case can be stated as follows.

Let S2 be bounded and A-John for some A > 1 and let 1 < p < q < np/ ((n - 1) A + 1 - p) (the last inequality may be improper for A = 1 or p = 1). Then (3) is true. Furthermore, q cannot generally exceed the given bound (see [85], Corollaries 4 and 5). These results can be partially converted. It has been shown by Buckley and Koskela [29] that if 0 supports (3) with p < q < q*, 1 < p < n, and if S2 has a so-called separation property, then SZ is s-John for s = p2(n - p)-1(q - p)-1

(clearly s > 1 and s = 1 if q is the Sobolev exponent q*). In particular, bounded simply connected domains in R2 satisfy the separation property [29].

Hence, if Q C R2 is bounded and simply connected, then a necessary and sufficient condition for inequality (3) to be valid for all u E C°°(S1) n LP(S2)

with 1 < p < 2 and q = 2p/(2 - p) is that 0 is John. We mention here generalizations of Poincare type inequalities for the norms generated by a finite number of first order differential operators, see e.g. Jerison [102], Franchi, Gutierrez and Wheeden [65], Garofalo and Nhieu [72]. Poincare type inequalities for functions on metric spaces have been studied in the papers by Hajlasz [83], Hajlasz and Koskela [84], Coulhon [46], Semmes [182], Heinonen and Koskela [94] and others.

1.12.2. It is well known that the validity of Poincare's inequality is equivalent to the solvability of the Neumann problem with right part in L2 (92) e 1 (see e.g.

Lions and Magenes [120], Necas [162]). Corollary 1.12.2/2 is due to Maz'ya

[127]. Lemma 1.12.2/1 for q = 2 can be found in the book by Lions and Magenes [120], Chap. 2, Sec. 9.1.

CHAPTER2

EXAMPLES OF "BAD" DOMAINS IN THE THEORY OF SOBOLEV SPACES

In the present chapter we collect counterexamples showing that some of the properties of Sobolev spaces, which have been studied in Chapter 1, may fail for unrestricted domains. Furthermore, we demonstrate the difference between Sobolev spaces of first and higher orders. The counterexamples given in this chapter concern approximation, extension and imbedding theorems.

2.1. The Property 81 = 8St does not Ensure the Density of C°° (SZ) in Sobolev Spaces It was established in Sec. 1.4.2 that for bounded domains Sl starshaped with

respect to a point and domains of class C the set C' (Q) is dense in the spaces LP(SZ), Wp(S2), V(0) with p E [1,00). A simple example given at the beginning of Sec. 1.4.2 shows that the set C°° (S2) may generally fail to be dense in Sobolev spaces. The domain in that example has the property 8S2 # B1l. It may appear that the equality 91l = 8S2 ensures the density of COO (Q) in Sobolev spaces. However, this conjecture is not true.

Example 1. We shall show that there is a bounded domain SZ C R" such that a = 8Q and WP (Q) n C(S2) is not dense in WP (Q) for any p E [1, oo).

Let n = 2 and let K be a closed nowhere dense subset of the segment [-1, 1]. By {8i}i>1 we mean the sequence of open disks constructed on adjacent intervals of K taken as their diameters. Put SZ = B \ Ui>113i,

where B is the disk x2 + y2 < 4. The set K can be chosen to satisfy the condition that the linear measure of I' = {x E K : JxJ < 1/2} is positive. The characteristic function of the halfplane y > 0 is denoted by X, and rl designates

a function in C01(-1, 1) such that 7 = 1 on (-1/2,1/2). Clearly, the function U defined by U(x, y) _ r!(x)X(x, y) 89

2. Examples of "Bad" Domains in the Theory of Sobolev Spaces

90

is in WP (12) for all p > 1. Suppose that there is a sequence {u; }2>' c C(SZ) n Wp (S2)

convergent to U in Wp (12). Then

uj (x, b) - ui (x, -b) =

f

ay (x, y) dy 6

for almost all x E r and for all b E (0, 1/2). Hence

f

< ff I pus (x, y) I dxdy, j > 1, r(6)

f-

where r(b) = r x (-b, b). Since uj -> U in W1(12), the integrals over r(b) are uniformly small. Thus, given any e > 0, there exists a b0 > 0 such that uj (x, -b) I dx < e

(1)

for all b E (0, b0). By Fubini's theorem, we have lim 2-'°°

f

0

6°

db

Jrr

1ui (x, b) - U(x, 6)1 + I ui (x, -b) - U(x, -b) I) dx = 0,

whence there is a subsequence of {uj} (which is relabled as {uj}) satisfying slim 00

=

fr

Iu, (x, b) - u3 (x, -b) I dx

f I U(x, b) - U(x, -b) Jdx = mess (t)

for almost all b E (0, b0). Now (1) implies mes1(r) < e which contradicts the positiveness of mesl(P).

Since 80 = 852, the required counterexample has been constructed for n = 2. In case n > 2, let 122 denote the planar domain just considered. One may put 12 = 122 x (0, 1)"-2 and repeat the above argument to obtain the 1 counterexample for n > 2.

2.1. The Property 011 = Of) does not Ensure the Density of CO°(1) ...

91

We now give another example showing that the property of the set C°° (SZ) to be dense in WP '(Q) need not simultaneously hold for all p E 11,00).

Example 2. Let 0 C R2 be the difference between the rectangle {x = (x1, x2) : x1 E (-1,1), x2 E (0, 1)} and the closed triangle with vertices (0, 0),

(-1/2,1/2) and (1/2,1/2) (see Fig. 5). We shall show that C°°(Q) is dense in Wp (1) for p E [1, 2], whereas the set C(ii) n Wp (0) is not dense in WP (S2)

forp>2.

SZ

0

-1

1

Fig. 5

Let p > 2. Consider a function f E Coo ([0, ir]) such that f (t) = 0 for t < 7r/4, f (t) = 1 for t > 37r/4. We introduce polar coordinates x = (e, 0) and put u(x) = f (0) if x E 0. Clearly, u E Wp (SZ). Let us check that u cannot be approximated in Wy (Q) by functions in C(SZ) n Wp (1). Assume that the opposite is the case, i.e., there exists a sequence {uj}j>1 C WP (S2) n C(1)

convergent to u in WP (SZ) as j -4 oo. If 0 E (0,7r/4), 0 < B < 1/cos 0, Holder's inequality yields

IUj(e,e)-uj(0)1= 1/ cos B

< c(p)

-

e 8uj Jo

ar

(r, 0)dr

P

rdr

Hence

Iluj - uj(o)IIP,s < c(p)IIVujIIP,s,

(2)

2. Examples of "Bad" Domains in the Theory of Sobolev Spaces

92

where S = {x

: xl E (0,1), x2 E (0, xl)}. Since u3IS -* 0 in Wp (S), inequality (2) gives uj(O) -* 0. However, a similar argument for the triangle {x : xl E (-1, 0), x2 E (0, -xi)} gives uj(O) -> 1, a contradiction. Let p E [1, 2). We introduce a function g E C°°([0, oo)) satisfying g(t) = 0 fort < 1 , g(t) = 1 fort > 2. For any u E WP (Sl) let

u£(x) = u(x)g(ole),

where e > 0 is a small parameter. First we show that £lim o IIu£ - u11w1(c) = 0.

(3)

Equality (3) is a consequence of the estimate IIu/PIIp,c < c(p)IIVuIIp,s,

(4)

in which u(o, 0) = 0 for o > 1/2. Let us verify (4). Fix any ray 0 = const, 0 E (0, 7r/4) U (37r/4,7r). An application of Hardy's inequality (1.1.2/7) yields

f 0

Iu(e,0)Ipof-pdo
Iue(P,0)Ipodo

(5)

o

Now (4) follows by integrating (5) with respect to 0. So (3) is valid.

Next we extend u£ to be zero on B£ \ Q. Then u£ E Wp (B£ U Q), and the function u£ can be approximated by functions in C°° (B£ U Sl) in view of Theorem 1.4.2/1. Thus, the set C°°(?!) is dense in Wp (Q).

In the case p = 2 one can obtain the same result if u£ is defined by u£(x) _ u(x)h£(o), where h£ is the same cut-off function as in the hint to Exercise 1.6

(for p = n - s).

2.2. Functions with Bounded Gradients are not Always Dense in Lp(0) Let Sl be a domain in Rn with finite volume and let U denote the set of bounded smooth functions on Sl with bounded supports in Q. Corollary 1.4.3 says that U fl LP(Q) is dense in Lp(Q) whenever p E [1, oo). The following example shows that the set UnLP(Sl) cannot be generally replaced by Ll(Q).

2.2. Functions with Bounded Gradients are not Always Dense in Lp(n)

93

Example. Let p E (2, oo) and {ai}i>1, {e;}i>1 be two sequences of positive numbers satisfying the conditions al + E1 < 1, a;+i + Ei+1 < at, i > 1, lima= = 0, and

ai -pEi < 00.

(1)

i>1

The planar domain S2 is the union of the square Q1 = (-1, 0) x (0, 1), the triangle

Il2={(x,y)ER2:xE (0, 1), yE (0, x)} and the passages

{(x,y):yE(ai,ai+E2), 0<x
u(x, y) =

0

if (x, y) E S21i

1

if (x, y) E SZ2,

x/y if (x, y) E SZ \ (01 U Q2)-

''Y 1

Q1

02

v 0 Fig. 6

Then u E LP(I) because of (1) (in fact u c C(SZ) n L,,(S2) n L1(S2)). We shall show that u cannot be approximated by functions in L , (Q) in the metric of the space LP(SZ) given by (1.5.3/1).

2. Examples of "Bad" Domains in the Theory of Sobolev Spaces

94

Let v E L' (1) be an arbitrary function. This function coincides with a function in C(1) a.e. on Q. Moreover, by Sobolev's imbedding theorem, there is a constant K > 0, independent of u and v, such that Ilvlioo,Q1 + IIv - 11100,02 < KIIv - UIIL;(Q)

Using the absolute continuity of v on almost all line segments with y E U:>1(ai,ai + e1),

we obtain Iv(y,y) - v(O,y)I =

f

Y

8x (x, y)dx s y llovlloo,o.

Thus I v(y, y) - v(O, y) I < 1/2

for sufficiently small y satisfying (3). Hence the left part of (2) is not less than 1/2, and the quantity Ilu - vIl LP(o) cannot be less than (2K)-1.

2.3. A Planar Bounded Domain for Which L2(0) n L00 (fl) is not Dense in Li (f)) According to Lemma 1.4.3, the subspace of bounded functions in Lp(SZ) is dense in L, (S2) for p E [1, co) and any domain SZ C R. It turns out that this property cannot be generally extended to Sobolev spaces of higher orders. In this section we give an example of a bounded domain 0 C R2 and a function such that f does not belong to the closure of Li(S2) n Lq(S1) in fE

the norm of L' (Q) with arbitrary q > 0. In particular, this implies that L2P (S2) n L.(Q) is not dense in LP(Q) for p < 2.

First we establish an auxiliary assertion. Below we identify functions in L2 with their continuous representatives (cf. Sobolev's theorem).

Lemma. Let G be a planar subdomain of the disk BR starshaped with respect to the disk Br. Then for any f c L2(G) the following estimate holds l f (xl) - f(Z2)1 < c (llo2f II1,G + Izl -

z21r-1-2/9llf

II9,G),

where z1, z2 E G and the constant c depends only on q and the ratio R/r.

(1)

2.3. A Planar Bounded Domain for Which L2 (1) fl L.. (S2) ...

95

Proof. It will suffice to consider the case r = 1 and then use a similarity transformation. By Theorem 1.5.2, there is a linear function P such that Ilf - ill 1,G <- c IIV2f IIl,G.

Hence, by the Sobolev imbedding L2(G) C C(G), we obtain two estimates

l (f - f)(zl) - (f - f)(z2)l < C IIV2f IIl,G, IIellmin{1,q},G <- C (Ilozf lll,G + IIf IIq,G)

Now (1) follows in view of the obvious inequality l1(zl) - e(z2)l <- c Izl - z21

Ilellmin{l,q},G.

This concludes the proof of the lemma.

0

We turn to the required example. Let {bi}i>o, {hi}i>o be two sequences of positive numbers satisfying bi < 2-i-2 and

hi < exp (- (1 + i)2/b; ), i > 0,

(2)

lim bi2ib = 0 for every b > 0.

(3)

i 400

Next, let {A1}i>o be the sequence of open isosceles right triangles with hypotenuses of length 21-i, placed on the lines y = Hi, where Hj = 21-3 + >(h, - ba), j = 0,1,... >j

We assume that all vertices of right angles lie on the axis Oy under the hypotenuses. Let ri denote the intersection of 8Oi with the half-plane y > Hi+l + hi. Clearly the distance between ri and r +1 is hi. By SZ we mean the complement of Ui>0I'i to the rectangle {(x, y) E R2 : lxl < 1, 0 < y < Ho}, (see Fig. 7, 8).

96

2. Examples of "Bad" Domains in the Theory of Sobolev Spaces

Let i E COI(-l, 1), and r)(t) = 1 for Itl < 1/2. Let f be defined on SZ such that for every strip

lIi={(x,y)E0:Hi+1 0, f is given on IIi n S2 by (IxI Ji) f (x, y) = sign x(1 + di log hi log (Ixi - bi + hi) 77 (2i+'x))JJ

for (x, y) Di, IxI > 8i, and f (x, y) = x/bi for the remaining points of IIi. Clearly IIi n supp (V2f) is placed in the set

{(x, y) E Hi \ Li : 6i < IxI < 2-i-1}.

2.3. A Planar Bounded Domain for Which L2([) n L00(1) ...

97

Furthermore, the following estimate holds for (x, y) E ni j02

fl < max{2i(i + 1), (jxi - bi + hi)-1} c

bi 1 log hi

(here and below in this section c is a positive constant independent of i). Hence IIV2f112

=

IIV2.f112

loghiI+(i+1)2


( log h1)2

i>O

i>O

which is dominated by c Ei>1 i-2 due to (2). We now prove by contradiction that f does not belong to the closure of L2(SZ) n Lq(SZ) in the norm of L2(SZ). Let Q0 = SZ \ (Vi>1Oi).

Since SZo has the cone property, Sobolev's theorem says that the space L1(SZ0) is imbedded into C(SZ0) n Lm(SZ0), and IIuiILoo(clo) < KIIuIIL2(n), K = const > 0,

(4)

for all u E L2(SZ). Suppose that there exists a function g E L2(SZ) n L9(SZ) subject to Ilf - 91ILi(O) <

(2K)-1.

Put AT = (+bi, Hi+1 + hi) , i > 0. Because f (Ai) _ -1, f (At) = 1, the estimate

is valid in view of (4). On the other hand, an application of the above lemma gives c

9(A1 )1

<_ JJV29liL1(oinni) + 2i+1+2i/9di IigiILq(oinn.)

This inequality in conjunction with (3) implies lim

9(A )i = 0,

and we arrive at the required contradiction.

98

2. Examples of "Bad" Domains in the Theory of Sobolev Spaces

2.4. On Density of Bounded Functions in L' (fl) for Paraboloids in R" We have seen in the preceding section that bounded domains with nonsmooth boundary may fail to have the property that the set LP (S2) fl L,, (S2) is dense in L2 (sl). It turns out that for unbounded sz C Rn this property may fail even when Sl is very simple with smooth boundary. Let f be a function in C'([0, oo)) such that f is positive on (0, oo), W WI < const and f (t) -4 oo as t -> oo . If f (0) = f'(0) = 0, we impose an additional

condition f(t) > 0 in the vicinity oft = 0. Let

92={(x,t) ERn:0
it

I FT

In this section c denotes various positive constants depending only on p, n, f . The following assertion gives a necessary and sufficient condition for the space LP (I) fl Loo(Q) to be dense in LP (S2).

Theorem. The space LP (S2) fl Loo (12) is dense in LP(Sl), 1 < p < oo, if and only if

00 ff(t)dt=oo.

(1)

(Clearly (1) is always valid for p > n).

The proof of this theorem will be given at the end of the section. First we establish three auxiliary assertions.

2.4. On Density of Bounded Functions in L2 (n) for Paraboloids in R"

99

Lemma 1. Let u E LP(Q) fl L,,, (Q). Put

I T= {(x, t) E S2 : t= T}, QT = {(x, t) : 0 < t < T, IxI < f(t)} and

w(T) = f(T)1-"

au

fr,

dx.

8t

Then for allT>0 r00

w(T)I

c (

l (P-1)/P f (t) T=F 1-7 dt f

/

\T

(L\OT

1 1fP IV2ulPdxdt I

(2)

f

Proof. We may assume that

ff(t)dt <

.

(3)

Since

w(T)

4I<1 8t

t-T

(f

we have for S > T

w(S) - w(T) =

a

Sdt f (ut(.f(t)f,t)) II<18t

ITT

a2u

s

f dt 41<1 (

at-2 (f (t)£,

t) + f'(t) (ox at) (f (t)£, t)) dC.

By Holder's inequality Iw(T) - w(S)I

< c(

f

S

\\ T

\ (P-1)/P

f (t) n--i dt I

f

if

S

\T

dt

1/p

f

(V2u) (x, t) I Pdx

Ixl
Thus, the limit d = limt.,,o w(t) exists and it suffices to deduce d = 0 in order to prove (2). Consider the function u on (0, oo) defined by

u(t) = f (t)1-n fxI
f

u(f (t) C, t)dd. I<1

2. Examples of "Bad" Domains in the Theory of Sobolev Spaces

100

Putting

t) = u(f (t)e, t), we find =

It

t)<-

w(t) +

41<1

(4)

Furhtermore,

f V vd = (1 - n) f vd + f ;l<1

CI<1

vdst, 1=1

1. Since v is bounded, where ds is the area element on the sphere the second term on the right in (4) tends to zero as t -4 oo. So we have limt-,, u'(t) = d, and d = 0 as far as u E Lo. (0, oo). This completes the

proof of Lemma 1.

1

Lemma 2. Let (1) be valid and let

fn- 'It. _ IT IT\I'IP + fPIco"!

F(OP)

1

(5)

Then inf F(W) = 0, where the infimum is taken over the set {cp : cp c C°°(R1), W(t) = 1 for t < 1, W(t) = 0 for large positive t}.

(6)

Proof. First let p > n. Since f (t) < c t for large t, it will suffice to prove the equality

infJ

(Ip'IP+tPIWo"JP)tn-1dt=0.

(7)

1

Consider the function ri E Cm(R1), 7j(t) = 1 for t < 1, rl(t) = 0 for t > 2. By setting W(t) = 77(t/N), N = 1, 2, ..., we obtain (7). Let p < n. Since

f lw'I 5 c (vi + I(f D-1 )'I ), functional (5) is majorized by

C(

n

-1

f D-1 !

I

-1 n-1 + f P-1 (f D-1 7' )/

P

It

y-1

Q (8)

2.4. On Density of Bounded Functions in LP(1) for Paraboloids in R"

101

on the set (6). After passing to the new variable T = T(t) given by t

T=f f(A)a

11d A,

1

the integral in (8) takes the form

G(0) =

0(r)=go(t).

foo(IG,IP +

0

It remains to observe that G(?PN) -+ 0, where V)N(T) = 71(T/T(N)), N = 1, 2, ..., and i is the function introduced above. This concludes the proof of the lemma. The proof of the preceding lemma contains the following assertion.

1

Corollary. If (1) holds, then there exists a minimizing sequence {cpN}N>1 for functional (5) such that q.' E C°°(R1), 9N(t) = 1 for t < N, WN(t) = 0 for large positive t.

Lemma 3. Let u E LP(cl). For any e > 0 there exist a linear function t and a function v E LP(S1) such that v(x, t) = t(x, t) for large t and Ilu - VIIL2(sl) < e.

Proof. Let

GN={(x,t)EcI: N
By Theorem 1.5.2, there is a linear function tN satisfying

IIV(u -

tN)IIp,GN +

f(N)-'Ilu - tN11p,GN

< c f (N) II V2uIIp,GN

(9)

We put

71N(t) _ ij(1 + (t - N)/f (N)),

VN = (u - tN)7IN + tN,

where 77 is the function introduced in the proof of Lemma 2. Clearly VN = u

fortN+f(N). Next,

c 11V2(u - VN)IIp,I 5 11(1- r)N)V2u11p,c

2. Examples of "Bad" Domains in tite Theory of Sobolev Spaces

102

+II IV(u - £N)I IV77NI IIp,S1 + 11(U - £N)IV21)NI 11p,62-

Since

IViNI < cf(N)-1, IV277NI < and in view of (9), it follows that

cf(N)-2

IIV2(u- VN)IIp,n < CIIV2UIIp,o\QN.

The last norm tends to zero as N - oo, and we can set v = VN for sufficiently large N. This establishes Lemma 3. 1 Proof of Theorem. Let (3) hold. We check that LP(12) f1 L.,, (Q) is not dense

in L2(1). Let u E LP (S2) f1 Lm(12) and u -+ tin the norm of LP (Q). By Lemma 1

frT at

dx < c(n,p, f,T) IIV2uvIIL(o) = o(1) as

v -oo.

(10)

We have u -4 t in L2

2 \ 121). Because 122 \ 121 E C°'1, the spaces LP and VP coincide for this domain. Thus, by Sobolev's theorem,

aui,

dx -4 mesn_ 1 FT

as

v -4 00

for almost all T E (1, 2). However, this contradicts (10). Suppose (1) is valid. It will be shown that an arbitrary function u E LP(1) can be approximated by functions in L2 (12) f1 Lm (12). According to Lemma 3,

it is sufficient to assume that u = w + t, where a is a linear function and w(x, t) = 0 for large t. Since any domain 12T is of class C, Theorem 1.4.2/1 applies, and w can be approximated in L2 (Q) by functions in C°° (S2) with bounded supports. It remains to approximate the functions t, x1,. .. , xn_1 Let {cpN} be the sequence from Corollary preceding Lemma 3. We set /Nt

VN (x, t) =

I N + J WN(s)ds t

for

t < N.

Clearly VN E L2 (12) f1 Lm (12) and II V2 (t - vN)IILp(o)

for

t > N,

2.5. Imbedding and Compactness Properties May Fail ...

=cJ

IWp

103

(t)IPf(t)"-'dt=0(1) asN -*oo.

0

To approximate xj, 1 < j < n - 1, we put WN(x,t) = xj c2N(t). Then IIV2(x; - WN)IIL,,(Q)

C

foo (IWNIp+fPIW/NIP)f"-ldt

and reference to Corollary completes the proof.

2.5.

Imbedding and Compactness Properties May Fail for the

Intersection of "Good" Domains Sobolev's theorem says that the restriction and imbedding operators mentioned in Remarks 1.8.2/2-3 are bounded if 1 is the union of a finite number of domains in EVP. However, this is not true if Q is the intersection of domains in EVP (or even in C°°). Consider the following example.

Example 1. Let 77 be a nonincreasing function in C°°([0,oo)) such that ra(t) = 1 for t E [0,1/2], n(t) = 0 for t > 1. We put

0ifx<0, f (x)

SZl

exp (-1/x)rt(x) if x > 0. = { (x, y) E R2 : y >

_f(X)I,

SZ2={(x,y) ER2: y< f(x)}. Let SZ = SZ1 n Q2, P E [1, oo) and 1 > 1 an integer. Define

u(x, y) = x1 exp((px)-1), (x, y) E Q. Then u E VP (Q) fl COO (Q), whereas u V Lq (SZ) for q > p and

ujy=o V Lq(SZ fl {(x, y) : y = 0})

for any

q > 1.

According to Lemma 1.10, the imbedding VP (SZ) C VP-' (1) is compact if SZ

is the union of a finite number of bounded domains in EVP. We now present

2. Examples of "Bad" Domains in the Theory of Sobolev Spaces

104

an example showing that the intersection of bounded domains 521, 522 E EV2' does not generally give a domain SZ for which V2 (52) is compactly imbedded

into L2(1).

Example 2. Let {hk}k>1 and {52k}k>1 be sequences of positive numbers satisfying CIO

0 < const < hk+1 hk 1 < 1, E hk = 1,

82k = (h2k)a < h2k+1

(1)

k=1

with a > 3. Put k

ek=Ehi, k=1,2,.... i=1

Let ii be the union of the square (0, 1) x (-1, 0] and the rectangles

Rt = (ej - hj, oj) x [0, hj/2),

Pt+1 _ [Pj, ej + hj+1] x [0, Jj+1/2),

where j = 1, 3, 5, .... By SZ2 we mean the image of Q1 under the reflection (x, y) H (x, -y). Clearly 521 i SZ2 are quasidisks (cf. the comments to Sec. 1.6.2) so that 521i SZ2 E EV21. The intersection 52 = S21 n Q2 is shown in

Fig. 10. It is the union of the "rooms" Rj and "passages" Pj+1 (j = 1, 3, 5, ...) which are obtained by unifying the rectangles Rt and Pj+1 with their mirror images under the reflection (x, y) ,-> (x, -y). y 82

---P2----R3--- P4--------

-i 1

h1

x

h2 Fig. 10

We shall check that V2 (52) is not compactly imbedded into L2(52). Consider the sequence {uk} (k = 1, 3, 5, ...) of continuous functions defined by hk 1

Uk =

I

in

Rk,

2.6. A domain for Which the Imbedding Wl(1) C C(l) fl L. (0) ...

105

Uk is linear in Pk_1 and Pk+1. Then I Vkul = (hkhk±l)-l in Pkfl, respectively. According to (1), we have IIUk II2, 1

- 1 + hk 2 (hk_16k_1 + hk+lbk+1) < 1 + const hk,

IIVukII ,sz = (hkhk-1)-2hk-lbk-1 + (hkhk+l) 2hk+lbk+l C const, and the sequence {uk} is bounded in V2 1(Q). On the other hand Ilui-ukU2,s1>_2

for

i#k,

and there is no subsequence of {uk} convergent in L2(ci).

2.6. A Domain for Which the Imbedding WP I(0) C C(1) fl Lc (11) is Continuous but Noncompact It follows from Theorems 1.8.2 and 1.10/3 that if (l - k)p > n, then the continuity and compactness of the imbedding operator: n(c) - Ck(c) n V, (S2) simultaneously hold provided S2 is a bounded domain having the cone property. However, this is generally not true.

Example. We shall show that the space WP 1(Q) (p > 2) is continuously imbedded into C(Q) n Lm(S2) for the domain SZ given in Fig. 11, but the imbedding is not compact. This domain is the union of the rectangle Q = (0,1) x (0,1) and the sequence of rectangles Qk, k > 1, adjoining to Q. Let Ek be the height of Qk, and bk = ElkAP-1) its base, where Ek -* 0. Suppose that the exterior of any neighborhood of the origin contains only a finite number of rectangles Qk.

Ek

Q

Q

0 Fig. 11

1

X

2. Examples of "Bad" Domains in the Theory of Sobolev Spaces

106

The continuity of the imbedding operator: WP (1) -+ C(S2) fl Lo,, (1) is a consequence of Sobolev's theorem and the following assertion.

Lemma. Let 0 < e < b < 1 and put QE,6 = (-b, d) X (0, E),

QE 6 = (0, d) X (0, E).

If p > 2, then C

IlUII.,Qc.6 <

IIuIIoo,Q +E-1/Pb1-1/PIIVuIIp,Qc,6

(1)

Q

for all u E WP (QE,6) with c = c(p) > 0.

Proof. With the aid of a similarity transformation, inequality (1) is reduced to the same inequality for d = 1 (in this case we write QE instead of QE,,).

For u E WP (QE) and x E (-1, 1), let u(x) be the mean value of u(x, ) on (0, e). Clearly (2) IIuIIoo,Q, 5 IluII.,(-1,1) + Ilu - ullo,QE An application of the estimate c IIkII-,(-1,1) <- Ik'IJ.,(0,l) + ll0IP,(-1,1)

with v = u yields C

s IItIIm,QE + E-1/PIIouJIP,Qe.

(3)

We now bound the last term in (2). One may assume that the number N = 1/e is an integer (otherwise we use the even extension of u across the left and right parts of aQE). Let

Q(=)_{(x,y)EQE:XE(ie,ie+e)}, -N
(4)

for each i = -N, ... , N - 1. This follows from the same inequality with e = 1 by contraction. Hence the last term in (2) is not greater than c IlVuIIP,QE. 1 Combining (4) with (2) and (3) gives (1) for 6 = 1.

107

2.7. Nikodym's Domain

We claim that the imbedding operator: WP (Sl) -+ C(S2) n L,,,,(S2) is not compact for the domain 1 in Fig. 11. Indeed, For each k = 1, 2.... define (1kak)-1/Px

uk(x,11) _

t0

if (x, y) E Qk,

if(x,y)Efl \Qk

Then the sequence {uk}k>1 is bounded in WP (1). At the same time

IIui - ukII.,n = 1 for i # k, and there is no subsequence of {uk} convergent in C(S2) fl L,,, (92).

2.7. Nikodym's Domain 2.7.1. A Domain with the Property L,(S2) ¢ Lq(S2) for l = 1, 2, ..., q > 0 and p E (1, oo)

Suppose we are given a decreasing sequence {ek}k o, such that o E (0, 11 and k -40. Let S2 C R2 be the union of the rectangles (cf. Fig. 12) Ak = {(x, y) : x E (C3k+2,e3k), Y E (2/3,1)}, k > 0,

Bk = {(x, y) : x E (6k+1,6k), Y E [1/3, 2/3]}, k > 0,

D= {(x, y) : x E (0,1), y E (0,1/3)}. y 1

Ak

If

Bk

D 0

1

Fig. 12

2. Examples of "Bad" Domains in the Theory of Sobolev Spaces

108

In case

6k =

2-k,

C 2-k - Ek, Ek E (0, 2-2-k), k > 0, (1) 6k+2 = 3 2-2-k, 6k+1 =

this domain 0 was used by Nikodym [169] to show the failure of the imbedding L'(Q) C L2(S2) for an appropriate choice of {Ek}.

Let {l;k} be defined by (1) with Ek = 2-3-k3, k > 0. We claim that then the inclusion LP(S2) C Lq(S2) is impossible for any p E [1, oo), q > 0 and l = 1, 2, .... Indeed, consider a function f E Cm([0,1]) such that f (t) = 1 for

t > 2/3 and f (t) = 0 fort < 1/3. Put 2k2f(y)

u(x, y) _

if (x, y) E Ak U Bk, k = 0,1, ... ,

0 if (x, y) E D.

Then

II VzulIP,, < c(l, p) E

2pka-k3

< W.

k

Hence u E Lp(Q). At the same time

IIulIq,0 > E

2gk2-k-2

_ CO,

k

and u V Lq(S2).

2.7.2. A Domain for Which VP (1) is Noncompactly Imbedded into Lp(1) for p E [1, oo] and l = 1, 2, .. .

Lemma 1.10 says that if a bounded domain S2 C Rn is the finite union of domains in EVp, then the imbedding operator: VP (Q) -+ Vy-1(Q) is compact. Also Vp(1) is compactly imbedded into Vq-1(S2) for 1 < q < p provided only 0 has a finite volume (cf. Corollary 1.10). However, the imbedding VP (S2) C Lp(S2) generally fails to be compact. We shall show that the imbedding VP(S2) C Lp(S2), p E [1, oo], 1 = 1, 2,..., is noncompact for the domain in Fig. 12 for any choice of Let f be the same function as in the preceding subsection. A sequence {uk}k>o is defined on S2 by uk (x, y)

ak 1/pf (y) if (x, y) E Ak U Bk,

0 otherwise,

2.7. Nikodym's Domain

109

where ak = mes2(Ak). Then {uk} is bounded in Vj(SZ). Furthermore, we have

if i

Iluk - uiIIP,O > 21/P

k.

Hence, there is no subsequence of {uk} convergent in LP(a). Thus, the imbedding VP (0) C LP(1) is noncompact.

Theorem 1.10/2 now gives that V(SZ) cannot be imbedded into Lq(Il) for the same Sl and q > p. However, this can be checked directly: if {uk} is the above sequence, then

Ilukllq,Q(llukIIVV(o))

cak/q-1/P

>

00

for q > p. So the continuous imbedding VP (SZ) C Lq(1) is impossible. 2.7.3. Equivalence of the Imbeddings LP(S2) C Lq(SZ) and L1(SZ) C Lq(SZ)

Here we again deal wth the domain described in Sec. 2.7.1 and shown in Fig. 12. We find necessary and sufficient conditions on the sequence {l:k} that

ensure the continuous imbedding L,(1) C Lq(SZ), p > 1, q > 0. It turns out that these conditions are the same for 1 = 1 and for l > 1. Positive constants c appearing in this subsection depend only on 1, p, q. Two

positive quantities a, b are called equivalent (denoted a - b) if c-1 < a/b < c.

Proposition. Let bj = t3j - 1;3j+2, Ej = S3j - S3j+1, j >- 0. The space LP()), p E [1, oo], is continuously imbedded into Lq(SZ) if and only if one of the following conditions holds: (i) p = q and M = supj>o(81Ej 1)1/P < oo;

(ii) 0 < q < p and K =

o

'00.

b

Moreover, the best constant C in the inequality inf{Ilu - Pllq,cz : P E P,_1} < C IIV,uIIP,n,

is equivalent to M for q = p and to 1 +

K1/q-1/P for

This proposition implies a direct consequence.

u E LP(SZ),

q < p.

(1)

2. Examples of "Bad" Domains in the Theory of Sobplev Spaces

110

Corollary. For the domain in Fig. 12, the space LP(1l) is continuously imbedded into Lq(I) if and only if L1(SZ) is continuously imbedded into L. (Q). We need two lemmas to verify Proposition.

Lemma 1. Let v E VP (0, b), 1 < p < oo. Then for any E E (0, 6) t-1

CII

V IIP,(O,6) :5 aI Ilv(I) IIP,(O,6) + (bE-1)1/P

E ak I

I V (k) I IP,(O,E).

k=O

Proof. It is sufficient to consider the case d = 1. We have cIv(s)IP <_ Iv(t)IP+J 1Iv'(T)IPdT, s E (0,1), t E (0, E). 0

By integrating with respect to s and t, we arrive at c

IIVIIP,(0,1)

E-1/P

IIVIIP,(O,e) +

IIv'IIP,(0,1),

p E [1, oo].

Now the required estimate can be obtained by iterating the last inequality. 1 Lemma 2. Let SZ be a bounded domain in R" and G a nonempty open subset

of Q. Suppose that q > 0, p > 1 and that (1) is valid with some constant C > 0. If u E COO (Q) and u I G = 0, then IIU1Iq,n < C (1 + y)C IIVzuIIP,n,

where y is the norm of the identity map of the space PI-1 with norm of Lq(G) onto the same space with norm of Lq(SZ).a

Proof. Let P E PI-1 be a polynomial which provides the infimum on the left of (1). Since uIG = 0, we have IIPIIq,G < C IIVIUIIP,n,

whence min{21-1/q,1}IIuIIq,n

IIPIIq,n + Ilu - PIIq,n

< (1 +'Y) C a for q E (0,1), II . Ilq,n and 11 - IIq,G are in fact pseudonorms.

IIVIUIIP,n.

1

2.7. Nikodym's Domain

111

Proof of Proposition. Let f E C ([0,1]), f I[O,1131 = 01

f L2/3 1] = 1.

For each k > 0 define Vk on S2 by

f f (y) if (x, y) E Ak U Bk,

vk(X,y) = Sl

0 otherwise,

(cf. Fig. 12). Clearly Vk E C°°(SZ), VkID

IIVlvkIIP,n < CE//P

0, IIVkjIq,Q

Suppose L,(12) is continuously imbedded into Lq(0). In view of Lemma 1.12.1,

(1) is valid for all u E L,(1) with some constant C > 0. By Lemma 2 IIvkIIq,Q <- cCIIVzvkIIP,S2, and

C > C bk/qEk 11P,

k = 0, 1,... .

Hence q < p and C > c M in the case q = p.

Let q < p. For N = 0,1, ..., we put UN =

Eko akvk with some positive

coefficients ak defined below. Since UN I D = 0, Lemma 2 implies IIUNIIq,I <_ CC IIVIUNIIP,l, N > 0.

Hence N

I

E akok) 1/9 < C C (E k=0

cek) 1/P' p< 00.

k=0

J

Choosing akbk = akEk, one obtains N

C > C (Y akl (P-q)6kl(q-P))

11q 1/p

, N = 0, 1,... .

k=0

Passage to the limit as N - oo yields C > cKl/q-1tP. For p = oo we arrive at the same result if we put ak = 1, 0 < k < N. The estimate C > c also

112

2. Examples of "Bad" Domains in the Theory of Sobolev Spaces

holds. This can be verified by Lemma 2 with a trial smooth function v on SZ such that v(x, y) = 1 for x < 1/3, v(x, y) = 0 for x > 2/3. Thus, the continuity of the imbedding operator: LP(SZ) -* Lq(SZ) implies either condition (i) or (ii). Now suppose that (i) or (ii) is fulfilled. We shall validate inequality (1) with C= c M for q = p and with C = c (1 + K1Iq-1IP) for q < p.

First consider the case u E L,(S2) and u(x, y) = 0 in the vicinity of y = 0. Here we check the Friedrichs inequalities IIuIIP,n < cm IIozuIIP,sa,

(2)

IIull.,Q < c (1 + K1/q-1/P)IIV,uIIRn, q < p.

(3)

Clearly <_

Cdk/q-1/PIIuIIP,Ak,

IIUIIq,Ak

p'

q

Put Gk = { (x, y) E SZ : x E (bk+1, S3k) }, k > 0.

Applying Lemma 1 to each section y = const of the set Ak and then integrating with respect to y gives

C IIUIIP,Ak < 8 jjVIUjIP,Ak + (bk

Ek

1/P

1-1

5 IIVjullP,GknAk. j=0

Hence 1-1

Cbk lI9EkIPIItIIq,Ak < IIVIUIIP,A,, +1: IIVjulIP,ck' j=0

Therefore, for every "axe" Tk = Ak U Gk the estimate !-1 CaklIgEkIPIIuIIq,Tk <

IIVjuIIP,Gk

IIo1UIIP,Tk+

j=0

holds true. Since u(x, y) = 0 in the vicinity of y = 0, we have IIVjuIIP,Gk 5 C

(4)

2.7. Nikodym's Domain

113

and (4) implies IIUIIq,A,,UB,, <

C6kI9Ek 11PIIoIUIIP,Tk,

qO.

(5)

Furthermore IItIIq,D < C IIVIUIIP,D, q <- p.

(6)

Let q = p. Raising (5) to the power p (for p < oo), summing over k > 0 and combining with (6), we arrive at (2). When p = oo, inequality (2) is a simple consequence of (5), (6). Let q < p. It follows from (5) that IUIq S2\D < C E 6kEk

4iPlloluIIP,Tk

k>O

An application of Holder's inequality to the last sum yields IIUIIq,fl\D < c K1/q-1/PIIVlulIP,s2.

(7)

Estimates (6) and (7) imply (3).

Turning to the general case u c LP(Q), we introduce a smooth cut-off function a on SZ such that u(x, y) = 1 for y > 1/3, u(x, y) = 0 for 0 < y < 1/6 and 0 < Q < 1. Clearly C Ilullq,s2 <_ II0'uIlq,S1 + IILIIq,D

The last term does not exceed c IIuUIP,D by Holder's inequality, whereas the quantity IIaullq,f, can be estimated with the aid of (2)-(3). Thus 1-1

IIuIIq,s2 < C (IIVIUIIP,c +:IIVjuIIP,D),

(8)

j=0

where C = cM for q = p and C = c(1 + K1/q-1/P) for q < p. By Theorem 1.5.2, there is a P E PI-1 such that I-1

Iloj(u-P)IIP,D 5 CIIVIUIIP,D j=0

To complete the proof of Proposition, replace u by u - P in (8).

114

2. Examples of "Bad" Domains in the Theory of Sobolev Spaces

2.7.4. The Neumann Problem for Nikod lm's Domain Here we give an application of Proposition 2.7.3 to the Neumann problem for an elliptic differential operator of order 21 for the domain 0 in Fig. 12 (see Sec. 2.7.1). Beforehand, we observe that the set C1 (Q) is dense in Lip (Q), l = 1, 2, ..., p E [1, oo). Indeed, let

SZ°=DUAoUBo, 1N=QN-1UANUBN, N=1,2,....

(1)

Introduce a bounded extension operator

E:Vp(D) -4 Vp((0,1) x (0,1)) (cf. Fig. 12). If u E Lp(S' ), define UN on Q by UN = u on ON and UN = E(uI D)

on 0 \ ON. Then clearly UN -4 u in L,(Q) as N -1 00. In fact, the equation UN = E(uJD) defines UN on some rectangle containing Uk>N(Ak U Bk) and disjoint from Ak U Bk for k < N. Hence UN belongs to VP (GN) for some domain GN E C°"1 with Q C GN. It remains to approximate UN in VP (GN) by functions in C°'(GN) (cf. Theorem 1.4.2/1). Let SZ be the same as above and consider the operator Aq of the Neumann problem for an elliptic differential operator of order 21 described in Definition 1.12.2. The preceding observation shows that Corollary 1.12.2/1 applies. Combining this corollary with Proposition 2.7.3, we obtain that the equation Aqu = f is solvable for every f E Lq,(SZ) e P,_ 1, 1/q' + 1/q = 1, if and only if one of the following conditions holds:

q=2 and

Sup Ek 1 ak <00 k

or

1 < q < 2 and

621(2-q)Ekl(q-2)

< 00.

k>O

2.8. The Space WZ (0) fl L (St) is not Always a Banach Algebra It was shown in Sec. 1.11 that if 0 C R' is a "nice" domain (say 1 is bounded with the cone property), then the space WP(1l) n L,(1l) is an algebra with respect to pointwise multiplication. However, this fact is generally not true.

2.9. The Second Gradient of a Function May Be Better Than the First One

115

Here we give an example of a bounded planar domain SZ such that W2 (1) n L... (1) is not an algebra. y

'- 2_k _ I

Pk

-aki I

Q

I

Sk

-

Fig. 13

Fig. 14

Let SZ be the union of the rectangle P = {(x, y) : x c (0, 2), y c (0, 1)}, the squares Pk with edgelength 2-k and the passages Sk of height 2-k and of width 2-,k, k = 1, 2,..., a > 1 (see Fig. 13, 14). Define 0 u(x,y)

on P,

23k/2(y - 1)2

=

i

on

Sk, k = 1,2,...,

2k/2(2(y - 1) + 2-k)

on

Pk, k = 1,2,...

One should merely compute to obtain IIV2u

II2,s' =

22+(2-a)k,

Jul < 4, 1iV2(u2) 112,Pk = 8.

Thus, if a > 2, then u E W2 (SZ) n L,,. (1), but u2 V W2 (SZ).

2.9. The Second Gradient of a Function May Be Better Than the First One It is obvious that we always have VP (SZ) C W, (1) C Lp(SZ). The inclusions become equalities for bounded domains with the cone property (cf. Corollary 1.5.2) or for finite unions of domains in the class C (cf. Exercise 1.9). The example considered in Sec. 2.7.1 says that generally W' (Q) # L , (Q). Here we show that the domain Q in Fig. 13-14 has the property W2 (Q) V2 (Q) for an appropriate choice of the parameter a.

2. Examples of "Bad" Domains in the Theory of Sobolev Spaces

116

Put 0 on P,

U=

4k(y _ 1)2 on Sk, k = 1,2,...,

2k+1(y_1)_1on Pk, k=1,2,.... Then 22+(3-c,)k,

lul C 3,

IIV2uII2,Sk =

IIVuII2,Pk = 2.

Hence IIO2uII2,o < oo if a > 3, but IIVuH2,o = oo. So W2 (Q) i4 V2 (Q). In fact, we have shown that L2 (Q) fl Lo. (Q) ¢ LZ (Q) for a > 3. It is interesting to note that for a > 5 L

0onP, v =

23k(y

_

22k+1(y

1)2 on Sk, k = 1,2,..., _ 1) - 2k on Pk, k = 1,2,...

Then IIV 2vlI2,Sk

= 22+(5-i)k,

IIVII2,Pk

_

1-

Therefore v E L2(0), but v V L2(Q).

2.10. Counterexample to the Generalized Poincare Inequality Let Q be the domain described in Sec. 2.8 and given in Fig. 13, 14. Here we find necessary and sufficient conditions on the parameter a E (1, oo) for the imbedding L2(Q) C L2(Q) to be continuous or compact. In particular, it will be shown that for some a this domain supports the generalized Poincare inequality

inf{Iloi(u - Q) 112,0 : Q E Pl} < C IIO2uI12,92, C = coast,

(1)

for i = 0 and all u E L2(Q) but does not support (1) for i = 1 and the same u. At the end of the section we consider the Neumann problem for elliptic equations of order 21 on the domain in Fig. 13 and give conditions for its solvability.

2.10. Counterexample to the Generalized Poincare Inequality

117

Proposition. The space L12(1) is continuously imbedded into L2(S2) if and only if a < 21 + 1. This imbedding is compact if and only if a < 21 + 1. We need a lemma for the proof of this result.

Lemma. Let S = (0, e) x (0, b) and r = { (x, 0) : x c (0, e) }. Then t-1

C IIUIIL2(S) < E b +1/2IIViuIIL2(r) + 61IIV1

IL2(S)

i=0

for any u E V2(S) with c=c(l)>0. Proof. It is sufficient to assume b = 1. We have 1

2-1 f I u(x, y) I2dy < Iu(x, 0)

12

+

0

f

1

I ut (x, t) I2dt, x c (0, e).

0

Integration with respect to x E (0, e) yields 112

IIU L2(S) < 2IIUI L2 (r) +2IIoIL L2(S)'

The result (with b = 1) follows by iterating the last inequality.

Proof of Proposition. Let bk = 2-k, k = 1, 2.... and let Tk denote the symmetric image of Sk with respect to the line y = 1 + bk (see Fig. 14). Suppose u E LZ(S2), ul P = 0. Then, by Lemma 2.7.3/1, we obtain

t-1

cbkIIUII2,Pk < bLIIVIUll2,Pk +bkllVjuII2,Tk

(2)

j=0

for any k > 1 (here and below in this section c designates various positive constants depending only on 1). Since D°ul8Skn&P = 0 for IaI < 1 - 1, the above lemma gives IIVjuII2,TkUSk < Cbk

'IIo1UII2,TkUSk.

Combining the last with (2), one arrives at IIUII2,SkUPk 5 CµkIIVIUIl2,SkUPk, k = 1, 2, ... ,

(3)

2. Examples of "Bad" Domains in the Theory of Sobolev Spaces

118

where µk =

Jk(1-a)/2

Let U E Ll (0) be arbitrary and let v be a smooth cut-off function on 1 such

that 0 < Q < 1, aIP =

0,

aIPk = 1, IViallsk < c5k

for all k > 1, 0 < i < I. If we put (cf. Fig. 13, 14)

1N+1 =SZNUSNUPN, N> 1,

01 = P, then

IIuII2,n

- IIUII2,StN +

IIUII2,Sk

k>N

IIUuII2,PkUSk.

+

(4)

k>N

With the aid of (3) (applied to au), the last term in (4) can be majorized by 1-1

C 1: Ak(I1olUI12,SkUPk +Ebk(t-`)IloiUll2,sk)

(5)

i=0

k>N

Quantity (5) is also a majorant for the second term on the right part of (4). Furthermore, the above lemma yields Cbk 1IIVjUll2,sk <_ IIVIUII2,Sk

1-i-1

+

6k+j-1+1/2IIVi+juIIL2(8SknaP).

j=0

Hence, expression (5) does not exceed !-1 C

µk II

c 1: Aj,

k> N

i=0

where

Ai =

62k(1+i)-aIIoiUIIL2(aSknBP)+

k>N

If a < 21 + 1, {µk}k>1 is a nonincreasing sequence. Therefore !-1

IIuII2,i !5

IUII2,S1N

+CµNHHVlUIl2,sz\c1N

+cAi/2 i=0

(6)

2.10. Counterexample to the Generalized PoincarE Inequality

119

To bound Ai, we consider the following two cases.

1. i < 1 - 2. By Sobolev's imbedding WW (P) C C'(P), one has jk,IIViuIIL2(&SknaP) < IIoiuIIC(8Skn8P) <
Thus 52(i+l) < c2-2NIIuIIW2(p).

Ai < CIIUIIW2(p)

(7)

k> N

2. i = I - 1. An application of Holder's inequality gives j21-a/2jjV1_1Ujj2

A!-1 k> N

Consequently

Ai 1 < IIoi-1uIIL,(ep) E 641-°. k>N

By Sobolev's theorem, the first factor on the right is dominated by c IIUIIj4y2(p), and

Ai 1 <

C2AN2(1-21)NIIUI14

(8)

W2(p)'

Combining (6)-(8), we obtain C IIu1I2,0 <_

IUII2,fN + /NIIVIU

I2,n\cN

+(2-N + pN2) IIoluII2,P,

(9)

where

a<21+1, N=1,2 .... ,

IN=

2-N(21+1-cti)/2

and u E L2(Q) an arbitrary function.

Note that ON E C°"1 for every N > 1. Therefore, if a < 21 + 1 (and hence AN < 1), (9) and Theorem 1.5.2 imply that there is a constant C > 0 satisfying inf{IIu - QII2,n : Q E PI-1} < C IIViuI12,n

for all u E L2(Q). It follows from Lemma 1.12.1 that L2(1) is continuously imbedded into L2(S2). Moreover, this imbedding is compact if a < 21 + 1, in which case AN -4 0 as N -* oo. Indeed, a set, bounded in L2 (1), is compact in L2(SON) for every N > 1. Hence, using (9) and a diagonal method (cf. the

120

2. Examples of "Bad" Domains in the Theory of Sobolev Spaces

proof of Theorem 1.10/1), we can select from any sequence bounded in L2(1) a subsequence convergent in L2(Il) We now check the necessity of the inequality a < 21 + 1 for the imbedding L2(SZ) C L2(1l) to be continuous. Consider a sequence {uk}k>1 C C°°(1) defined by Uk = 1 on Pk, Uk = 0 outside PkUSk and IOtuk l < c dk 1, k = 1, 2, ... If the imbedding just mentioned is continuous, then there is a constant C > 0 such that .

Iluk112,o 5 C IHVlukII2,S2

for all k > 1. Hence 8(21+1-a)/2 <

c C and a<21+1.

To conclude the proof of Proposition, we check that the imbedding L2(1) C

L2(Q) is noncompact if a = 21 + 1. Let vk = 6k luk, where {uk}k>1 is the above sequence. Then {vk} is bounded in L2(Sl). On the other hand llvk - vj 112,0 >_

llvk 112,Pk + Ilvj

112,P, =

2,

k z j,

and there is no subsequence of {vk} convergent in L2(1).

1

Remark. By using the same trial sequence {vk } as above, one obtains that the space WZ(SZ) is noncompactly imbedded into L2(1) if a > 21 + 1.

1

In case 3 < a < 5 Proposition gives L22 (Q) C L2(Q) but L2(Q) ¢ L2(1). Furthermore, it follows from Sec. 2.9 that LZ(SZ) fl L,,,, (SZ) ¢ L21 (0).

In view of Lemma 1.12.1 and Theorem 1.5.4, we can state the following assertions concerning 1.

Corollary 1. Let SZ be the domain in Fig. 13, 14 with 3 < a < 5. There is no positive constant C satisfying (1) for i = 1 and all u E Lz(1) n Lm(1l), whereas (1) holds for i = 0 and all u E L2(SZ). Corollary 2. If SZ is the same as in Corollary 1, then Theorem 1.5.4 is valid for W2 (1) but is not valid for W2 (SZ).

1

Let Q be the domain in Fig. 13, 14 and let A = A2 denote the operator of the Neumann problem for an elliptic differential operator of order 21 (see Definition 1.12.2). An argument similar to that in Sec. 2.7.4. shows that C°°(Q) is dense in L2(1). Hence Corollary 1.12.2/1 applies. Combining this corollary and above Proposition, we state that the equation Au = f is solvable for every f c L2(SZ) e Pi_1 if and only if a< 21+1.

Let B = B2 be defined by (1.12.2/11). Then the equation Bu = f is uniquely solvable for any f E L2(Q). B has discrete spectrum if and only if

a<21+1.

2.11. Counterexample to the Sharpened FYiedrichs Inequality

121

2.11. Counterexample to the Sharpened Friedrichs Inequality In this section we consider generalizations of the Friedrichs inequality

L which was established by Friedrichs [66] under the sole assumption that Sl is a bounded domain supporting the Gauss-Green formula. Let SZ C R" be an open set, u E C(8S2) and put 1/r

IIuIIL, (asl) _ (f lulyds)

r E [1, oo),

,

n

where s is the (n -1)-dimensional Hausdorff measure. Let Wpr(1, 84 p, r E [1, oo), denote the completion of the set of functions in Lp(SZ)nC°°(0)nC(S2), having bounded supports, with respect to the norm IIuIIWP r(Q,ao) = IIouIILp(s1) + IIuIILr(esz)

(1)

We are concerned with the inequality IIUIIL,(s) < C IIuIIwp r(sz,as1),

C = const,

(2)

for u E Wp r(Q, 8S2). In contrast to the Sobolev theorem for domains in C°,1, the term II IlLr(as2) in (1) does not always play the role of a "weak perturbation" for the Lp-norm of the gradient. It turns out that for "bad" domains the exponent q in (2) may depend on r. We begin with the following assertion. -

Lemma. Let 1 C R" be an open set. The inequality n-1vn 1/nIIullw, 1(I,8I), vn = mesn(Bi), holds for all u E W11, 1

(3)

, O Q) with exact constant.

Proof. Let u be a function in C' (1) nC(SZ) with bounded support. By using (1.9/4) and Minkowski's inequality, we arrive at IIuIILn,(n-10) C

f

0 00

mesn(Mt)(n-1)/ndt,

2. Examples of "Bad" Domains in the Theory of Sobolev Spaces

122

where Mt = {x E SZ lu(x)I > t}. Note that the classical isoperimetric inequality (1.9/1) is valid for arbitrary Lebesgue-measurable sets G C R' :

with finite volume (see De Giorgi [49, 50], Burago and Zalgaller [32], Sec. 2.3.1,

3.3.3). An application of (1.9/1) to Mt gives IuIILn/(n-1)(o) < n-lvn

1/n

(f°°

s(E)dt +

J0

s({x E 8: Ju(x)I > t})dt f

/

: lu(x)I = t}. The former of the last two integrals equals IIVulll,o by Lemma 1.9.2, whereas the latter is

with Et = {x E 1

oo

J

dt

J

X[o,lu(x)I](t)dsx = f Iu(x)I dsx.

Thus, (3) is valid. That the constant in (3) is sharp follows from its sharpness for functions u E Co (92) (see Sec. 1.9). Inequality (3) implies the following assertion.

Corollary 1. Let 92 C Rn be an open set. If 1 < p < n, then there is a constant c > 0 depending only on p, n, such that (4)

IIVIILn,/(n-p)(B)
for all v E Wp p(n-1)/(n-p) (H) 892).

Proof. Let v be a function in Lp(92)nC°°(92)nC00(S2) with bounded support. lvlp(n-1)/(n-p) into (3), we obtain By inserting the function u = IIVIILq(I2) < cn

(rf vlr-llovldx+ fvlyds)

,

where q = np/(n - p), r = p(n - 1)/(n - p), cn = n-lvn 1/n. inequality yields 1/p / Ivr-llvvlI f IDvlpdx) (flvldx) \o o

f and

IIVIILq(n)

Cn/r (rl/nIIVIILq(n)/rIlVVIILp(o) + IIvlILr(8Sl))

Holder's

2.11. Counterexample to the Sharpened Friedrichs Inequality

123

An application of the parametric inequality a(''-1)/rb1/r < ea(r - 1)/r +

E1-rb/r, c > 0, a,b > 0,

leads to (4) with c independent of v and Q. 1 In a similar way, replacing u by Iulr in (3) and using Holder's inequality, we arrive at the next corollary.

Corollary 2. If I C R' is an open set with finite volume and 1 < p < n, the sharpened Friedrichs inequality (2) is valid for all u E WP r (1, 8I ), where

1 < r < p(n - 1)/(n - p) and q = rn/(n - 1).

1

We now check that the exponent q = rn/(n - 1) on the left of (2) cannot be improved unless I is subject to additional restrictions.

Example. Let I be the union of the semiball B- _ {x E R" : IxI < 1, xn < 0}, the sequence of balls B(k), k = 1, 2, ..., and thin pipes Sk connecting B(k) with B- (see Fig. 15). Let Pk be the radius of B(k) and hk the height of Sk. We denote by Uk a continuous piecewise linear function equal to unity on B(k)

and to zero outside B(k) U Sk. Suppose there exists a constant C > 0 such that

Fig. 15

IIkk!ILq(S1) < C (IIoukIIL,(S2) + IIukIILr(8S2))

for all Uk. Hence

[mes,, (B(k))]1/e < C (hk 1[mes"(Sk)]hi' + [s(aB(k) U ask)]llr)

The first summand on the right can be made arbitrarily small by diminishing O(ekn-1)/r) the width of Sk so that 0k19 = and q < rn/(n - 1).

124

2. Examples of "Bad" Domains in the Theory of Sobolev Spaces

2.12. Planar Domains in EVp Which are not Quasidisks Let S2 C R2 be a domain bounded by a quasicircle, i.e., by a Jordan curve such that ix - zI < constIx - yl, (1)

where x, y are arbitrary points in 8S2 and z an arbitrary point in that are of 8S2 connecting x to y whose diameter is smaller. We pointed out in the comments to Sec. 1.6.2 that such domains (quasidisks) belong to the class EVp

for all l = 1, 2.... and p E [1, oo). Moreover, quasidisks are the only planar simply connected domains contained in EV21. We may ask whether the class of Jordan curves bounding domains in EVP, p # 2, is exhausted by quasicircles.

According to a theorem due to Gol'dshtein [77], the simultaneous inclusion of a planar simply connected domain S2 and the domain R2 \ Sz in EVP implies that 1X is a quasicircle. So this result suggests an affirmative answer to the above question. However, we show that in general the answer is negative.

Theorem. There exists a domain S2 C R2 with compact closure and Jordan boundary such that: (a) 8S2 is not a quasicircle ; ()Q) 8S2 has finite length and is a Lipschitz graph in a neighborhood of all but one of its points; (ry) S2 is in EVP for p E [1, 2);

(6) R2\S2 is in EVp fort 0} is the image of

the domain G (see Fig. 17) under the transformation G D x = (x1, x2) H 4Dx = (x1, x2 + x1)

(2)

Here G is the difference between the rectangle R = (0, 1) x (0, 1/3) and the union T of the sequence of isosceles right triangles {tk}k>o. The hypotenuse of tk is the segment [2-k-1, 2-k] Furthermore, we assume that upper and lower "teeth" are symmetric with respect to the xl-axis. Positive constants c appearing below in this section may depend only on p. We need two auxiliary assertions to prove statement ('y).

2.12. Planar Domains in EVy Which are not Quasidisks

125

x1

Fig. 16

Lemma 1. Let S be a sector defined in polar coordinates by the inequalities

0 2. Then Ilu/rllp,s < c (IlVullp,s + a

1Ilullp,s), p c [1, 2),

and Ilu/rllp,s < c liVullp,s, p > 2.

These estimates follow from Hardy's inequality (1.1.2/7).

Lemma 2. There exists a linear continuous extension operator

E1:VP(G) -.VP(R), 1
Proof. Since G E C°'1, there exists a bounded extension operator E VP (G) -* Vp (R). Let X2 = W(x1) be the equation of the broken line which

2. Examples of "Bad" Domains in the Theory of Sobolev Spaces

126

consists of the legs of the triangles tk, k = 0, 1, .... We introduce a cut-off function 77 on R by

1 ifxEG, r7(x) _

x2/co(xl) if x E R \ G.

Clearly Iv77I < C (dk 1 + dk+1)

(3)

on tk, where dk(x) is the distance from x to the point x1 = 2-k, x2 = 0. The required extension operator can be defined by

Vp (G) 3 u' E1u = r?Eu. To prove this, one needs to verify the inequality II V (71v) IIP,T < C II V II V; (R), V E VP (R).

(4)

We have II VvIIP,T + IIvo77IIP,T.

II V (hlv) IIP,T

In view of (3) IlVV7lII T < c

(Ildk 1vIIP k>O

tk + Ildk+1vIIP,tk ),

where tk and tk are the right and left halves of tk. By Lemma 1 Iidk 1vlip,tk < c (IiVvllp,tk + 2kIIvIlp,tk ).

The same estimate holds for Iidk+1vllp,tk . Therefore IIvonIlp,T < C (IIVVIIP,T + II IXI-1tIIP,T).

An application of Lemma 1 yields II

Ixl-1vIIp,R < C (IIvIIP,R+ IIVVIIP,R)

Thus, (4) is valid and Lemma 2 is proved.

D

2.12. Planar Domains in EVP Which are not Quasidisks

Now let u E V P (SZ),

127

1 < p < 2, and let Q+ = {x E Q : x1, x2 > 0}.

Since G can be mapped onto SZ+ with the aid of transformation (2), w = (El(u o

o (P -i is the space-preserving extension of u from SZ+ onto

4tR = {x : x1 E (0,1),x2 E (xl, xl + 1/3)).

Furthermore, w(x) = 0 almost everywhere on the line x2 = x1. Applying the same reasoning to {x E S2 : x1 > 0,x2 < 0}, we can construct a spacepreserving extension of u onto the domain

Q U {x : xl E (0,1), Ix2I < 1/3 + xi },

(5)

the extension being zero on the set {x : xl E (0, 1), Ix2I < xi}. Since domain (5) is in Co,', reference to Theorem 1.6.2 completes the proof of statement (y) of the theorem. Let us now turn to statement (8).

Lemma 3. Let D be a bounded planar domain starshaped with respect to the disk B6. If p > 2, then sup Iu(x) - u(S)I !5 ca-1d2-2/PIIVuIIp,D

(6)

x,{ED

for all u E VP (D) n C(D), where d is the diameter of D.

Proof. We observe that by Theorem 1.5.1/1 sup I mi(x) - uW I

x,{ED

V u(y) I y I

supD !D

-Fx --Y I

Estimate (6) follows by using Holder's inequality.

1

Let T = Uk °tk (cf. Fig. 17). We introduce the space VP (T), p > 2, which consists of functions u E VP (T) satisfying the condition: the limit values of u out of the triangles tk and tk+1 coincide in their common vertex for k = 0, 1, ... (note that VP(tk) C C(tk) by Sobolev's theorem). The space VP (T) is endowed with the norm of VP '(T). The following assertion implies statement (8) and concludes the proof of the theorem.

Lemma 4. Let A = {x : x1 E (0, 1), 0 < x2 < xl/3}. There exists a linear continuous extension operator

E2:VP(T)-4 Vp'(A), 2
2. Examples of "Bad" Domains in the Theory of Sobolev Spaces

128

Proof. Consider the rectangle Q = (0, a) x (0, b) with vertices 0(0, 0), A(a, 0), B(0, b), C(a, b).

Let w E VP (OAC) and w(O) = 0. We shall construct a linear extension operator w F-+ f E VP (Q) such that f (0, y) = 0 for y E (0, b) and IIWIIL,o(OAC),

llfIIL-(Q)

IIVIIILD(Q) <_ c(p,a/b)IIVwIIL9(OAC)

It is sufficient to assume that Q is a square. Let w be the even extension of w across the diagonal OC to the triangle OBC. By A we denote a smooth function of the polar angle such that 0 < A < 1, A(O) = 1 for 0 < 7r/4 and A(7r/2) = 0. Since ti (O) = 0, Lemma 1 implies dl V IIP,Q,

Ir-1w1IP,Q

where r is the distance to the point O. So f = aw is the required extension of w.

Let Qk, k = 0, 1.... denote the components of the set A \ T (cf. Fig. 17). Using the described procedure, we can construct an extension vk of u E VP (T) to the triangle Qk such that Vk(2-k, y) = u(2-k, 0) and Ilvklioo,ak <-

I

llVvkHHP,ak 5 cll

where k = 1 , 2, ...

.

IUIloo,tkUtk_1,

Vullp,tkutk_1,

For k = 0, one can construct an extension vo of u

satisfying similar inequalities with tk U tk_1 replaced by to. We define an extension u F-+ v to A by v = u on T, v = vk on ak. Clearly Ilovkllp,A + IItIloo,n <_ C (IIVUIIP,T + IILIIoo,T).

(7)

An application of estimate (6) to each triangle tk yields IIulIm,T <_ C (IIVILIIP,T + IlulIm,to).

Thus, the right part in (7) is equivalent to the norm in VP (T). This completes the proof of Lemma 4. 1

2.13. Counterexample to the Strong Capacitary Inequality ...

129

According to Gol'dshtein's result [77] mentioned at the beginning of the present section, we can refine statements (o!) - (8) of the theorem. The domain SI in Fig. 16 does not belong to EVp for p > 2 and its exterior does not belong to EVP for p E [1, 2].

2.13. Counterexample to the Strong Capacitary Inequality for the Norm in L2 (fl) Let SZ be a domain in R" and F a relatively closed subset of Q. By the capacity of F generated by the norm in LP(0) we mean the set function cap (F; Lp(sz)) = inf{IIUIILD(Q) : u E C°°(SZ), uIF > 1}.

(1)

Similar capacities are frequently used in potential theory, partial differential equations, theory of function spaces, probability etc. (see Maz'ya [136], Heinonen, Kilpelainen and Martio [93], Adams and Hedberg [3]).

Here is, for example, a simple application of capacity to the integral inequality C IIUIIL;(cl),

C = const,

(2)

where p E (1, oc), p is a Borel measure on H and u c C0(1) an arbitrary function. Take, as before, any relatively closed subset F of 0 and any function u from the above definition of cap (F; L,(1)). Then (1) implies

µ(F) < D cap (F; LP(1l)),

D = const.

(3)

Thus, the capacitary isoperimetric inequality proves to be a necessary condition for inequality (2). It is easy to show that condition (3) is also sufficient for (2) provided a so called strong capacitary inequality 00

cap ({x E 0 : Iu(x)I > t}; L,(1))d(tP) < c II fill LP(s) has been established for all u E CO° (cl) . Indeed, since I u(x) I P =

J

"O XN, (x)d(tP),

(4)

2. Examples of "Bad" Domains in the Theory of Sobolev Spaces

130

where

Nt = {x E SZ : Iu(x) I > t},

(5)

Fubini's theorem gives fn I u(x) IPdp =

f(Nt)d(t").

(6)

By applying (3) and (4), one arrives at (2) with CP = c D, c being the same as in (4). The role of the strong capacitary inequality (4) is analogous to that played by the coarea formula in Lemma 1.9/2 for p = 1 (cf. the proof of Theorem 1.9). We now remark that (4) is really true for l = 1, 2, .. ., p E (1, oo) and "nice"

domains S2, say, Q E Co". In this case (4) is a consequence of extension Theorem 1.6.2 and the validity of a strong capacitary inequality for the norm in Vp(R"), see e.g. Maz'ya [136, 8.2.3], Adams and Hedberg [3, 7.1].

It turns out that if l = 1 and p E [1, oo), inequality (4) holds for all u E C°° (1) without restrictions on Q.

Theorem. Let S2 be a domain in R'. If U E C°° (1) n L, (1), 1 < p < oo, then (4) is valid with l = 1 and c depending only on p.

Proof. We fix a set w CC SZ in definition (1.5.3/1). Let Nt be given by (5). Since cap (Nt; Lp(1k)) is a nonincreasing function of t (which follows from the definition of capacity (1)), the left side in (4) does not exceed 00

S=(2P-1) E 2P"cap(N2;;L1(1)). j=-00

Given any e E (0, 1), let A. be a function in C0(Rl) such that 0 < aE < 1, 0 < AE < 1 + c, ) = 0 in a neighborhood of (-oo, 0] and AE = 1 in a neighborhood of [1, oo). Putting ui(x) = A,(2'-'Iu(x)I - 1),

we observe that u3 E C°° (92), uj (x) = 1 for x E N2; , supp uj C N2, -1 . Hence

S < 2p-1(2P - 1)(S1 + S2), where OK)

S1 = > 2P3 3=-00

f

N27-1\N2i

IVujIPdx,

2.13. Counterexample to the Strong Capacitary Inequality ... 00

f

S2 = E 2P1 nN2, j=-CO

131

I ujI Pdx.

(7)

Clearly IDujI < (1+e)21-iIVu1, and CK)

Si < C

I

IVulPdx = c IlouIIP

=-0o N2i-1\N2i with c = (1 + e)P2P.

Turning to estimation of the sum in (7), we note that luj I < 1 and that the t H mesn(w n Nt) is nonincreasing. Therefore, the general term of the sum in (7) is not greater than function (0, oo)

Zi -1

2P(1 - 2-P)-1

(w n Nt)d(tP). f-mes, ' 2

Thu s 00

2-P(1 - 2-P)S2 < fo

mesn(w n Nt)d(tP) =

f

,

Here formula (6) with µ = mesn I has been used at the last step. By letting e tend to zero, one arrives at (4) with 1 = 1 and c = 23P-1. 1

In this section we show that the capacitary inequality (4) for l > 1 fails unless some restrictions on SZ are imposed. We describe a bounded domain Q C R2 and a Borel measure p on Q such that there is no constant C > 0 for which inequality (2) with p = l = 2 holds for all u E C°°(Q) in spite of the fact that the estimate µ(F) < const cap (F; L2 2(Q))

(8)

is true for all sets F C f closed in Q. According to what has been said at the beginning of the section, the capacitary inequality in L2(SZ) fails for the same domain. It should be noted that the validity of (2) for all u E C°°(Q) is equivalent to its validity for all u E C(Sl) n L,(1) (cf. Remark 1.4.1/2). Before we proceed to the construction of SZ, we prove two auxiliary assertions. Below in this section c designates various absolute positive constants.

Lemma 1. Let

Te={(x,y)ER2:IxI
2. Examples of "Bad" Domains in the Theory of Sobolev Spaces

132

and let uE W2(T6), where 0 < 2E < 6 < 1. Then IIvuII2,T< < CE I log

Ell/2(o-2

IuII2,T6 +

IIV2uII2,T6)

Proof. It will suffice to consider the case b = 1. In this case the required estimate is a consequence of the inequality IIv1I2,T < CEI log E W2 (Ti), v(x, y) = 0 for r = (x2 + y2)1/2 > 1. Passing to the polar coordinates (x, y) = (r, 0), we observe that

\2 v(r, 0)2

=

Y

v0(e, 8)do l

/

< I logrl

v"(,0,0)2 odp

if r E (0, 1), 0 E (7r/4, 37r/4). Thus, 3or/4

IIvII2,T < f/4 do] n

of

1

rllogrldrv(,0,0)2ede

0

fo

and the result follows.

1

Lemma 2. Let 0 < 2E < 6 < 1 and let u E W2(T6). Suppose that II V2uII2,T6 < 1,

IIull-,T <- 1,

(9)

IIVuII2,T. < El logEI1/2.

(10)

Then for all (x, y) E T6 I u(x, y)) < c (1 + yI

logell/2).

Proof. There exists a linear function £(x, y) = ax + by + d such that the generalized Poincare inequality 6-2 11U

- £II2,T6 + 8-1IIV(u - e)II2,T6 <- c IIV2uII2,T6

is valid. Therefore, by the Sobolev imbedding and Lemma 1 IIu-tII"'CiT6
2.13. Counterexample to the Strong Capacitary Inequality ...

133

Ilo(u - e)II2,T< < CEI logE1112.

In view of (9), (10), we obtain 11111.,T. <- c,

IIV II2,T < CEI logEl1/2.

In other words, Ial + IbI < c I log_I1/2, Idl < c.

By using (11), we arrive at the desired estimate thus concluding the proof. I }y

Fig. 18

We now turn to the counterexample. Let {a i }i> 1 be a sequence of open nonoverlapping intervals of length 21-i lying in interval (0,4) of the real axis. For definiteness, Qi are supposed to concentrate near the origin, i.e., for any small e > 0, interval (E, 4) contains only a finite number of Ui. By {Di}i>1 we denote the sequence of open right isosceles triangles situated over the axis Ox with hypotenuses u j. 8

21-`

Fig. 19

Let bi be the triangle symmetric to Ai with respect to the line y = 2-1 Ei E (0, 2-i) being specified below. Furthermore, let R = [-1, 4] x [-1, 0]. We

2. Examples of "Bad" Domains in the Theory of Sobolev Spaces

134

define S2 to be the interior of the union of R and all the triangles Ai and 8i (see Fig. 18, 19). To introduce the measure p, we use the points

Aij=('Yi,2-'-ei+23h;), j=1,2,...,2'; i=1,2,..., where yi is the abscissa of the middle point of vi, hi = 11092 Ei I-1/2,

loge 11092 Ei i = 2i+2

Let F be an arbitrary Borel subset of S2 and XF its characteristic function. The required measure p is defined by 2'

IL(F) =

E

2-2j-i/2XF(Ai,j)

i>1 j=1

Let us verify (8). To be more specific, we introduce the norm in L2(1) by IIUIIL22

(sl) = IHV2UIIL2(s1) + IIUIIL2((-1,4)x(-1,0))

(cf. Corollaries 1.5.2 and 1.5.3/2). We deduce (8) by using the estimate cap ({Aij}; L2(f2)) > c2 -2j, j = 1, 2, ... , 2',

(12)

which will be checked later. Let F be a relatively closed subset of SZ and let

m(F) be the minimum value of j such that there exists a point Aij in the set F. Put m(F) = oo if there are no points Aij in F. Then (12) and the definition of p imply

cap (F; L2(12)) > c 2-2m, p(F) < c 2-2m Hence p satisfies (8). Inequality (12) is a consequence of the estimate

Iu(Aij)I < c2-,

(13)

where u is an arbitrary function in L2(11) normalized by IIUIIL2(Q) = 1. Let G

be the interior of the union R U°_1 L. Clearly, G C Sl and G possesses the

Exercises for Chapter 2

135

cone property. By Lemma 1 and the Sobolev imbedding L2 (G) c L. (G), we have IIVuII2,ti < ceil logei1112, IIuIIoo,ti <- c,

where ti = ai fl 0i. Next, Lemma 2 applied to the triangles ai and bi fl { (x, y)

y < 2-' - ei} yields Iu(x,y)I <-c(1+Iy-2-t+2eij Iloge, 1112) for all (x, y) E bi. Since the ordinate of the point Ai, equals 2-i - ei + 2'hi, (13) follows.

We now define a function f such that f E Lz (1Z) n C(c), f V L2 (0, A)

Put f (x, y) = 0 for y < 0 and f (x, y) = 2 sI4hi77(2ty)(y + ei - 2-') 1092 (2-' - y)

on the set

Ti={(x,y)EAi: y<2-t-ei}, where 7 E C°°(0, oo), i7(t) = 0 fort < 1/4, y(t) = 1 fort > 1/2. The linear extension of this function f to the set {(x, y) E bi, y > 2-i - ei} will be also denoted by f, i.e.

f (x, y) = -2-i/4hi 1(y -

2-t

+ ei)

for (x, y) E bi, y > 2-i - ei. Clearly supp V2 f is placed in

the

Furthermore, the estimate

IV2fI S c 2-t/4hi max{2ti, (2-t - y)-1} holds for (x, y) E Di. Therefore IIV2fIIL2(n) =

IIV2fIIL2(oi) i>1

< c E2 -i/2 h? (22 i>1

2-i/2 < oo,

+ 11092 ei I) < c i>1

set Ui>1Ti.

2. Examples of "Bad" Domains in the Theory of Sobolev Spaces

136

and f E L2(S2). At the same time 22'

IIf

IIL2(fl,/U)

E 2-i12-2jf (Aij)2 = i>1 1 =

i>1 j=1

Exercises for Chapter 2 2.1. Let S2 be the domain in Fig. 5 (see Sec. 2.1). Prove that the space of functions in C°° (S2) with bounded gradients of all orders is dense in n (Q) with l = 1, 2, ... and p E [1, 00).

2.2. Let ] be the planar domain presented in Fig. 7-8 (see Sec. 2.3) with 1)2-2i-2, i > 0, while hi > 0 satisfy (2.3/2) for i > 0. Show that the bi = (i + set Lq(S2) n L2 (S2) is not dense in Wp (S2) for 1 < p < 2 and q > 2.

Hint. Check that the function f constructed in Sec. 2.3 is in W2 2(Q). Next proceed as in Sec. 2.3.

2.3. Let S2 be the domain in Fig. 12 (see Sec. 2.7.1). Prove that if LP(Q) is continuously imbedded into Lq(Q) with q < p, then this imbedding is compact.

Hint. Let ON be given by (2.7.4/1). By using (2.7.3/5) (and the notation in Proposition 2.7.3), show that C IIUIIq,cz <

IIUIIq,cN +

SN(IIVIUIIP,c + IIUIIP,D)

for all N > 0 and arbitrary u E L,(Q) with SN =

\

[ apl (P-q)Ek/(q-P))

11q-11P

k> N

2.4. Let S2N be the same as in the preceding exercise. Show that the best constant CN in the inequality inf{IIu - PIIq,cIN : P E 7I-1} < CN is equivalent to max {81/qek 1/p}

O
IIVIUIIP,ON,

u E LP(QN),

Comments to Chapter 2

137

if q 2 p, 1 - 2/p + 2/q > 0 (q < oo for p = 2, 1 = 1), and if q < p, CN is equivalent to /1 1 + E N g1(p-q)Ek1(q-p) 11q 1/p l

k=0

2.5. Let P, Pk Sk be the same as in Fig. 13-14 (cf. Sec. 2.8). Put

01=P, QN+1=IZNUSNUPN, N>1. Let CN denote the best constant in the inequality inf{llu - PII2,stN : P E P1-1} < CN JPVIUII2,ON,

Prove that the expression by positive constants independent of N.

CN1(1+2-N(2t+1-a)/2)

u E L'2(1N)-

is bounded above and below

2.6. Let S2 be the domain in Fig. 13-14 (see Sec. 2.8). Show that if VIA _ L2(0) for some l > 1, then this equality holds for all 1 = 1, 2, ... . Hint. Prove that the above equality with some l > 1 implies a < 3 and use Proposition 2.10.

2.7. Let SZ be the same as in the preceding exercise. Prove that if 1 < a < 2, then W22(9) is a Banach algebra. Hint. Check that Lz(S2) C L4(Q) and L2(S2) C Lm(S2) for these a. Next use Exercise 1.19.

2.8. Show that domain SZ in Fig. 5 (see Sec. 2.1) is in EVE for p E [1, 2) and Il EVE forp> 2. Hint. Use the extension procedure described in Lemma 2.12/4.

2.9. Prove that domain f in Fig. 16 (see Sec. 2.12) does not belong to EVp for p E (1, 2). Hint. Verify that VP 2(Q) 0 C2-21p(S2).

Comments to Chapter 2 2.1. Example 2.1/1 is due to Kolsrud [109]. Earlier Amick [8] constructed an example of a bounded domain Q C Rn satisfying a1 = asp and such that VP (52) fl C(Q) is not dense in VP (Q) if pl > 2. Example 2.1/2 is analogous to that of Amick for n = 2.

138

2. Examples of "Bad" Domains in the Theory of Sobolev Spaces

2.2. In connection with the contents of Sec. 2.2, we mention the following result. Let Cb (SZ) denote the set of functions in Cm (Q) with bounded gradients of all orders. If fZ C R2 is a bounded domain which is either weakly starshaped with respect to a point or satisfies the interior segment condition (see the comments to Sec. 1.4), then Cb (f2) is dense in VP (SZ) for p E [1, oo) and l = 1, 2, .... This theorem is due to Smith, Stanoyevitch and Stegenga [185]. In particular, it gives sufficient conditions for the space V1 (0) n L. (Q) to be dense in V1 (Q), fZ C R2. The result just mentioned fails for multidimensional domains. Example 2.2 is analogous to Example 7.1 in the paper by Smith, Stanoyevitch and Stegenga [185] which was constructed to show that the space Cb (fZ) is not always dense in WP '(Q).

2.3-2.4. In Sec. 2.3 and 2.4 we follow the paper by Maz'ya and Netrusov [141].

2.5. The domain in Fig. 10 of "rooms and passages" type was used by Fraenkel [63] to demonstrate the failure of some properties of Sobolev spaces for domains with unrestricted boundaries.

2.6. Example 2.6 is taken from the paper by Maz'ya [131] (see also Sec. 5.5.3 in the book by the same author [136]), where the continuity of the imbedding operator: WP (1l) -4 Lm (fl) fl C(SZ) was justified by a different argument.

2.7. The domain described in Sec. 2.7.1 with parameters given by (2.7.1/1) (cf. Fig. 10) was used by Nikodym [169] to show that generally L'(SZ) ¢ L2(S2).

2.8. The counterexample in Sec. 2.8 is taken from the paper by Maz'ya and Netrusov [141]. The domain in Fig. 13 resembles that in Courant and Hilbert [47, p. 521].

2.9. Another example of a domain for which W2 2(Q) # V2 2(Q) can be found in the book by Maz'ya [136] (Sec. 1.1.4).

2.10. For l = 1, Proposition 2.10 was proved in Sec. 4.10.3 of the book by Maz'ya [136].

A multi-dimensional analog of the domain in Fig. 13 (resembling that in Fig. 15) was considered by Berger [18] who obtained necessary and sufficient conditions for the continuity and compactness of the imbedding LZ C L2 for such a domain. We observe that domain SZ in Fig. 13-14 is a-John (cf. the definition in the comments to Sec. 1.5.2). So that the conditions for the continuity of the

Comments to Chapter 2

139

imbedding Lp(ul) C Lq(I) from the paper by Hajlasz and Koskela [85] apply (cf. the comments to Sec. 1.12.1). Corollary 5 and Theorem 5 from this paper is compactly imbedded into _L2(Q). give: if a < 3, then

2.11. The contents of Sec. 2.11 can be found in the book by Maz'ya [136] (Sec. 3.6, 4.11).

Much attention has been paid to Friedrichs type inequalities for functions

with zero boundary conditions. For example, in the case lp < n, p < q < np/(n - lp), the criterion for the validity of the estimate IIUIiLq(n) <_ constllVluIIL,(n),

u E Co (SI),

is the existence of a number r > 0 such that inf cap (Br.(x) \ SZ; L, (R")) > 0,

xER^

see Maz'ya [136, 11.4] (cf. also Carlsson and Maz'ya [42]). Here the capacity cap (F; LP(G)) of a compact subset F of an open set G C R" is defined by cap (F; LIP" (G)) = inf { II VivIILP(n) : v E Co (G), VI F > 1 } .

Capacitary isoperimetric conditions for the validity of the inequality II u II Lq (n,µ)

< const

Jn

f (x, u, V u)dx,

u E Co (S2), q> p,

where µ is a Borel measure on SI and Ii a function satisfying certain assumptions, were obtained by Maz'ya [136, 2.3.2]. In particular, the Hardy type inequality

f

(1H(!flPdx
J

na(xn

l

DuI Pdx

bO x_dist(x 8SZ),

O1

holds for all u E Co (SZ) if and only if dx

< const cap (F; LP0))

iF b(x)P

for all compact F C Q. Conditions for the validity of (1) and its generalizations were also studied in the works by Ancona [9], Lewis [118], Wannebo [214],

140

2. Examples of "Bad" Domains in the Theory of Sobolev Spaces

Kufner and Opic [113], Nystrom [172], Mikkonen [157], Cianchi, Edmunds and Gurka [45], and others. For example, Lewis [118] proved that (1) holds for all u E Co (Ii) with p E (1, n) if the set Rn \ SZ is uniformly fat in the sense that

inf

{rP-n cap (R,-

xER^,r>O

r(x)

\ c; LP(Rn)) } > 0.

The inequality IIUIIP,Bi <_ C (IIVuIIP,B + IIV1utIP,Bj,

(2)

where C = const, p c (1, oo) and u a function on B1 often turns out to be useful for various applications. The following result is due to Maz'ya.b

Let F be a compact subset of B1 and let u be an arbitrary function in C0(B1) satisfying dist(suppu,F) > 0. Then (2) is valid with C = c [cap (F; L1P(B2))]-1/P, c = c (n, p, l) > 0. Conversely, assume that 0 < cap (F; L1P(B2)) < -y, where -y = y(n, p, 1) is a sufficiently small constant. If (2) holds for all u E C°° (B1) with dist(supp u, F) > 0, then the best constant C in (2) admits the estimate C > c [cap (F; L1P(B2))]-1/P. When cap (F; LP(B2)) > -y, one has to formulate two-sided estimates for C in terms of the so-called (p, l)-inner diameter of the set B1 \ F (see Maz'ya [136, 10.2]). Conditions for the validity of the inequality IIUIIP,Bi

C (II 7ktIIP,B + IIVZUIIP,Bj,

1 < k < 1,

(3)

can be found in the books by Maz'ya [136], Chap. 10, Ziemer [221], Chap. 4,

and by Adams and Hedberg [3], Chap. 8. A weighted version of (3) with weights satisfying the Muckenhoupt AP condition was studied by Turesson [203]. Estimates of the form (3) play an important role in the paper by Hedberg [89] (see also [3, Chap. 9]), where the problem of spectral synthesis in Sobolev spaces was solved. 2.12. Theorem 2.12 was proved by Maz'ya [136, 1.5.2].

2.13. Theorem 2.13 is due to Maz'ya [126, 130]. The counterexample to the strong capacitary inequality for the norm in L2 (S2) is taken from the paper by Maz'ya and Netrusov [141]. b V. G. Maz'ya, The Dirichlet problem for elliptic equations of arbitrary order in unbounded regions, Dokl. Acad. Nauk SSSR, 150 (1963) 1221-1224 (Russian). English translation: Soviet Math. Dokl. 4 (1963) 860-863.

Part II Sobolev Spaces for Domains

Depending on Parameters

CHAPTER3

EXTENSION OF FUNCTIONS DEFINED ON PARAMETER DEPENDENT DOMAINS

Introduction This chapter deals with extension operators acting in Sobolev spaces on domains depending on small or large parameters in such a way that the limit domains do not belong to EVP. We are concerned in the speed with which these operators degenerate when the parameters tend to their limit values. We begin the study with estimating norms of arbitrary extension operators E : VP (Ql :) -4 VP (Ge), F : VP (Go \ SlE) - V' (GQ); 1 < p < oo, 1 = 1, 2, ... .

Here o > 0, a is a small positive parameter,

1 _{ex:xE1l}, Ge=fox: xEG}, SZEeGe, SZ and G are bounded domains in R". The following results are obtained.

1. Let R" \ S2 be in EVP (cf. 1.6.2) and let dist(Q , 8Ge) > ce, where c = const > 0. Then there is a linear extension operator F with norm uniformly bounded in e, g. 2. If SZ E EVP, the relation holds e-"/P min{Qn1 inf IIEII

e -' min{et,

,

e"/p-I}

I

e-"/p min{e"/p, 1}

if 1p < n,

logel('-r)/n} if lp

= n,

if lp > n.

The symbol - designates the equivalence uniform with respect to e, e.

Let VP(GQ) be the closure of the space Co (GQ) in VP(R") and let Eo VP (Q2) - VP (Ge) denote an arbitrary extension operator provided sz E EVP. 143

3. Extension of Functions Defined on Parameter Dependent Domains

144

Suppose, furthermore, that dist (SZE, 9Ge) > cE, c = const > 0. The following relation is established in Sec. 3.2: E-1 if pi < n, inf IIE0II '

E-1 max{ (log(1

+

[JE-1))-1+1/p,

I loge1-1+l/p} if pl = n,

E-n/p max{60-1+n/p, 1} if pi > n.

In Sec. 3.3 we construct an extension operator E : V1 (Q,) -+ VP (Rn) with

the least possible norm. In particular, it is shown that for p = 2, 1 = 1 and n > 3 any extension operator E satisfies JJEJJ

> _

(sn(n_2)caPII)h/2 mes( )

1

E

and there exists a linear extension operator E such that EII < _

(Sn(fl - 2)cap1/2 1+0(1) mesn (S2)

E

Here sn is the area of the sphere Sn-1, cap is the Wiener capacity in Rn and 0(1) a positive infinitesimal as E -+ +0. Analogous results are proved for extension operators to the exterior or

interior of a thin cylinder. Put

c ={(y,z)ERn+8:y/eEwCRn, zERe},

Ge={(y,z)ER+8:y/pegCR', zERe}, where w and g are bounded domains, w E C°'1. Let c be a small positive parameter, a, c G. and let E : VP '(Q,) - VP (GQ),

F : VP (Ge \ QE) --> Vp (GP)

be arbitrary extension operators. Then the above assertions 1 and 2 are valid.

The asymptotics for the norm (as e -4 +0) of the best extension operator: Vp (Q,) -- VP (Rn+s) is obtained for (l - 1)p < n. A combination of results mentioned above enables us to estimate the norms of extension operators for domains of complicated configurations. Some examples of domains depending on small parameters and the estimates for the

3.1. Estimates for the Norm of an Extension Operator ...

145

norms of corresponding extension operators are given in the last section of Chapter 3. Theorems on small domains and narrow cylinders will be used in Chapter 5 to construct extensions of functions from domains with cusps.

3.1. Estimates for the Norm of an Extension Operator to the Exterior and Interior of a Small Domain In this section we obtain two-sided estimates for the norms of extension operators: VV (QZ) - VP (Ge), VP (Ge \ QE) -* VP (GO).

Here a is a small positive parameter, o > 0, Ti, C Ge, Q, = {ex : x E I}, Ge = {ox: x E G}, S2 and G are bounded domains in R. The symbols c, co, cl,... denote positive constants depending only on n, p,1, S2, G. The equivalence a " b of positive quantities a, b is meant in the sense that co < a/b < c1. Such quantities are also called comparable. If X is a set in R" and A E R1, then AX = {Ax: x E X}. For brevity, we write II IIp,1,G -

instead of II

- IIV, (G)

3.1.1. Generalized Poincare Inequality for Domains in EVp We begin with a version of Theorem 1.5.4 for the space VP (1l) where Q is the finite sum of domains in EVp (cf. Definition 1.6.2).

Lemma. Let SZ be a domain in R" which is the union of a finite number of bounded domains in EVp. If F(u) is a continuous seminorm in Vp(S2), such that F(P) # 0 for any nonzero polynomial P E Pi_1i then the norm in VP (S2) is equivalent to the norm F(u) + I VzuI ,,c . Proof. We need to verify the inequality IjuIIp,j,n < c (F(u) + IjVluII,i)

(1)

for any u E Vp(1). If (1) is not true, there is a sequence {uk}k>1 C VP(1l) such that IIukHHp,I,s1 = 1 and

F(uk) + IIVtUkIIp,n < 1/k, k = 1, 2, ....

(2)

By Lemma 1.10, there exists a subsequence of {uk} (which we relabel as {uk}) convergent in Vp-'(1). Let u be the limit of Uk in Vp-1(S2). By (2) Uk -* U

146

3. Extension of Functions Defined on Parameter Dependent Domains

in VP (1) and u E PI-1. In view of the continuity of F, we have F(u) = 0 and hence u = 0. However, this contradicts the condition IukIIP,1,c = 1. The proof of the lemma is complete. 1 This lemma implies the following assertion.

Corollary. Let SZ be the same domain as in Lemma, Q, = e St, E E (0, oo), and let Pe E PI-1 be the polynomial defined by (1.5.4/6).Then for any u c VP (Q,,) the inequality IIV8(n - PE)IIP,i

C CE '-'11V1U11P'nE

(3)

holds with s = 0, 1, ... , l - 1. Proof. It will suffice to consider the case e = 1. Since the mapping u H P1 is a continuous projector of VP (SZ) onto Pi-1 (cf. Corollary 1.5.4), we may put F(u) = IIP1IIP,s1 in Lemma 1. The result follows. 3.1.2. An Extension from a Small Domain to Another One

First we state a simple assertion on extension with dilation which will be frequently used in the sequel.

Lemma 1. Let .Q C Rn be a domain of class EVE' for some l = 1, 2.... and some p E [1, oo]. If ci = E SZ, E E (0, co), then there exists a linear extension operator

EE : VP(1l) - V'(Rn) such that the estimate t

IIVj(EEU)IIP,Rn < C

EEk-7IokuhIPA

(1)

k=0

is valid f o r any u E V P (cie) and j = 0,1, ... ,1.

Proof. By Definition 1.6.2, there exists a linear extension operator E : VE' (SZ) -* VP (Rn). It is easily checked that the required extension operator can be defined by

EEu = (E(u o 4D)) o 0-1 with 4i : Rn E) x H ex.

(2)

3.1. Estimates for the Norm of an Extension Operator ...

147

The constant c in (1) is the same as in the inequality IIVJEvIIP,Rn


0

Remark. Suppose 11 satisfies the hypotheses of Theorem 1.6.2 and let E be Stein's extension operator from Q onto R. Since E is a bounded extension operator: VP (Q) estimate holds

VP (R") for any nonnegative integer j, the following

II Vj (Ee-u) IIP < c(j, p, n, 1) E

Ek-i

j = 0, 1, ... ,

II V kuII P,Q,,

k=O

where E. is given by (2). We now prove one more auxiliary assertion to be used later.

1

Lemma 2. Let G be a domain in R" with compact closure and SZ its subdomain of class EVp. For any e E (0, 1), there exists a linear extension operator: V7 (e 1) -+ VP '(e G) with norm uniformly bounded in e.

Proof. We put SZe = E 0, GE = e G. Let EE : VP '(Q,) -> VP (R") denote an extension operator with dilation subject to (1) and let

VP(Qe)DuHPeEP(-1 be a linear mapping satisfying (3.1.1/3). Put

Eu=Pe+Ee(u-Pe), uEVp(Qe). Clearly, E is an extension operator: VP (1e) -+ VP (Ge). By (3.1.1/3) and (1) IlokE.(u - Pe)IIP,Rn < C E'- c UViuIIP,ci

,

0 < k < 1.

Next, by using the estimate IIQIIP,Ge

C

IIQIIP,ne, Q E P!-1,

we obtain IlokPeIIP,GE 5 C II VkPell P,oE

< C1 (IlokuIIP,flE +Ei-k IViUIIP,nE) , 0 < k < 1.

148

3. Extension of Functions Defined on Parameter Dependent Domains

Thus E is the required extension operator. 1 Note that if 0 E C°"1, the conclusion of the preceding lemma is true for any e E (0, oo). Indeed, the existence of an extension operator l; : VP (SZE) -4 Vp (Ge), E > 1,

with 11611 < c follows from the above remark.

3.1.3. The Interior of a Small Domain

Let S and G be bounded domains in Rn(n > 2). Suppose that G contains the origin. Put SZE = E 0, Ge = oG, where E E (0,1/2) and o > 0. The case o = oo is also taken into consideration and then Gm = R. We call SZf a small domain.

Theorem. Let 0 be simply connected and Rn \ S E EVp for some p E [1, CO] and some positive integer 1. Suppose Q, is a small domain such that StE C G. with dist (SZE, 8Ge) > c°e. Then there is a linear extension operator: VP (Ge \ 1 e)

VP (Ge)

whose norm is uniformly bounded in e, o.

Proof. By assumptions, dist (Sl, BGeE-l) > co. Hence, there is a domain g c R" with compact closure and smooth boundary such that SZ C g, g C GeE-1 and g does not depend on the parameters e, o. Note also that g \ S2 is in EVP. Put gE = eg. Since Sle C ge C G. (cf. Fig 20), it is sufficient to construct a linear extension operator: Vp (gE \ 0E) -+ VP (g-)

with norm uniformly bounded in E. Reference to Lemma 3.1.2/2 completes the proof.

Fig. 20

3.1. Estimates for the Norm of an Extension Operator ...

149

3.1.4. Inequalities for Functions Defined on a Ball

Let [ C R" be a bounded domain and put SZe = e Q. Suppose Q C Be. Here we establish some e, p-dependent estimates for functions in Lp(Be) which will be used in Sec. 3.1.5.

Lemma. Let Q. C Be. If u E L,(Be) and u(x) > 1 for almost all x E Sle, then

Elp-nlloluIlp,B,

+ -nIIuIIp,BQ >- C, lp < n,

(1)

and log(PE-1)IP-lIIOIU'IIP,BQ

+ P-nIInIIp,BQ > C, lp = n.

I

(2)

Proof. Put A = 2 sup{ Ixl : x E SZ}. If L< E max{A, 2}, then P-n II uII p,BQ > (max{A, 2}e) -n IIuIIp,S2. > c,

and estimates (1), (2) are valid.

Let P > E max{A, 2}. In this case Q. C Bae/2 C Bae C Be. We make use of Sobolev's integral representaton (1.5.1/3) in the ball Be. The polynomial in this representation can be taken in the form Q(x) =

lP

P-n

(

Ipl
P Cx)

(3)

0p(y/P)u(y)dy,

Be/2

where cp E Co (B112) are standard functions for III < l - 1. Integrating the inequality

c < IQ(x)IP+

I=l C

ID°u(y)Idyl Ix yIn-l P J, X E Qe,

J Be

-

we arrive at the estimate

f" Cen < IIQIIp,S2, + E1 fE dx (j I

I-

`

P.

aU()I

dy

I

Ix

.

)

An application of Holder's inequality to each integral in (3) yields IQ(x)Ip < CO-°IIUIIp,Bo/2, x E Be.

(4)

IN

:4

I;xtousion of Functions Defined on Parameter Dependent Domains

Hence IIQIIp,Q, <_ c1E"IIQIICO,BQ < C2(E/e)"IIUIIp,BQ/2.

(5)

To bound the sum in (4), we fix a multi-index a, j al = 1, put v = I Daul and separately estimate the integrals

I1 =

dx UXF

v(y)dy 1 P ix - yI "-`)

P

I2 = f dx (fnQ\BA. v(y)dy IX -

E

yl"-!)

We extend v(y) to be zero for Iyl > Ac in the interior integral of I. The change of variables y - x = z and application of Minkowski's inequality result in 1 < IIv( + z)IIL,(Rn) < C E'jIvIIP,Be (6)

J BZaa z I" I

'

To estimate 12, we first consider the case p > 1. Note that l y x c BILE, y c Be \

B,\,. Holder's inequality gives

- xj ? I yI /2 for

p-1

I2 < C E"IIVIIp,BQ (LQ\BA.

IyI('-n)P/(P-1)dy

and hence

cE1PIlvllp,BQ, lp
CE" (log AE)

P-1

(7)

II vlJP,BQ, lP = n.

The last estimate also holds true for p = 1. Indeed, l y - x j > c E if y E Be \ BaE, x c SZE, and the integrand in the interior integral of I2 does not exceed c e'-"v(y). Thus

fE dx fBQ\Ba

v (dl y

1

c E'IIvII1,BQ,

l < n.

Ix

Now (4)-(7) imply (1) and (2).

3.1.5. The Exterior of a Small Domain We preserve the notation and assumptions stated at the beginning of Sec. 3.1.3. The following theorem is the principal result of this subsection.

3.1. Estimates for the Norm of an Extension Operator ...

151

Theorem. Let Q,, be a small domain, Q. C Ge, and suppose 1 E EVp for some p E [1, oo] and some l = 1, 2, ... . (i) The norm of any extension operator E : VP (S2E) -+ VP (GP)

satisfies the inequality

e-n/P min{Qn/P, en/P-1} if lp < n, c ITCH ?

e-1 min{Q1, I

logEJ(1-P)/P} if lp

e-n/P min{Qn/P,1}

= n,

if lp > n.

(ii) There exists a linear extension operator E with norm satisfying the inequality opposite to that in (i). 1 Two auxiliary assertions will be established to facilitate the proof of the theorem.

Lemma 1. Let S2 be a bounded domain of class EVP and SZE C Be, E E (0, 1/2). There exists a linear extension operator E : VP (Q,) -+ VP (Be) with IIEII

c E-n/P min{Qn/P, 1}.

(1)

Proof. We can assume without loss of generality that Sl C B1 (otherwise

we write Q = (\

1

)AE for SZ C BA and use A-1S2 instead of SZ). By Corol-

lary 3.1.1, there is a linear mapping VP(QE)

U H PE E P1-i

such that IIVk(u - P.) 11p,11, 5 Ce1-kIIVluIIP,ne, 0 < k < 1.

Let Q < 1 and consider the extension operator E(1)

: VP (QE)

(2) VP (Be)

defined by E(1)u = PP + EE(u - PE),

(3)

where EE : VP (QE) -+ VP (Rn) is a linear extension operator subject to condi-

tion (3.1.2/1). We claim that IIE(1)ullp,I,BQ

C

(Qe-1)n/PIIuIIP,1,n,.

(4)

1 52

3. Extension of Functions Defined on Parameter Dependent Domains

Indeed, from (3.1.2/1) and (2) it follows that IIo3EE(u -

PE)IIP,Rn < C

Et-sljVIUIlp,n, 0 < s < 1.

(5)

W e now obtain an upper bound for II PE II P,1,BQ Fix a multi-index a, Ial = s <

l - 1, and note that II D'PEIIP,BQ <- C gf/P

C:

of"IID`+APE(0)I

/Q!.

101<1-1-8 Since

IQIO)I <- CE-n/PIIQIIP,n, Q E Pt-1,

the following inequality holds IID°`PEIIP,BQ 5 C

(ee-1)n/P

IIDc+APEIIP,n,.

IPI
The general term of the last sum is dominated by II Da+aulIP,o. + II D+a(u - PE)IIP,nE, and t

IIVsPE1IP,BP < c

(ee-1)n/P

IIVkuIIP,ne,

S < l -1,

(6)

k=s

in view of (2). Now (5) "and (6) imply (4).

Let 1 < p < oo. We introduce a cut-off function 77 E Co (BI), ri Bl/z = 1 and put E121u = r,PE + EE(u - PE), U E VP (S2e),

(7)

where PE and EE have the same sense as above. By assumptions, Q. C BE C B1/2i and E(2) is a linear extension operator: VP (1 ) -4 VP (R"). Furthermore, estimates (6) (for p = 1) and (5) imply that IE(2)

uIIP,1,R^ < C E-n/PIIUIIP,t,n,'

(8)

If we set E = EM for e < 1 and E = E(l) for p > 1, then (1) holds. This completes the proof of Lemma 1.

1

3.1. Estimates for the Norm of an Extension Operator ...

153

Lemma 2. Let S2 be a bounded domain in EVP and let n, C Be, E E (0, 1/2).

If lp < n, there exists a linear extension operator E : VP '(Q,) -+ VP (Be) satisfying

IIEII <- c min {(,,-l)n/p,

-I).

(9)

If lp = n, then there is a linear extension operator E : Vp (SZE) -* Vp (Be) such

that IIEII

C E-1 min {Q1, I

logEJ(l-P)/P}.

(10)

Proof. Let EE and Ell) be the operators introduced in Lemma 1. Note that (3.1.2/1) implies the estimate C E-`IIuIIP,t,ne, U E VP(S2E).

(11)

Put E = E(l) if Q" <_ en-IP and E = EE otherwise. In view of (4) and (11), inequality (9) is true. Turning to the case lp = n, we construct a linear extension operator E(3) : Vp(SZE) -> VP(Rn)

subject to IIE(3)II < Ce-`I

logEl(1-P)/P

(12)

It will suffice to assume SZ C Bl. Let cp E Cm(Rl), W(t) = 0 for t < 1/3, W(t) = 1 for t > 2/3. Put (E(3)u)(x) = uv(x) + (EE(u - u)) (x), x E Rn,

(13)

where

v(x) =' (log IxI/loge), u E VP (Q) and u is the mean value of u on J LE. Since S2E C B. and v I B = 1,

it follows that E(3)uln. = u. Next, v E Co (Bl) and hence IIVIIP,i,R^ < CIIVIVIIP,B.

so that IIVII,1,R^ < c1

f JBl\B

IxI-'Pdx < c2I logell-P.

(14)

3. Extension of Functions Defined on Parameter Dependent Domains

154

Combining (3.1.2/1), (14) and the estimates lul( mes (Q ))1/P <

Ilullp,ci

,

Ilu - uIIP,QE < CE IlVullP,QE,

we arrive at I


cE`11£(3)U11P,l,Rn

E1lVsull,,nE. S=1

Therefore (12) holds. Now set £(3) if on > min{1, 1logEI1-P},

£=

£(1) otherwise,

where £(1) is defined by (3). Then (10) is true by (4), (12), and the proof of the lemma is complete. I

Proof of Theorem. (i) Let ry(E, o) = inf { II ul1 p,1,G, : u E VP (Ge), u = 1 a.e. on Q. }

.

(17)

Since

1J£11 ? 'Y(e, o)/ [mes (1l)J1/p

the required lower bound for the norm of £ is a consequence of the following inequality which is proved below min{on/p, En/P-'} if 1p < n, c'Y(E, o) >_

min{ol, I loge1(1-P)/P} if lp = n,

(18)

min{on/p, 1} if lp > n.

Fix a positive number r0 such that Bro C G (we recall that G contains the Broe, then clearly e > co. On the other hand, the inclusion Q, C Ge implies E diam (S2) < o diam (G). Thus E - o, and the right part in (18) is equivalent to en/p for E E (0, 1/2). At the same time

origin). If 0,

'Y(E, o) ? [ mes (cE)]1/P ,., en/P

3.1. Estimates for the Norm of an Extension Operator ...

155

and (18) follows.

We now check (18) when 1l C Broe. Fix a positive number r1 such that 0 C Br,. If rte > roe, then a is comparable to e, and in this case (18) has been already verified. Let rte < roe. One may assume without loss of generality that r1 = ro = 1 and then n., C BE C Be C Be C G. The first inequality (18) is a consequence of Lemma 3.1.4. To establish the second one, we distinguish two cases. If lp = n and e < e < 1, then (19)

7(E, e) >_ inf{IIujIP,t,B, : ulne =1} and

7(E, e) ? c min{e', I

logel(1-P)/P}

by Lemma 3.1.4. If lp = n, e > 1, one has K!, C B1 C Be for e c (0,1/2) and 'y(e, e) > inf {IIuIIP,i,B,

: uln = 1}, e > 1.

(20)

An application of Lemma 3.1.4 to a function u E Vp (B1), ul., = 1, yields 7(E, e) > c I log 61 (

1-P)/P.

Thus, the second inequality (18) is true.

Turning to the case pl > n, e > 1, we again use (20). By Sobolev's imbedding V(B1) C Lm(B1), the estimate -y(e, e) > c holds. If e < e < 1, lp > n, then (19) takes place. The similarity transformation y = x/e E B1, x E Be, leads to 7(E, e) > e" /P inf {

IIvIIP,l,Bl

: v l1ZE/o =

11.

The last infimum is bounded below by a positive constant c in view of Sobolev's imbedding mentioned above. Now (18) and statement (i) of Theorem follow.

(ii) Fix a positive number r such that G C Br. Lemmas 1 and 2 imply the existence of an extension operator E : VP (Q,) -+ VP (Bre) with norm subject

to the inequality opposite to that in (i). Since Ge C Bre, e > 0, the same operator E satisfies the conclusion of statement (ii). This completes the proof of Theorem. I

3. Extension of Functions Defined on Parameter Dependent Domains

156

Remark. The theorem just proved admits a shorter though less explicit formulation: inf 11C II - E-n/P [ cap (P.; 4'(G,,)) ]

1/P

,

1 < p < oo,

(21)

where g is an arbitrary extension operator: VP1 (Q,) -+ VP '(G,,) and

cap (F; VI (D)) = inf {Ilullp,l D :

UE

Cm(D), UIF > 1}

for open sets D and relatively closed F C D (cf. (2.13/1)). Relation (21) is a consequence of Theorem and the following assertion. Lemma 3. Let SZ and G be bounded domains in R', G containing the origin.

If cie c 2, Ge = oG, E E (0,1/2), 0 < e < oo, S2f C Ge, then I' (e, e) = [cap (K ; VP (Ge))]

1/p

is equivalent to the right part of (18).

Proof. One should merely repeat the proof of inequality (18) in order to dominate the right part of (18) by cr(e, e). The opposite inequality is established by choosing a suitable trial function u E VP (Ge) n COO (G,,) such that ul?je = 1. We can assume without loss of generality that sz C B1. Let 77 E Co (B1), i? In = 1,

a c Cp (B1), UI B112 = 1

and v be the function constructed in Lemma 2. Then one of the functions

u=1, GeDxHu(x)=77(x/E), u = v or u = a serves as a trial function.

3.2. Extension with Zero Boundary Conditions Let S2 and G be bounded domains in R", G containing the origin. As above, by iE we denote a small domain E S2 (with e E (0, 1/2)) and assume that

QC C Ge, where Ge = PG; e E (0, oc). Sharp two-sided estimates for the norm of an extension operator: VP (ci)

VP (Ge) are obtained in this section

provided S2 is in EVP. Here 1 < p < oo, 1 > 1 and Vp(D) is the closure of

3.2. Extension with Zero Boundary Conditions

157

the set Co (D) in VP (Rn) for p < co. The space V1(D) is the subspace of V. (Rn) consisting of functions with supports in D. We preserve the notation introduced in the preceding section.

Theorem. Let Q. be a small domain, Q. C Ge, and let S2 be in EVP for some p E [1, oo] and some l = 1, 2, ... . (i) The norm of any extension operator

5o:VP(Q.)-*VP(GO) satisfies the inequality

e-I for pl < n, c IIEoII >

e

-I

max { (log(1 + BE-1))" , I loge) lp } for pl = n,

(1)

E-n/p max { e-1+n/P,1 } for pl > n.

(ii) If dist(BGe, SZE) > coE, there exists a linear extension operator Eo with norm satisfying the inequality opposite to that in (i).

Proof. (i) Let 'Yo(E, o) = inf I IUIIP,I,GQ : u E VP(Ge), u = 1 a.e. on c

}.

(2)

Clearly (3)

IIEoII ? -(o (E, o)/ [ mes (Q.) 1 /P

for any extension operator Eo : VP(QE) - VP (G.). Thus (1) is a consequence of the inequality

E-I+n/p if pl < n, max{(log(1+eE-1))(1-P)/P,

c7o(E,P)

IlogEl(1-P)/P}

if pl=n,

(4)

Max { -I+n/P, 1 } if pl > n.

Let pt < n and let y(., ) be defined by (3.1.5/17). Since yo (E, e) > y(E, oo) and in view of (3.1.5/18), the first inequality (4) holds.

3. Extension of Functions Defined on Parameter Dependent Domains

158

Consider the case lp = n, o < 2e. To obtain a lower bound for yo (e, o), we use the Friedrichs inequality (5)

IIVzuIIP,GQ >- C o-`IIullp'Gp,

1. Then

where u E Vp(Ge),

1'o(E, o) > c o-t [ mes (11)]l/P > Cl, o < 2E.

(6)

Turning to the case lp = n, o E (2e,1], we introduce a positive number r = r(G) such that G C Br. An application of Lemma 3.1.4 to a function u E Vp (Ge), u I n. = 1, yields I log(rQE-l)IP

llloluIlp,Bre

(7)

+ e-nIIUIIP,B,Q > C.

Since o > 2e and in view of (5), the left hand side in (7) does not exceed c

(log(oe-'))P-l II V1UIIP,G,

Thus (log([JE-1))(1-P)/P

2E < o < 1.

'YO (e, o) > c

(8)

Let lp = n, o > 1. It follows from (3.1.5/18) and the estimate -yo (El o) > y(E, oo) that 'YO(E, o) > C I

logEl(l-P)/P

A combination of (6), (8) and the last inequality lead to 70(E, o)

c max { (log(1 + of-l)) (1-P)/P,

I

logel(1-P)/P}

,

lp = n.

If lp > n, o > 1, then -yo(e, o) > -y (e, oc) > c by (3.1.5/18). Consider the case lp > n, o < 1. Here we have 70(e, o) ?

o-l+n/P

lnf {IIOIZI P,G : V E Vp(G), vIS2c/e

1}

By Sobolev's imbedding VP(G) C L, (Rn) and the inequality IHvHHp,1,G cIIVtvIIP,G, the last infimum is bounded below by a positive constant c. Inequality (4) and statement (i) are established.

3.2. Extension with Zero Boundary Conditions

159

(ii) Let EE be a linear extension operator: VP '(Q,) -+ Vp (R") such that (3.1.2/1) and (3.1.5/11) hold. Since dist (lie, 8Ge) > c,, e, E. can be con-

structed to have the property supp (E,,u) C G. for all u E V(ll) (see Lemma 3.1.2/1). Thus, a linear extension operator EE : VP (l E) -4 Vp (Ge )

is defined and IIEEIIVI(SI,)IVp(G) < c e-l.

(9)

To construct a linear extension operator co : Vp(Q2) -+ VP(GO)

satisfying the inequality opposite to (1), we introduce positive numbers ro = ro(S2), Ti = rl(G) such that Sz C B,.(,, B,.1 C G, and consider several cases. 1) lp < n. Here (in view of (9)) it is possible to put go = EE.

2) lp > n and rie < 2roe. Then e - o (because diam (le) < diam (G,)) and in this case the right hand part of (1) is equivalent to a-' So it is again possible to choose Eo = E. 3) lp = n and r1o > 2roe. We may assume without loss of generality that

rl = ro = 1. Then SZeCB.CB2ECWO CGp.

(10)

Let E(3) be defined by (3.1.5/13). Since vI Be = 1 and v E Co (Bi), E(3) is an extension operator: VP(QE) -4 VP (G p) for > 1. Furthermore, estimate (3.1.5/12) holds. Suppose o < 1, and let cp be a function in C' (R'), W(t) 0

for t < 1/3, w(t) = 1 for t > 2/3. Put

E)u = uh +EE (u - u), u e VP (Q.), where u is the mean value of u on li and h(x) = co (log(Plxl-1)/

log(oe-1))

, x E R".

(11)

Then hl B, = 1 and h E Co (Bp), and thus E) : VP (li) -* VP (Gp) is an extension operator. The following inequalities are readily verified IIhII,I,R.° < c IIVzhllP,Rn < C1 (log

e )-P IBQ\B. IxI" =

C2 I log

g) 1-P

160

3. Extension of Functions Defined on Parameter Dependent Domains

Next, (3.1.2/1) and (3.1.5/16) imply that IIE£(u - u)IIP,1,Rn < CE1-`I1UIIP,i,S]c.

(12)

Hence by using (3.1.5/15), we arrive at IIEo1)uHIP,z,R^ < Cl--' (log(1 + 0E-1))(1-P)IP IuIIP,i,fj,, 2e < o:5 1.

Now let go = E(3) if o > 1 and go = Eo1) if 2e < p < 1. Then IIEoII < CE-1 max { (log(1 + 'OE-1)) (1-P)IP

I logel(1-P)/P1 .

4) lp > n and r1p > 2roe. We can again assume without loss of generality that ro = r1 = 1 and that (10) holds. Let o > 1 and let g(2) be the extension operator: VP (S2£) -+ V(R) defined by (3.1.5/7). The support of the first term on the right part of (3.1.5/7) is in B1 while the support of the second term is in Ge. Therefore E(2)u E VP (Ge) for all u E VP (S2£). Furthermore, estimate (3.1.5/8) is valid. In case o < 1 we set (Eo2)u)(x) = rl(p-1x)P£(x) + (E£(u - P£)) (x),

where u E VP (S2£), x E Ge and 77, P£ have the same sense as in (3.1.5/7). In view of (10) and the definition of r) rl(p-1x) = 1

for

x E S2£, r7(o-1x) = 0

for

Ixl > p.

This means that Eo2) is an extension operator: VP (S2£) -4 Vp (Ge). Inequalities (3.1.5/5-6) imply IEo2)UIIP,I,Go S C

(p/e)"/Pp-`IIuIIP,,,i .

Choosing Co = E(2) for o > 1 and go = E0(2) for o < 1, we arrive at IIEoII < c E-nIP max f o'/P-1, 11.

The proof of Theorem is complete.

I

3.3. On the "Best" Extension Operator from a Small Domain

161

Remark. In case p < 1, 1 < p < oo, the conclusion of Theorem can be written shorter though less explicitly as inf IIEo ll - E-n/P [ cap (QE; LP(Ge))]

11P

(13)

where eo is an arbitrary extension operator: Vp (QE) -+ VP (Ge) and the capacity cap (F; LP(D)) is defined by cap (F; LP(D)) = inf {IIVzuIIP,D : u E Co (D), ul F > 1}

for open D and compact F C D. Relation (13) is a consequence of the above theorem and the following assertion.

Lemma. Under the assumptions of Theorem En-"P if pi

cap (fiE; LP(GQ))

I

(log(1 +

< n, eE-1))1-P

if pl = n,

(14)

Bn-lP if pl > n. provided P < 1, 1 < p < oo.

Proof. By repeating the argument of the theorem leading to (4) and by using the relation IItIIvp(GQ) - IIVlUII,,,GQ, u E Co (G0), e < 1,

we obtain the required lower bound for the left part of (14). The upper bound is established by choosing a suitable trial function u E Co (Ge), uI sa, =

1.

Let D denote co-neighborhood of SZ, where co is the constant defined in state-

ment (ii) of Theorem. If Q E Co (D), aln = 1, then the function

Ge D x " u(x) = a(xle) can be taken as a trial function for lp < n or for lp > n and e - e. Let lp = n and p E [2e, 1]. We can assume without loss of generality that (10) holds. Then the function defined by (11) serves as a trial function. Finally, in case

162

:i

Extension of Functions Defined on Parameter Dependent Domains

lp > n one may put u(x) = r/(e-lx), where 77 E Co (B1),

77IB112 =

1 (here we

again assume that (10) holds).

3.3. On the "Best" Extension Operator from a Small Domain We preserve the notation introduced in the previous sections. As was pointed out in Theorem 3.2, the norm of any extension operator £o : VP (Q.) VP (Ge) is subject to inequality (3.2/3), where -yo(., ) is defined by (3.2/2). It turns

out that if (l - 1)p < n, £o can be constructed to have the norm satisfying the inequality opposite to (3.2/3) up to the factor 1 + o(1) on its right part. Namely, the following assertion holds (we adopt the convention VP(R") _ VP (R")).

Theorem 1. Let QE be a small domain obtained by contracting from a bounded domain Q C R", 11 E EVP. Suppose that 52E C Ge, e < oo, and that dist (Q,, 8Ge) > coe fore < oo . Then the norm of any extension operator £o : VP(f2E) -p VP(Ge)

satisfies the inequality II£oII ? 'Yo(E, e) [mes

(QE)]-1/P ,

and in the case l - 1 < n/p there exists a linear extension operator £o such that I£oll < (1 + o(1)) lo(e, e)[ mes (1E)]-1/P,

(1)

where o(1) is a positive infinitesimal as e -> +0 and 'yo(E, p) is defined by (3.2/2).

Proof. We should only construct an extension operator £o subject to (1). Consider a function f with the properties f E VP(GO), f Isz = 1, IIf IIP,I,G0

(1 +e)'Yo(E, e)

Let E. be the extension operator: VP (SZE) -4 VP (Ge) introduced in Theorem 3.2. In particular (3.1.2/1) and (3.2/9) hold. Put

£ou=uf +EE(u-u),

3.3. On the "Best" Extension Operator from a Small Domain

163

where u E VP '(Q,) and v, is the mean value of u on Q,. Clearly E° is a linear extension operator: VP(1) Vp(Ge). Let us check (1). By using (3.1.5/15) and (3.2/12), we obtain IlEoll < (1 + E)yo(e, p)[ mes (Q,,)]-'/P + C E1

l

(2)

According to Theorem 3.2 (and to Theorem 3.1.5 in case o = oo) E-l+n/P if lp < n, max{(log(1+oE-1))(1-P)/P,Ilogel(1-P)/P}

'Yo(E,0)

Max 1,0-1+n/p, 1 }

if 0 < l - n/p < 1.

if lp=n,

1

Consequently e1-1 [mes (Sle)]1/P[.yo(e, o)]-1 -a 0 as e -+ +0.

Hence and from (2) it follows that (1) is valid. Moreover, (1) can be refined by

0(e) iflp
O(EI

logel(P-1)/P)

0(e1-1+n/P)

if lp = n,

(3)

if 0 < 1 - n/p < 1.

The proof of Theorem 1 is complete.

0

We now examine the case p = 2, 1 = 1, o = no in detail. Suppose Q is a planar bounded simply connected domain of class C°'1 containing the origin. Let F = 852, 52(e) = R2 \ Sz and let IIF I denote the length of r. We need an auxiliary inequality with a small positive parameter e.

Lemma. The inequality (1 - o(1))

Ir1I loge) IItLl L2(r) <_ IIVUIIL2(Qw)

(4)

holds for any function u c V2 '(Q(')) satisfying u(x) = 0 for Ix1 > e-1

Proof. Let ur denote the mean value of u on the curve r. By Theorem 1.5.4 Ilu - urlIL2(r) < c IIDullL2(n(e)),

3. Extension of Functions Defined on Parameter Dependent Domains

164

whence Iur12

IIUIIL,(r) = lur12Irl + IIu - urIIL,(r)

+ c IIVU112 L2(0(.))

Therefore (4) is reduced to the inequality (1

0(1))

1 log EI

lUrl2 -- IIouIIL,(n(-)).

(5)

Computing the variation of the functional F(u) = IUr1-211VullL2(n('))'

we find that the function providing minF(u) is the solution of the following boundary value problem Du = 0 in B11E n D(e),

au

av r

= 1, uI

eBliE

= 0.

(6)

Here v is the unit normal to r directed into 0. We represent the solution of the problem (6) in the form u = v + w, where v = A log(EIxI), A = const, and v is normalized by (7)

Then A = -Irl/27r and w is a solution of the boundary value problem Aw = 0 in B1te n Q (e), w

IaB,1,

= 0, aw

8v r

= cp

(8)

with

W(X) =1 + 2r1 a log Ixl. Moreover, (7) implies that jPr = 0. Putting for brevity G(E) = Q(e) n Bll,, we obtain IIDuII2,G(E) 11ir12

> (IIVvII2,c(<) Ilowll2,c(°))2 (Ivrl + Iwrl)2 (Irl(2ir)-1/21 logEl1/2 (1- 0(1))

- IlVwll2,c(E))2

(Ir1(2ir)-11 logEl (1 + o(1)) + Iwrl)2

3.3. On the "Best" Extension Operator from a Small Domain

165

Thus, to verify (5), one should check that the quantities IIVwlI2,G(e) and Iwrl are uniformly bounded with respect to E. Since qPr = 0 and in view of (8), it follows that fG() I vwl2dx

=

f

r

w aw

8v

d1 =

r=

Jr

Jr

(w - wr)dr.

Hence IIVWIIL2(G(-)) <_ Iiw - wrIIL,(r)IISIILZ(r) < C IIowIIL2(G(-))IIWIIL,(r),

and IIVwIIL2(G(°)) is uniformly bounded in e.

We now turn to the estimate (9)

Iwrl <_ c. Since

f aloglxl r

av

and according to Theorem 1.5.4, one obtains c Iwrl <_ IIVwIIL2(G(l)) +

f (0loglxl r

l

av

) wdI'Z

,

so (9) is reduced to

f

r

(aloglxl av

)

wdI' < c.

(10)

By Green's formula 8

log IxI) wdsx = faGW

JOG(') C a

(where ds is the length element on aG(e) and at x), we have

aw log I xI dsx

the outer unit normal to OG(£)

f aloglxIwaT= JlogIxlff+IlogeIJaids. r

av

B1, a

(11)

3. Extension of Functions Defined on Parameter Dependent Domains

166

Now the identities

f av dI' = Jr or = Pr v i 0 and

f Owdx=J8G(e)a aw ds=0 ( e)

imply that the last integral in (11) equals zero. Let r) be a smooth function on R2, 77 = 1 in a neighborhood of IF and supp 77 C Br with some r = r(S2). It follows from (11) that

Jr

r (77 log IX 1) dr 8v r

log IxI av dI'

(e)nBr

O(r7logIxI)Vwdx < c

Thus (10) is true. The proof is complete.

IIowlIL2(s2(e)nB,/e).

0

The following assertion is obtained from Lemma.

Corollary. The inequality 2,7r

2 e2IIvIIL2p`)) + II°vIIL2(Q(`)) > IrII logEl (1 - o(1)) IIvIIL2(r)

is true for anyvEV2(1 )). Proof. Let o E Co (B1), QIB112 = 1, 0 < a < 1 and u,(x) = u(ex). By inserting u = a ,v into (4), we find that the right hand side in (4) is equal to

J

or,(Vv)2dx ( e)

0(e)

D'E(DUE)v2dx,

whereas u I r = v I r for sufficiently small e> 0. Thus

cc211V11+ IIVVIIL2(n(e))

L2(O(e))

2" (1- o(1)) IIVIIL2(r) Irlllogel

3.3. On the "Best" Extension Operator from a Small Domain

167

It remains to replace E by c 1/2E in the last inequality.

1

We are now in a position to state a theorem on the extension operator: V2 (SZE) -> V2 (R2) with minimal norm, where Q, is a small domain.

Theorem 2. Let SZ be a planar bounded simply connected domain in Co,' containing the origin and let Q, = E 0, where E is a small positive parameter. Then the estimate 27r

1/2

uEII2 (mes2())

1 - 0(1) E1 logeI1/2

holds for any extension operator E : Vz (Qt) -* V2 (R2), and there is a linear extension operator E with 27r

IIEII

1/2

(mes2())

1+0(l) EI1ogE11/2

Proof. According to Theorem 1, it will suffice to check the inequality 1 - o(1) < -y(E) (I logEI/27r)1/2 < 1 + o(1),

(12)

where 'y(E) = inf {IIu1I2,1,R2

: uln. = 11.

The left estimate (12) follows from Corollary by dilation. To verify the right inequality (12), we put A = sup{Ixl : x E 1} and define the trial function

UEV2(R2)by 1

u(x) =

for IxI
log(.V 11x1)/ loge for AE < IxI < A,

0 forIxI>.A. Then uIn, = 1 and 0(11ogE1_1).

I1L112,R2 + 11ou112,112 = (27r/1 logE1)1/2 +

The result follows.

I

3. Extension of Functions Defined on Parameter Dependent Domains

168

We now turn to the multi-dimensional case. Here the norm of the best extension operator: V2 (SZE) -+ V2 (Rn) will be characterized in terms of the Wiener capacity (see e.g. Landkof [115]). Let 52 be a domain in Rn, n > 3, and let cap

S2=sn1(n-2)-l inf{11VU1IL,(R^): uEV2(Rn), u1n=1},

where sn is the area of the sphere

Si-1

Theorem 3. Suppose SZ C R" (n > 3) is a bounded domain of class EV2 and E a small positive parameter. If Q, = E 0, then the estimate 114 _ >

(sfl(n_2)caP1\h/2 mes(1l)

1 E

is valid for any extension operator E : V2 (Q,) -+ V2 (Rn), and there exists a linear extension operator E satisfying 1 +0(1) I

E II

_

mesn (1) (sfl(n_2)cap1)'/2

E

Proof. By Theorem 1, it will suffice to verify the inequality 1 < a'e2 -n. ' (e)2 < 1 + o(1),

where a = sn (n - 2) cap a and ^/n(E) = inf

{IIulI2,1,Rn

:

Ill SZe

= 1}.

With the aid of a similarity transformation we find that E2-n-Yn(E)2 = inf { (EIIuII2,Rn + IIVui12,Rn)2 : uln = 1}.

Therefore E2-nryn(E)2 > a and, moreover, E2-n1'n(E)2 -4 a as e -+ +0. This completes the proof of the theorem.

3.4. The Interior of a Thin Cylinder In the present section we obtain estimates for the norms of the extension operators: V1

I (G,, \ 1E) -4 Vp (GO),

3.4. The Interior of a Thin Cylinder

169

where SZE C G pi Q, is a thin cylindrical layer of width comparable to e and

Go is a cylindrical layer of width comparable to e. 3.A.1. An Extension Operator with Uniformly Bounded Norm

Let w and g be bounded domains in R", n > 2, and let w be in C°'1. We assume that g contains the origin and introduce the domains wE = e w, go =

P9, where e E (0,1/2), 0 < e < oo, g,,. = R'. Let s > 1 be an integer. Put

Q,=wEXR8CR"+e, Ge=gexReCR"+e In what follows c, co, c1, ... denote positive constants depending only on n, s, p, 1, w, g. We now state the principal result of this subsection. Theorem. Let w be simply connected, wE C go and let dist (w6, R"\ge) > c°e.

Then for all 1 < p < oo and all l = 1, 2.... there is a linear continuous extension operator. VP' (Ge \ Q.) -4 VP (Ge) whose norm is uniformly bounded in e, e.

Proof. Construction of the extension operator. Let d C R" be a bounded domain in

C°,1 such that

w c d, d c gol,

and d does not depend on the parameters e, e. Putting dE = ed, we introdu the cylinders DE = dE X R8, TE _ (d \ wE) x R8. Since wE C dE C dE C go, it suffices to construct a linear extension operator E : VP(TE) -> VP(DE) with norm uniformly bounded in E.

Let {rlj}jEZa denote a smooth partition of unity for R8 subordinate to the covering {Qj}, where

Qj ={zER8: Izk-jkI<1, k=1,...,s}, jEZ8. W e may assume I Vrgj I < c for j E Z8, r = 0, ... ,1. Put T(i) = (d \ w) x Qj C R"+e, T(j) = ET(j)

Note that T(j) is in EVp for 1 < p < oo, 1 > 1 (see Remark 1.6.1/2 and Theorem 1.6.2). By Lemma 3.1.2/1, there exists a linear extension operator EEi)

:

VP(T(i)) -4 V'(Rn+8)

170

3. Extension of Functions Defined on Parameter Dependent Domains

satisfying t

Ilok(EE7)f)IIp,Rn+s < C Eei_dIIoifJIpT(j)

(1)

i=0

for any f EVP(T.''))and 0
Vp(TEj)) E) uHPE . EP(nl8) 13

IlVk(u-PE,j)IjpT,(n
(2)

We introduce a mapping VP (TE)D u ,-> Eu by Eu = v + w,

(3)

v(x) = E 77j(z/e)PEa(x),

(4)

w(x) = > rlj(z/E) (EEj)(uj - PE,j)) (x),

(5)

where

jEZ

jEZ°

uj = UIT(3), x = (y, z), y E D, z c R9. Clearly, the mapping u H Eu is linear and EuI TE = U. Estimation of the norm of the extension operator. We now verify that IIEuIIp,i,DE -< C IIuIIp,j,TE, u E VP(TE).

(6)

Let A C Z+ be the set of multi-indices having components with only two values 0 or 1. If

IIj={zERB: 0
71j+a(z) = 1, z E IIj. aEA

(7)

3.4. The Interior of a Thin Cylinder

171

Combining (3) and (7), we obtain v (x)

Pe,j (x') + E /j'+. (.7i/--) (Pe,7+a W - Pe,j (X) aEA

for the points

x=(y,z)ED(j), D(j) = e (d x IIj). Therefore

CIIVkvIIPD,(i) <

IIVkPC,jIIPD,(i)

k

Et-k II Di (Pe,j+a - Pc,i) II P D(i) , 0 < k < 1.

+

(8)

aEA i=0

Put

S(i)={(y,z):y/cEd\i, z/eEIIj}.

We observe that IIQIIP,DE') <- C IIQIIP,s'(i) e) for any Q E P1 n1 and hence it is possible to replace II IIp,Dacn by II ' P,S(i) on the right hand part of (8). Furthermore, the following estimates hold II

IIOkPe,jllp Sti) <

IIVk(U - Pe,i)IIpTi),

IIVi(Pe,j+a - Pe,j)II P Si)

IIDi(u - Pe,j+A)IIPT(i+A)

where a E A (the latter is true because SEA) C TES) for any a E A). This results in r- Ilokvllp D,(i)

IlokuhIP,TEi) k

+

'

11 ei_k II V (u - Pe,i+a) IIP T(i+a).

aEA i=0

In view of (2), the general term of the last sum is dominated by CE 1-kIIDiuIIP'TP+o) I

3. Extension of Functions Defined on Parameter Dependent Domains

172

and so C IIVkvllp,D(i) <- II V, uHHP TT') + > IIVIUIIP,Tui+o), CEA

(9)

where j E Z8, k = 0, ... ,1. Since the multiplicity of the covering {T£J) } depends only on s, (9) implies that (10)

IIVIIP,I,D< <- C Ilvllp,l,TT.

We now bound the quantity IlwHHp,l,DE Fix a k, 0 < k < 1. It follows from

(4) and (7) that k

IIVkwllp,D(j) <- c

E

Em-k11V.E(j+a)(uj+.

- Pe,7+cz)IIp,Rn+..

aEA m=0

By (1) and (2), the general term of the sum on the right is not greater than C

E!-klloluIIP,T(i+o)

and therefore Ilw11p,1,D< <_ c IIV1uIIp,TT

The last along with (10), (5) implies (6), and operator (3)-(5) is the required extension operator E : Vp (TE) - VP (DE). The proof of Theorem is complete.

3.4.2. The Case n = 1 The case n = 1 is not covered by the preceding theorem. We will examine it separately.

Theorem. Let E E (0, 1/2) and let D. = (R1 \ [0, e]) x R8 c R8+1 (i) For any extension operator E:Vp(De)-+VP(Rs+1) the following estimate holds 116,11 >- C E-l+liP

3.4. The Interior of a Thin Cylinder

173

(ii) There exists a linear extension operator E satisfying IIEII <_ c E-1+"P

Proof. (i) Let x = (y, z) be a point in R'+l, where y E R', z E R'. We introduce functions cp E Co (B2('1) and ti E C°°(6, oo) such that e l) = 1, V'I(2,.) = 0.

'IB(') = 1, ?b

Clearly, the function

DE

x

0 fory<0, zER',

uo(x) _

j W(z)ti(y) for y > e, z E R',

is in VP (DE). If v = Euo, then IIEII >_ IIVIIP,1,Ra}1/IIUOIIP,1,Dc

and fE

dz

IIEII.>-c

B

0

(f1

a1v

ayl(y,z)

(1)

Note that E

1 = v(e, z) = J dyl 0

< el-1/P

(l

I

Y1

dye ... dy!-1 f o

o

E alv ayl (y, z)

P

1

yI-181v

a (y, z)dy y

1/P

dy)

for almost all z E B1. Thus, the right part in (1) is not less than cE-1+11P. (ii) Let

DE ={xEDE: y>e}, DE =DE\DE (see Fig. 21) and let u E VP(DE). Put u+ = UIDf , u- = uI D- . First we separately extend the functions u+, u- to R8+1

174

3. Extension of Functions Defined on Parameter Dependent Domains

Fig. 21

To this end we consider a linear extension operator E+ : VP (DE) - VP (Rs+1)

with norm uniformly bounded in E. Let 77 E Cm(R1), i1(t) = 0 for t < 0, ,q(t) = 1 for t > 1. Define

v(y, z) = rl(y/E)(E+u+)(y, z), y E R1, z E R.

(2)

Clearly, v E VP (R8+1), v- = 0, v+ = u+. The inequality IIvlIP,1,R-+i < C E-1+1/PIIU+IIp ! D+

(3)

is a consequence of the estimate C E-'+11P IIu+IIP,(,D± ,

IIvIIP,1,nc

where H. = {x E R8+1 : y E (0, E)}. Let us verify the latter. It follows from (2) that k

IIVkvIIP,11E

C EEt-kIIVi(E+u+)IIP,n., k<1

(4)

i=0

If i = k = 1, then the corresponding term is not greater than c IIv,+IIp,i,D . If Ial = i < 1, then the estimate II(D°E+u+)(y, -)IIP,R' < c IIE+u+IIP,1,R.+'

3.5. A Mollification Operator

175

is valid for almost all y E (0, e) by Sobolev's imbedding theorem. Hence IViE+u+Ilp,rl _< C el/PIIU+IIp,t,D

Thus, the right side in (4) does not exceed the right side in (3) and therefore (3) holds. Now we have constructed the extension

VP(DE) u+HvEVP(R'+1) such that v- = 0 and (3) is true. In a similar way an extension VP (DE)

u- H w E Vp (R'+1)

is constructed to satisfy w+ = 0 and G C E-t+1

IIu

Ilp t D_ .

The required extension operator E can be defined by Eu = v + w which concludes the proof of Theorem 2. We point out the following special case of the theorem.

Corollary. There exists a linear extension operator E : Vrl(DE) -4 Vll(R'+1)

with norm uniformly bounded in E.

3.5. A Mollification Operator Let d be a bounded domain in R" and let K E Co (R'). For a function v defined on the cylinder D = d x R' C R"+', we put (Tv)(x) = f K(t)v(y, z + l yl t)dt, x = (y, z) E D.

(1)

This mapping v " Tv will be used in the next section to construct the extension operator from a thin cylinder to its exterior. Here we study properties of the mapping (1). In this section c denotes various positive constants independent of v, d and y.

3. Extension of Functions Defined on Parameter Dependent Domains

176

Lemma 1. Let l > 1 be an integer. Suppose that

f K(t)t"dt = 0

(2)

for all multi-indices v E Z+, I vI < 1 - 1. If v E C°°(D), then the estimate

Iy(Tv)(y,')I n,x. < c ID

yI1-171II(Vzv)(y,')IIp,R'

(3)

holds for y E d \ {0}, I7I < 1, p > 1. Moreover, if n = 1 and the set d contains the point y = 0, then

lo

a yk

(Tv) (y, z) = 0, k = 0, ... ,1 - 1, z E R8.

(4)

Proof. First consider the case ry = 0. Applying the Taylor formula to v(y, ) and using (2), one obtains

(Tv)(y,z)=1IyI1 f K(t)dt

L

1

a. CI=1

By Minkowski's inequality, II(Tv)(y,')IIp,a, < c IYI1II(Vzv)(y, )IIp,R',

and thus (3) is valid for ry = 0. We continue the proof by induction on 1. Let l = 1. It is easy to see that

l o(Tv)(y, z) _ (Tv) (0, z) = 0

if n = 1 and z E R8. Furthermore, aTv _ ayi

2 y.

E f t.K(t)az.(y,z+IyIt)dt j 8

IyI ,j=1

+

f K(t)ayi(y,z+Iylt)dt=Tvy;+

IyI

ETjvz',

(5)

3.5. A Mollification Operator

177

where 1 < i < n and Tj is the operator of the form (1) with the kernel K3(t) = tjK(t). Minkowski's inequality implies aTv 11

ay,

(y, ') 11p

,R'

< C llov(y, -)

I1p,Ra.

The case l = 1 is exhausted. Let 1 > 2 and let the conclusion of Lemma 1 hold for orders not greater than l -1. We now verify (3) for 0 < aryl < 1. Note that D7 = D0 ays for some 1 < i < n and a E Z+, lal = l'Yl - 1 < 1 - 1. In view of (5), one obtains Dy T v = Dy (T vy,) + E Dy (yt l y l -'Tj v.,) . j=1

(6)

The following estimate holds for each term in the last sum: IIDy (yilyl l(Tjv=,)(y, -)) II p,R'


(7)

IyI161-1c'1IID'(Tjv.j)(y,')IIp,Re.

a
Here b < a means ba < ai for i = 1, ... , n. The kernel Kj of Tj satisfies (2) for l vl < 1 - 2, and so the induction hypothesis applies to the operator Tj and

the function vi, . This results in II Dy(Tjv.,)(y, -)Ilp,R, < c lyll-1-16'I1(Vl-lvzj)(y, -)IIp,R,

The induction hypothesis can be also applied to the first summand on the right side of (6). We have lI Dy (Tvy,) (y, -)llp,R, < C

lyll-l-b0III(o1-1vy,)(Y,

-)lRRa

The last two estimates along with (6) and (7) imply (3). Let n = 1 and 0 E d. Let us verify (4) for k = 1 - 1. If y > 0, (6) yields al-_Tv

al 2Tvy

8

ayl-1 - ay1-2 + E j=1

a1

-2Tjyzi ayl-2

3. Extension of functions Defined on Parameter Dependent Domains

178

If y < 0, the sign + on the right should be replaced by -. Now the equality 81

lim 5V:-yI T v (y, z) = 0, z E Re,

y-)O

y

follows from the induction hypothesis (with respect to (4)) applied to vy and 1 V,,, This completes the proof of Lemma 1.

In the next lemma we prove the continuity of the operator T : Lp(D) Ll (D) under certain conditions on the kernel K. Lemma 2. Let (2) be fulfilled for all v E Z+, 1 < IvI < 1 - 1. Then operator

(1) is bounded as an operator: L,(D) -+ Lp(D) for 1 < p < oo and the following estimate holds (8)

IVITvUIp,D
Proof. If l = 0, then (8) is easily verified by Minkowski's inequality. The same argument leads to the continuity of the operator T : Lp,l°,(D) -+ Lp,loc(D). For l > 1 and v E Cm(D) fl Lp(D), we establish (8) by induction on 1.

Let l = 1. Since 8Tv/8zj = T(8v/8zj) for j = 1, ..., s and in view of (5), the result follows for n > 2 by the continuity of T in Lp(D). If n = 1, we have to check the absolute continuity of the function d = (a, b) E) y H (Tv)(y, z), where z E R9. Only the case 0 E (a, b) needs to be checked. In this case the function y H (Tv)(y, z) is smooth for y # 0 and also yllimo(Tv)(y, z) =

z) = v(0, z)

J

K(t)dt.

The absolute continuity follows, and the case l = 1 is concluded.

Let l > 2 and let the inclusion Tv E LP (D) and inequaltity (8) (with l replaced by k) hold for all k < I - 1 and all smooth functions v E LP(D). We now check (8) for v E C°°(D) fl L,(D). Note that DyDOTv = Dy(TDpv), where y

0, /3 E Z+, ry E Z.

Let IQI+h'I =l. If'y=0, then IID13TvIIp,D = IITDavIIp,D < CIIVIVIIp,D

(9)

3.5. A Mollification Operator

179

by the continuity of the operator T : Lp(D) -> Lp(D). If 0 < IyI < 1, then by (9) and the induction hypothesis (applied to DQv), we find IIDYDZTvJIp,D
Let IQI = 0, I-'I = 1 and let DyY = Dy eye In view of (6), (7), the estimate

IID' (Tv)(y,')IIp,R < IIDy(Tvyj(y,')IIp,Rs S

+c

1y115 1-1°I IID6(Tjv=,)(y,')IIp,R.

j=1 6
is valid, where Tj has the same sense as in (5). Note that the kernel Kj of the operator Tj satisfies (2) for 0 < IvI < 1 - 2, and thus Lemma 1 is applicable to Tj and the function v.,,. Hence the general term in the last sum does not exceed II (ot-lv=;) (y, ') Ilp,R° By integrating with respect to y E d, we arrive at IID°Tvyi IIp,D + C IIOIV II

IID'TvIIp,D C

p,D.

According to the induction hypothesis, the first summand on the right is not greater than c Ilol-1Vyi IIp,D. So (8) is true for smooth functions. The following equality should be established in the case n = 1 and 0 E d: l

1

I

1

yllimo ayl T v (y, z) = ylimo ayt

Tv

(y, z), z E R9.

(10)

Indeed, let y > 0. Combining (5) and (6), we obtain al-1

al-2

at-2

ayl 1Tv = ay1_2Tvy +

ay1_2TjvZj.

(11)

j=1

In the case y < 0 the sign + on the right should be replaced by -. Lemma 1 (applied to each Tj and vi,) implies that the sum on the right part of (11) tends to zero as y -* 0. Thus the left side in (10) is equal to 81-2Tv

lim y (y, z), y-+-o ayl-2

whereas the right side is lim

81-2Tv

y-++o ay

1_2

y (y, z).

180

3. Extension of Functions Defined on Parameter Dependent Domains

The last two limits coincide by the induction hypothesis applied to vy. The inclusion Tv E L,(D) and estimate (8) are now verified for any smooth function v E ' (D). Since smooth functions are dense in L,(D) with p < oo and the operator T : Lp,loc(D) -p Lp,loc(D)

is continuous, the inclusion Tv E Lp 10c(D) and inequality (8) are valid for arbitrary v E L,(D), p < oo. When p = oo, one should repeat the previous argument for a function v E L'.(D) in order to obtain (8). This concludes the proof of Lemma 2. 1 The following lemma gives estimates for the difference v - Tv in case K is a mollifier (cf. 1.2.1).

Lemma 3. Let l > 1 and (2) be fulfilled for 1 < I vI < 1 - 1. Let, furthermore,

f K(t)dt = 1. Then the estimate holds Ilok(Tv - v)Ilp,D < cr'-kl iVIvllp,D,

(12)

where 0 < k < 1, r = sup{IyI : y E d} and v an arbitrary function in L,(D).

Proof. We first check (12) for v E Lp(D) n C-(D). An application of the Taylor formula to v(y, ) gives

(Tv)(x)-v(x)I =fK(t)(v(y,z+ iyit) -v(x))dt


I(Dov)(y,z+IylTt)I dT.

101=1

Now Minkowski's inequality implies IITv - VIIp,D < c r'Ilowllp,D.

(13)

The proof will be continued by induction on 1. Let 1 = 1. For k = 0, inequality (12)' has been already verified. If k = 1, (12) follows from Lemma 2. The case l = 1 is exhausted. Let l > 2 and the conclusion of the lemma be valid for orders not greater

than l - 1. Let us choose milti-indices 3 E Z+, -y E Z+ such that IQI >

0, I/jl+l I=k<1. Then Dy DQ(Tv - v) = Dt(TDOv - Dpv)

3.5. A Mollification Operator

181

and the induction hypothesis (with respect to T and DQv) yields II DI D7(Tv - v)Il p,D < c rl-IQI-171IIVj-IQI DavII P,D.

In the case IQI = 0, 0 < Iryl = k < 1, we put Dy = Dy ey, for some i = 1,...,n. Identity (6) gives IID7(Tv - v)IIP,D 8

< IIDa(Tvy,

-vy,)IIP,D+IIDa(y=lyl-1T.jvZ,.)IIP,D,

(14)

j=1

where Tj is the operator of the form (1) with kernel Ki(t) = t3K(t). To bound the general term of the sum in (14), we use inequality (7) and then apply Lemma 1 to Ti and v=, . The term is thus dominated by c r1-1-IaI II

V1-1vz,, II P,D

Furthermore, by the induction hypothesis, IID°(Tvy: - vy.)IIP,D < c

r1-1-ICtIII01-1vy:IIP,D,

and so IID7(Tv - v)IIP,D < C rl-dIIVivIIP,D.

The proof of inequality (12) is concluded for smooth functions. By density of smooth functions in LP(D), p < oo, and by the continuity of the mapping v H Tv in LP,i,,c(D), inequality (12) is true for all v E L,(D) and 0 < k < 1.

In order to verify (12) for p = oo, one should repeat the argument of Lemma 3 for a function v E L' (D).

1

The following two lemmas deal with functions v that do not depend on y. We may assume that v is defined on R'.

Lemma 4. If v c LP(R'), p > 1, then the estimate Iy(Tv)(y,')IIP,Rs < C IyI-I7II VIIP,R, ID

holds for -y c Z+ and y c Bi") \ M.

(15)

3. Extension of Functions Defined on Parameter Dependent Domains

182

Proof. If 1-yj = 0, then (15) is a consequence of Minkowski's inequlity. Let 1-yj = 1. Then

ayiTv

ay:

(F0 1 fK(tlylz)v(t)dt/

=-y=IyI-2(sTv+Tv), 1
(16)

Here t is the operator of the form (1) with the kernel e

k(t) = > tjKt1 (t). i-1 Applying again Minkowski's inequalty, we obtain (15) with 1-yj = 1.

Let k > 2 and let (15) hold for all derivatives Dy of orders 1-yj < k - 1. If a I'YI = k, then D7 y ay; for some i = 1, ... , n, and (16) yields y = D' I Dy (Tv) (y, -) I ip,R°


The result follows by the induction hypothesis applied to each summand in 1 square brackets. This completes the proof of Lemma 4. The last lemma of this section refines estimate (15) if the kernel K has zero moments of orders 1, . . . , 1 - 1.

Lemma 5. Let condition (2) hold for 1 < I vI < 1 - 1. Then

Iy(Tv)(y,')IIP,R°
(17)

where y E B(') \ {0}, ry E Z+, 1-yj > 1 > 0 and v E LP(R9) an arbitrary function.

Proof. We will prove (17) by induction on 1. If 1 = 0, then (17) is a consequence of Lemma 4. Let 1 > 1 and suppose that the inequality IIDy (Tv)(y,')IIRR° < C

IyI`-1-'O'lio!-1vIIP,R°

3.6. Extension to the Exterior of a Thin Cylinder

183

holds for v c LP i(Rs), 1,31 > 1 - 1 provided the kernel of T satisfies (2) for

1 1, then D7 = Da a for some i = 1, ... , n. We have Y 8Y; Y

y+


(Tjvzf)(y,'))

D

(18)

E(IyI

P,R'

where Tj is the operator of the form (1) with the kernel Kj(t) = tjK(t), a

si =

IyI161-IaIIID'(Tjvz,)(y,')IIP,R-, j=1 6
and S2 is the sum with the same general term for the indices 1 < j < s, b < a, 161 > 1. Since Kj satisfies (2) for Ivj < I - 2, the following estimate is valid for

181

(19)

The last estimate is verified in the same way as inequality (3) has been proved

in Lemma 1. The induction hypothesis (applied to Tj and v,,) implies that (19) is also true for 161 > 1. Now (17) follows from (18), (19), and this completes the proof.

3.6. Extension to the Exterior of a Thin Cylinder The goal of the present section is to obtain two-sided estimates for the norm of the extension operator: VP (Q,) -+ VP (Ge) from a thin cylindrical layer to

a larger one. Here Q, = We x R9, Ge = ge x R9 are cylindrical layers in Rn+s, wE = e w, ge = e g, iE C ge, w and g are bounded domains in R", w is in C°'1 and g contains the origin, s E (0, 1/2), 0 < e < oo, g,,,,, = R". For the simplicity of presentation, we assume in what follows that the closure of " w is contained in Bl) A point x E Rn+s will be written as x = (y, z), where y E R", z E Rs. The symbols C, co) Ci, ... denote positive constants depending only on 1, n, s, p, g, w;

a - b means that c1 < a/b < C2. As above, we write for brevity II instead of II ' IIV;(D).

' IIP,i,D

3. Extension of Functions Defined on Parameter Dependent Domains

184

3.6.1. Three Lemmas on Functions Defined in a Thin Cylinder

Let 0 E Co (W) and let f ii(y)dy = 1. With every function u E VP '(Q.) we associate the function of the form PE(x)

=E -"

1

(1)

'yEZ+,lil
we

where x = (y, z) E R"+s.

Let K E Co (B(8)). Assume that f K(z)dz = 1 and that (3.5/2) holds for all v E Z+, 1 < IvI < l - 1. By T we mean the operator given by (3.5/1) with kernel K. Lemma 1. The mapping VP 1010 E) u H TPr is a continuous linear operator: VP (Q.) -+ VP (GQ), 1 < p < oo, ge C Bi"1. Moreover, the following estimate holds t

k=0,...,1.

IIVk(T1e)Ilp,Gs -<

(2)

i=k

Proof. The function defined by (1) can be expressed in the form

(

PE (y, z) = E-"

7 Ey)

I7I
f,

0-1

(11) u(r,, z)drj E

with /i.y E CO '(w) independent of u. So the mapping VP (Q,) 3 u HP, E VP (G(?)

is linear and continuous, and hence the first conclusion of the lemma follows from Lemma 3.5/2. To verify (2), we observe that Pe(x)

=E-" E ai1 Ic I
(-1)la-6I

6
a

,/1 (b)y6

Dnu(rl,z)rla 6w(E)d

Let f be a function defined on Q,. For a, d E Z+, a > 8, put ('Da,6f)(z) =

E-"

f f (77, z) W (71/e) d77

(3)

3.6. Extension to the Exterior of a Thin Cylinder

185

with

'P(rl) = Da 6(y"v)(ij)) Then Pe takes the form

P.(y,z) _ 1 E ( " ) y6(-I'.'6v)(z), Ick«

6
where

v(y, z) = (Dyu)(y, z) Note that for Q E Z+, ry E Z+, 1)31 + I'YI = k < 1, the inequality II D- DQTPE II p,cQ 5 C

II D7

Il p,ce

(4)

IuI
holds. To bound the general term of the last sum, we distinguish three cases. 1. Let Io < IryI. Then DOT-D,,6v = TO,,,6(Dpv) = T-D,,,6(D'DQu).

Putting w(z) =

one obtains II Dy (y6(Tw)(y, )) IIp,R. IyIl61-IaI IID' -a(Tw)(y,


-)IIp,R.

A
for y E ge \ {0}. Since w E

LP71-161(Re),

Lemma 3.5/5 implies that

I1 y (y6(Tw)(y, .)) IIp,R. < C IIV171-161wHIp,R-

< c1

II Dc,b(Dvu)IIp,R" < C2E-"'pIIVkU Ip,Q., k = IQI + kvl. IvI=k

Here v is a multi-index in Z"+e of length k. Furthermore, the estimate (-%,6f) (z) 5 C E-" Ip 11 f ( . , z) II p,..

(5)

3. Extension of Functions Defined on Parameter Dependent Domains

186

has been applied. Now the inequality IIDv (y6DQT4)a,6v) I lP GQ

C (P/e)" /pjjVkuhIP,ce

(6)

follows by integration with respect to y E ge.

2. Let IbI > k = IQI + I-yl Put w(z) = (T)a,6v)(z). If T' is the operator of the form (3.5/1) with the kernel D'K, then

D1Tw = (- IyI-IQI)T'w. We have IIDy (y6(DATw)(y,'))IIP,R'

< c E I Dv(IyI-IAly6)IIIDb-A(T'w)(y,.)IIP,R a
< C1 E

Iyl161-IQI-Ial IIDv-'\(T'w)

(y, ) IIP,R°,

y E ge \ {0}.

A
By Lemma 3.5/4, each term of the last sum is not greater than IYI161-kIIwIIP,R

which does not exceed c IIwIIP,R" for IyI < 1. In turn, (5) implies that IIWIIP,R < CE-n1PIID6uIIP,Q,.

Integrating with respect to y E ge, we dominate the left part in (6) by c (e/E)n/PIIo161uHIP,ce.

3. Let I'yl < 161 5 k = IQI + I'yl. We fix a multi-index p c Z+ such that p < Q and IpI + 161 = k. If t is the operator of the form (3.5/1) with the kernel DO-AK, then DO (T I ,6v) = DA-'Tw = (-IyI)lµl-IPITw,

y E ge \ {0},

where w(z) = (4%,6DZ v) (z). Thus IIDv (y6 (DgT a 6v) (y, ')) IIP,R.

= IIDv (y6Iy1

A
I'r1-161(Tw)

(y, ')) IIP,R-

IylIY1-IahII(Dv-aTw)(y,')IIP,R,.

3.6. Extension to the Exterior of a Thin Cylinder

187

Applying Lemma 3.5/4, we dominate each term in the last sum by c lwI1p,R.s

which is not greater than c E-n/p1IDz Dau11 p,n.

in view of (5). After integrating with respect to y E ge, one obtains (6). So, the general term of the sum in (4) does not exceed the right part of (2). The proof of Lemma 1 is complete. I The following assertion is the "generalized Poincare inequality" in a thin cylinder.

Lemma 2. Let u E VP l(92,) and let Pe be defined by (1). Then (7)

Iloi(u - PE) Ilp'nE < c E'-tllViullp'n,

foralli=0,1,...,1-1. Proof. It will suffice to establish (7) for E = 1 and then use a similarity transformation. For e = 1, we put P = P1, S2 = SZ1 = w x R8. Let

uEVP(S2), aEZ+, QEZ+, lal+IQI=i<1. Then

Dy Dp(u - P) = Dy (DQu - H) -

DOS,,

(8)

where the sum is taken over all ry E Zn such that ry > a and l > 1'YI > 1 - 101, S7 (y, x) _ (Y

1

a)!

f i1

H(y, z) _ 171<_1-1-IRI

7

Dnu(rl, z) (y - r))7 '',(r))drl,

D'Dpu(rl, z)(y - rl)7')(rl)drl

Since the mapping VP-IA,

f

(W) i) v H

D7v(rl)(y - rl)70(,q)dn

ry'

is a continuous projector on Pinl01-1, it follows from the theorem on equivalent norms in Sobolev spaces that II D° (Dpu(, x) - H(', z)) II p, < c 11 (Vi-IAI DRu)

z) IIp

,

3. Extension of Functions Defined on Parameter Dependent Domains

188

Here 0,_1p1 denotes the gradient of order l - IQI in variables yl, ..., y,,. Integration of the last inequality with respect to z E R' yields IIDy (DAu - H)IIP,n < c IIDjuUIP,ci

(9)

To bound the quantity II D,6S7IIP,n, we fix a multi-index v E Z+ such that

v< y and

Ivi-l-I,3I. Then

p

(-1)17-VI

DZ S7(y, z)

- a)!

f

D77DOu(77,

z) D77

((y

- 77)7`0(77))d77.

An application of Minkowski's inequality leads to IID1'S7IIP,i

J

II(Vlu)(77,')IIP,R°IIf(77,')IIP,,dij,

where f (77, y) = D7-" ((y - 77)7-`,0(r7)) . Since Vi E Co (w), the second factor in the integrand is bounded uniformly in 77 E w. So 71

IID1S7IIP,n < c IIozuhip,o.

Combining the last with (8) and (9) concludes the proof of Lemma 2.

1

We introduce a linear extension operator EE : VP (SZE) -* Vp (Rn+s)

satisfying IIokEEVIIP,Rn+e < C

6";jjVjVjjP,n,

(10)

i=0

for any v E V'(1), 0 < k < l (see Lemma 3.1.2/1). In Lemma 3 stated below an auxiliary extension operator is constructed which will be used in the proof of the principal result of Sec. 3.6.

Lemma 3. Let SZE C Ge. F o r all l = 1, 2, ... and all 1 <_ p < oo there is a linear extension operator E,B ' VP (HE) -4 VP (GQ)

3.6. Extension to the Exterior of a Thin Cylinder

189

such that IIEE,QII

CC` /P min{Qn"P,

1}.

Proof. Let VP (SZE) E) u F4 P. and PE H TPE be the mappings introduced at the beginning of the subsection. We put EEu =

TPE + EE(u -

TPE),

u c YP (ALE),

where E. is an extension operator: VP '(Q.) - Vp (Rn+s) satisfying (10). Let us verify that IIeEUIIP,I,G" C C (Q/E)njPIIUIIP,,,,e

(11)

if g,, C B(In). Since IIV (u - TPE)IIP,s2< < IIV1(u - Tu)IIP,n. + IIV T(u - PE)IjP,nE,

Lemmas 3.5/2-3 along with the preceding one imply the estimate

IIV (u-TPE)IIP,n.

cE" IIVIUIIP,o., 0
(12)

By Lemma 1, (12) also holds for i = 1. Combining (10) and (12), we arrive at IIVkEE(u - TPE)IIP,Rn+. < C E'-kjIViUIlP,ne, 0 < k < 1.

(13)

Now (11) follows from (13) and Lemma 1. Let E : VP (Bin) X Rs) -+ VP (Rn+s)

be a linear continuous extension operator. IfEE1)u is the restriction of EEu to the cylinder B(n) x Re, then the norm of the extension operator VP (Q,) E) u F-a E(E,(') u) E VP (Rn+s)

is dominated by c E-n/P. Thus, the required extension operator EE,e can be defined by EE,e = EE if p < Qo and by EE,e = EEE1) if o > Qo, where Qo = (sup{IyI : y E 9j)-'-

190

3. Extension of Functions Defined on Parameter Dependent Domains

The proof of Lemma 3 is complete.

1

Remark. Let

R++B={(y,z):yER", zER8, z,>0}. The operator EE,e in Lemma 3 can be constructed to satisfy the condition Ee'auIGnR++' = 0

if uIS2.nR++' = 0.

a

To this end, one should require that supp K C {z E B(g) : z, > 01,

where K is the kernel of T (see the beginning of Sec. 3.6.1). We should also impose the following conditions on the extension operators E. and E which appear in Lemma 3: (Eu) (y, z) = (EEu)(y, z) = 0 for z, > 0

if u(y, z) = 0 for z, > 0. 3.6.2. An Extension Operator from a Thin Cylinder

We preserve the notation introduced at the beginning of Sec. 3.6. Theorem stated below is the principal result of this section.

Theorem. Let i i C ge. (i) The norm of any extension operator

E:VP(Qe)-+VP(GQ), 1
E-i min{e', I logEl(l-P)/P} if lp = n, E-n/P min{,o'/P, 1} if lp > n.

(ii) There exists a linear extension operator E with norm satisfying the inequality opposite to that in (i).

3.6. Extension to the Exterior of a Thin Cylinder

191

Proof. (i) First we verify the estimate (mesn(WE))-1/P,

IIEII > 'Yn(E, Q)

(1)

where 'Yn(E,,o) = inf{IIuIIP,i,9Q

: u E V'(ge), ul,,e = 1}.

0 and define vb on Q by

Let 71 E Co (R8), 77

ve(x) = 68/Pij(6z), x = (y,z),

6 > 0 being fixed. The estimate II(Eva)(',z)IIP,1,9a >_ 7n(E,Q)5s/Plrl(6z)l

holds for almost all z E R8 and hence II6voIIP,l,GQ >_ -Y. (E, Q)II77 IIP,R' Since

IIell >_

IIEvoHIP,i,Gv/IIvoIIP,i,oE,

if follows that 114

'Yn(E, Q)

II71IIP,R'

(mesn(we))1/P C-`k_o

dkllok77llP,R.

Passage to the limit as 6 -4 +0 yields (1). Now statement (i) of Theorem is a consequence of the relation min{[Jn/P, En/P-'l 'Yn (E, Q) ,,,

if lp < n,

min{Ql, I log e1(1-P)/P}

if lp = n,

(2)

min{Qn/P,1} if lp > n,

which was proved in Theorem 3.1.5.

(ii) In view of Lemma 3.6.1/3, it will suffice to construct an extension operator 9 : Vp (SZE) - Vp (GP)

3. Extension of Functions Defined on Parameter Dependent Domains

192

for each case lp < n or lp = n such that c c -' -'

if lp < n,

and 11611 5 c e 11 log EI11P if lp = n.

We shall prove the following stronger statement: if 1 - 1 < n/p, then there exists a linear extension operator Vp(S2e) -+ Vp(Rn+s)

subject to IIEII <- -rn(E, oo) (mesn(We))_1/p (1 + 0(1)),

(3)

where o(1) is a positive infinitesimal as a -* +0. and z E R' let v(z) denote the mean value of v(., z) on We. For v E We have IIvhlp,R- (mesn(We))1/p < IlvllP,Q,-

If Vv

E LP(SZE),

(4)

then also 11V - vll

p,cf

<- CE 11VV11p,s2f

(5)

Let we E Vp (Rn),

W,

and Ilwellp,Z,R^ < (1 + E)'yn(E, 00)-

We introduce a linear extension operator EE : VP '(Q,) - VP (Rn+s) such that (3.6.1/10) holds. Put

(Eu)(x) = u(z) we(y) + (Ee(u - u))(x),

(6)

where u E VP (n,), x = (y, z) E Rn+' Clearly, (6) defines a linear continuous extension operator: VP (Ste) -+ VP (Rn+'). Let us verify (3). From (4), (5) and (3.6.1/10) it follows that IIEe(u - U)Ilp,i,R^+, < C

E1-`IIuhIp,i,Qf

(7)

3.6. Extension to the Exterior of a Thin Cylinder

193

Next, it is readily checked that IIuweIIp,I,Rn+a -

IIWEIIp,I,R"IIhIIp,I,Rs.

Since

Mu)(z)I <_ IVkul(z), Z E R8, and in view of (4), we have IIVk'ullp,R- < Ilokullp,sl. (mesn(wE))

1/p

k = 0, ... ,1.

Thus ll1

1IWEllp,I,R.nllulp,j,Q,(mesn(wE))-1/p.

(8)

Now a combination of (6)-(8) yields IIEII

1n(E, oo)(1 +

e)(mesn(we))-1/p

+

Furthermore, (2) implies that lim e1

E-,+0

(mesn(wE))1/p(1n(E,oo)1-1=0

provided 1-1 < n/p. So (3) holds for the operator given by (6), and the proof of the theorem is complete. I

Remark 1. We may assume that the operator E defined by (6) has the property

(Eu) (y, z) = 0 for z8 >0 if u(y, z) = 0 for z9 >0 (cf. Remark 3.6.1).

1

The proof of the above theorem also contains the following assertion on the best extension operator: VP(S2E) --* Vp(Rn+e).

Corollary. Let QE = wE x R8 be a thin cylindrical layer in Rn+e. The norm of any extension operator E : Vp(1 ) -1 Vp(Rn+9)

satisfies the inequality IIEII ? 1'n(E)(mesn(WE)) 1/p,

3. Extension of Functions Defined on Parameter Dependent Domains

194

and if l - 1 < n/p, then there is a linear extension operator E such that IIEII < 7n(E) (mesn(wE))-1/p (1 + o(1)), where

1'n(E) = inf{IIuIIp,i,Rn : u E Vp(Rn),

1}

and o(1) is a positive infinitesimal as E -4 0.

0

Remark 2. When p = 2, 1 = 1, we have grEl 10

(1- o(1)) < 72(E) <

I

loge)

('+o(')),

and if n > 3, then E(n-2)/2 < ((n - 2)sn cap w)-1/2 ryn(E) < E(n-2)/2 (1 + 0(1))

,

where sn is the area of the sphere Sn-1 and cap is the Wiener capacity in Rn. The inequalities just mentioned were proved in Theorems 3.3/2-3.

Remark 3. The infinitesimal o(1) in the above corollary can be refined by (3.3/3).

3.7. Extension Operators for Particular Domains Extension theorems which have been proved in preceding sections enable us to obtain two-sided estimates for norms of extension operators for concrete domains depending on parameters. In the present section we consider examples illustrating possibilities arising.

3.7.1. Examples of Extension Operators for Domains Depending on a Small Parameter

In what follows E E (0, 1/2), and the symbol - denotes the equivalence of positive quantities uniform with respect to E. Example 1. Consider the "hat" Sl(E) in Rn : Q(E) = B+ U G(E) (see Fig. 22). Here B+ = {x E B( n) : Xn > 0} and

G(E) = {x = (x', xn) E Rn : xn E (-E, 0), Ix'I < 2}.

3.7. Extension Operators for Particular Domains

195

Fig. 22

Let E : VP(S21E1) --> VP(Rn)

(1)

be an arbitrary extension operator. We shall check that inf IIEII - E-11P

To obtain a lower bound for IIEII, we consider a function f E Co (1, 2) such that f(t) = 1 for 4/3 < t < 5/3. A trial function v E VP(SZ(E)) is defined by v(x',xn) = f(Ix'I). Then, for any extension operator E as in (1), II(Ev)(x',')IIR1,R1 > C(Al) > 0

for a.e. x' E A, A = {x' E Rn-1 : 4/3 < Ix'I < 5/3}. Hence IIEvIIP,1,Rn 2 IEvH1P,1,AXR1 >- C, and IIEII

IIEVIIP,1,Rn/IIVIIP,1,f21E> > CE-1/P

We now construct a linear extension operator E subject to IIEII <_ CE-1/P

(2)

To this end, first we extend a function in VP(G(c)) from G(E) to the thin layer Rn-1 x (-E, 0). This can be done with the aid of a finite order reflection (cf. 1.6.1) with respect to each radius of the ball Ben-1). Namely, let G(C) E) x = (r, 0, xn), where (r, 0) are the spherical coordinates in Rn-1. If u is a smooth function on G(E), define u on D(e) = {x = (r, 0, xn) : r < 2 + 1/1, 0 E Sn-2, xn E (-E, 0)}

196

3. Extension of Functions Defined on Parameter Dependent Domains

by u = u on G(e) and t

u(r, 0, x") = E cju(2 - (r - 2), 6, x") i-1

on

D(E) \ G(E),

(3)

where the coefficients c3 satisfy (1.6.1/3). It is readily checked (cf. Theorem 1.6.1) that then u E C'-'(D(E)) and that IIuIIP,z,D(E) <- c(l,p, n) IIiIIP,I,c(E).

Thus, the map u i--p u can be uniquely extended to a linear bounded operator: V'(G(E)) -+ VP (D(E)). Multiplying ii by a smooth cut-off function depending only on r, having support in [0, 2 + 1/1) and equal to 1 for r E [0, 2], we obtain a linear continuous operator E1 : V'(G(E)) -4 VP '(QE),

QE =

R"-1 x (-E 0)),

whose norm is uniformly bounded in e.

Put R" = {x : x" < 0}. By Theorem 3.6.2, there is a linear extension operator E2 : VP '(Q,) -> VP '(R

with norm subject to IIE211 <

cE-'Ip.

Let u E Vp(Qfr)) and let ul denote the restriction of u to G(e). Define v on R" U B+ by v = u on B+ (cf. Fig 22) and v = E2E1u1 on R" . Then V E VP(R" U B+),

vl,(,) = u

and

IIVIIP,1,R^UB+ < cE_11PlluIIp,i,sl(.).

It remains to extend v from R" U B+ to R" by Theorem 1.6.2. If w is such an extension, then VP(Q(E)) 9 U"Eu = w E VP(R") is the required operator satisfying (2).

3.7. Extension Operators for Particular Domains

197

We now turn to extension from R' \ Q() onto R" for QW in Fig. 22. Let .F : VP(R" \ Q(--))

Vp(Rn)

be an arbitrary extension operator. We claim that inf II.FII - E-1+1/p

A lower bound for II.FII is verified with the aid of a smooth trial function which

is equal to 1 on the set

{(x', xn) : 4/3 < x'I < 5/3, Xn E (-1, -E)}

and 0 over the "hat brim". The argument is quite similar to that in Theorem 3.4.2 (i). To establish (5), we have to construct an extension operator F as in (4)

with II.FII <

cc-'+'IP.

Clearly, there is a linear extension operator

F1:Vp(Rn\1 )) -* Vp(Rn\Gie)) with norm uniformly bounded in e (see Theorem 1.6.2). By Theorem 3.4.2, there exists a linear extension operator F2 : Vp(RT \ S2e) - V'(Rn),

1l =

Rn-1

x (_E, 0),

satisfying IIF2II <- cc-1+11P. Given any u E Vp(R' \ QW), define

u1 = (F1u)IRn\se , Then

v E Vp(Rn \ G(E)),

v = Flu - F2u1.

vIR^\il = 0

(6)

(cf. Fig. 23). We now construct a linear map v F-a F3v transforming a function subject to (6) into a function F3v E Vp(Rn) with

F3v = v on Rn \ G (E),

C IIvIIP l Rn\G(E) .

(7)

Once such a map has been defined, the required extension is given by .Fu = F2u1 + F3(Flu - F2u1).

3. Extension of Functions Defined on Parameter Dependent Domains

198

So let v satisfy (6) and let g(E) = B3"-1) x (-e, 0). Applying a radial finite order reflection analogous to (3), we can extend v from the annulus g(e) \G(E) onto a wider annulus g(e) \ h(E), h(E) = {x E G(E) : Ix'I < 2 - 1/l} G(E)

IF-- h(E) -->1

11E

g(e)

Fig. 23

(see Fig. 23). We relabel the extended function as v. Now F3v subject to (7) is defined on G(E) by F3v = 0 on h(E) and F3v = tlv on G(E) \ h(e), where 77

is a smooth function depending only on Ix'I, g = 1 for Ix'I > 2, g = 0 for

Ix'I<2-1/l.

Example 2. We shall check the relations inf

.- min{l, n/P} for lp # n, .

e-og e-orp ,J lI(1P)/P f = n, l

(8)

and

infIIFII - 1

(9)

for arbitrary extension operators

E : v'(G(E)) P

v'(Rn+1), F P

:

V'(Rn+1 \ G(E)) -4 vi(Rn+1), P P

(10)

where n > 1 and G(E) is the (n + l)-dimensional "dumbbell" shown in Fig. 24.

Fig. 24

3.7. Extension Operators for Particular Domains

199

With the aid of a finite partition of unity, the proof of (8), (9) is reduced to the model domain G(E)

= R++1 U {x = (x', xn+l) :

where R++1 = {x E Rn+l

E BEn) Xn+l E (-1, 01 }

xn+l > 0} (see Fig. 25).

2e

Fig. 25

Turning to (8), we observe that a lower bound for IIEII is verified by using

a smooth trial function depending only on xn+1 with support on the rod B(n) x (-1, 0). The argument is similar to that in Theorem 3.6.2 (i). We now construct a linear extension operator u --* Eu whose norm is equivalent to the right part of (8). Clearly, the general case u E VP(G(E)) is reduced

to the case u(x) = 0 for x E R++1 We may assume that the rod has the form x (-oo, 0) (see Theorem 1.6.1). Let 0E = BE n) x R1 and let E : Vp(QZE) - Vp(Rn+1)

be the extension operator constructed in Theorem 3.6.2 (ii). According to Remark 3.6.2/1, we can assume that Eu(x) = 0 for x E R++1. Putting (Eu)(x) = u(x) for x c SAE U R+ 1 and (Eu)(x) = (Eu)(x) for the remaining points x E Rn+1 one obtains the required extension of u. To prove (9), we should construct a linear extension operator .7 as in (10) with norm uniformly bounded in e. Let De = BEn) x (-1, oo). Applying a finite order reflection (cf. Theorem 1.6.1) across the plane xn+1 = 0 and then multiplying by a smooth cut-off function, one obtains an extension v E VP (Rn+l \ DE) of a function u E VP (Rn+l \ G(r)) such that IItIlp,I,Rn+l\D, C C IJUJIp I Rn+t\G(c)

200

3. Extension of Functions Defined on Parameter Dependent Domains

It remains to extend v from the exterior of D. to R"+1. Let QE = Since there is a linear extension operator:

B(En)

x R1.

V1(Rn+1 \ -QE) 4 VP '(R n+1)

with norm uniformly bounded in a (see Theorem 3.4.1), we can assume that v = 0 in the exterior of f2,. In this case the required extension of v from Rn+1 \ DE to the cylinder DE can be given as a finite order reflection across the plane part {x : x' E B(En), xn+1 = -1} of BDE, multiplied by a smooth cut-off function. 3.7.2. Extension from a Domain Depending on Two Small Parameters

Let b E (0, 1/2) and e E (0, 8/2). We introduce the n-dimensional small and narrow cylinder GE,6 =

B(En-1)

x (-8, 8).

Let

E:Vp(GEVp(Rn)

(1)

denote an arbitrary extension operator. The appearance of the second parameter increases the number of variants. It will be shown that e-1

if lp < n - 1,

e-i (log(8/e))

(1-p)Ip

if lp = n - 1,

if n - 1 < lp < n,

inf IIEII ^

E(1-n)/p8-IIpI

logB1(1-p)Ip

(2)

if lp = n,

E(1-n)/p8-1/p if lp > n.

Suppose that v E VP (Rn), p E [1, oo), and that v(x) = 1 for a.e. x E G. Then v can be approximated in VP by functions in Co (Rn) which equal 1 in a neighborhood of Ge,6 (a simple proof of this fact follows from the starshapeness of GE,6 with respect to the origin). Therefore, the norm of any extension operator E mentioned above satisfies (Cap (0E,6, VP)/mesn(GE,6))1'

3.7. Extension Operators for Particular Domains

201

where the capacity Cap (F; Vp) is defined for any compact F C R' as inf {IIuHHp,l,Rn

:

u E Co (R"), u = 1 in a neighborhood of F}.

(3)

If F C B1, one can easily show with the aid of a smooth cut-off function supported in B2 and equal 1 in B1 that Cap (F; V) is comparable to another capacity Cap (F; L1P(B2))

= inf {IIVzullp,B2 : u E Co (B2), u = 1 in a neighborhood of F}. Hence c IIEII ?

(E1-nb-'Cap

(Ge,6;

Lp(B2)))11P

We now refer the reader to the book by Maz'ya [136] (Sec. 9.1) where it was shown, in particular, that the right part of the last inequality is comparable to the right part of (2). This means that relation (2) can be written shorter though less explicitly as inf IIEII - (Cap (Ge,6;

1/p, 1 < p < oo.

To prove (2), we should construct a linear extension operator E as in (1) with norm dominated by a constant times the right part of (2). We need some lemmas (which will be also used in Chapter 5).

Lemma 1. There exists a linear map Ll (Ge,6) 3 U H P E Pl_1 such that

IIVk(u - P)IIP,co,a < c

61-klloluIJRGe.s,

k < 1.

(4)

Proof. It will suffice to verify (4) for l = 1 (see Lemma 1.5.2). Let u c LP(Ge,6) and let u be the mean value of u on G. We denote by u(t), ain fact, the infimum in (3) is comparable to the same infimum over the set {u E Co (R^) : UIF > 1}. See Maz'ya [128], [129], [136, 9.3] for p > 1, Netrusov [163], Carlsson and Maz'ya [42] for p = 1.

3. Extension of Functions Defined on Parameter Dependent Domains

202

t E (-b, b), the mean value of the function BE('-')

x' H u(x', t) on

BEn-1)

Clearly IIu - ujlp,GE,e C cc(n-1)iPllu - uHIP,(-6,6) 1/p

+I J6 6 IIu(',xn) - u(xn)Il,B,dxn)

.

(5)

Since ii is the mean value of the function (-b, b) E) xn H u(xn), we have IIu - ullp,(-6,6) < C bllu IIP,(-6,6)

Furthermore, by Holder's inequality

IB (xn)I = IV._

-1)

Hence the first term on the right of (5) does not exceed cb lloulip,G,,,. Ap1), plying the Poincare inequality to the function u(., xn) in we dominate B(En

the second term on the right of (5) by cE IIDullP,G,,s. Now (4) follows for 1 = 1

and P = u.

I Bbn-1) x (_b,

Lemma 2. Let G6 =

b) C Rn and suppose P c p(n)

The

following estimate holds !-1 C(bE-1)(n-1)/PE

llPlIP,Gs

k=0

Proof. If S is a polynomial in Rn-1, then IS(O)l <- C

E(1-n)"PIISIIP,B.(n-1),

(6)

where c depends only on p, n and the degree of the polynomial. Fix a z c

(-b, b). Using (6), we obtain IIP(-, z) llp,B(n-1) <- C E I DaP(0, z) Iblal+(n-1)/P lal<1-1

<

c(b/E)(n-1)IP E blal lIDaP(.,

laIU-1

z)II p,B

3.7. Extension Operators for Particular Domains

203

z) denotes the differentiation with respect to first n-1 variables. The result follows by integration over z E (-6, 6). 1 where

Lemma 3. Let GE,6 = B(en-1) x (-6, 6), e E (0, 6/2). If lp # n - 1, then there is a linear extension operator EE,6 : VP (GE,6) _+ Vp (Rn)

satisfying IIVi (EE,6u) IIP,Rn (6E_1)min{l,(n-1)/P}


i

E 6k-illVkujIp,Ge,b, i < 1, k=0

for any u E VP (GE,6). In case Ip = n-1 there exists a linear extension operator EE,6 acting as above, such that IIVi(EE,6u)IIP,Rn

< c (6E-1)' (log (6E-1)) (1-P)/P E 6k-i II VkuIl p,Gc.6,

i<1,

k=0

for all u E VP (GE,6)

Proof. With the aid of the transformation Rn E) x H O(x) = x/6 we reduce consideration to the case 6 = 1. Clearly there is a linear extension operator: VP (GE,1)

VP (f2E),

QE =

Ben-1) X R1,

with norm uniformly bounded in e (see e.g. Theorem 1.6.1). Therefore, the desired result follows by reference to Theorem 3.6.2 for p = oo. Hence, in each

case lp # n - 1 or lp = n - 1 there exists a linear extension operator EE : VP(GE,1) -4 Vp(Rn)

subject to cE-min{!,(n-1)/p}

116-11 5

if lp # n - 1,

IIEEII < c,--'l logEl(1-P)/P

if lp = n - 1.

The required operator EE,6 can be defined by EE,6 u = (EE/6(u o 0-1)) o . 1

3. Extension of Functions Defined on Parameter Dependent Domains

204

Lemma 4. Let G6 be the same cylinder as in Lemma 2. If lp > n - 1, then there exists a linear extension operator 4,j: VP (GE,6)

whose norm is dominated by c

VP (G6)

(6/E)(n-1)/p

Proof. According to Lemma 3, there is a linear extension operator VP (Ge,6) E) u H Ee,6u E Vp (Rn) such that

I-1 (6E-1)(n-1)/P [, 8k-il1VkUIIP,G..6,

IIVi(Ee,6U)IIP,Rn < C

(7)

k=0

where i < 1. Let Vp (GE,6) D U H P E PI-1

be a linear map satisfying (4). Combining (4) and (7), we find (8E-1)(n-1)/Pat-iJjVIUIIP,G8.6'

IIVi(Ee,6(u - P))IIP,Rn < C

(8)

Next, by Lemma 2,

IIViPIIP,G6 <- c

(6E-1)(n-1)/P E

69IIVi+8PIIP,GE,6.

8=0

Since IIV +8PIIP,GE.6 5

IIVi+eullp,GE,6 +c

IIVIUIIP,Ge.6,

it follows that I

IIV1PIIP,G6 <- c

(6/e)(n-1)/P

i < 1. k=i

The last estimate along with (8) implies that the required extension operator can be defined by

4'6u = P + Ee,6(u - P). This establishes Lemma 4.

Comments to Chapter 3

205

We now turn to construction of a linear extension operator E acting as in (1), whose norm is majorized by a constant times the right part of (2).

1. lp < n - 1. By Lemma 3.1.2/1, there exists a linear extension operator E for which IIEII < cE-1.

2. In each case lp = n - 1 or n - 1 < lp < n we can put E = EE,6, where EE,6 is the extension operator constructed in Lemma 3. Then IIEII < cE-1 (log(8/E))(1-P)/P

if lp = n - 1,

IIEII < C6-l(6/E)(n-1)/P if n - 1 < lp < n.

3. lp = n. Let G6 be the cylinder introduced in Lemma 2 and let

.F:Up(G6)-1Vp(Rn)

(9)

be a linear extension operator satisfying II.FII < c 61 log 6I(1-P)/P

(cf. Theorem 3.1.5). If EE,6 is the operator constructed in Lemma 4, then E = F E,,6 : VP(GE,6) -+ vp(Rn)

(10)

is the required extension operator with (6E-1)(n-1)/P6-1I log 61(1-P)/P

IIEII < c

4. lp > n. By Theorem 3.1.5, there exists an extension operator acting as in (9) such that IIEII < c6-n/P. Let EE,6 be the extension operator constructed in Lemma 4 and let E be defined by (10). Then IIEII < C

E(1-n)/P6-1/P

The last case does not exclude p = oo.

Comments to Chapter 3 Lemma 3.1.2/1 is well-known (see Sobolev [190], Smirnov [183]). The contents of Sec. 3.1.3-3.1.5 and Sec. 3.2 are taken from the paper by Maz'ya and Poborchi [143]. Theorem 3.3/1 was proved in the same paper.

3. Extension of Functions Defined on Parameter Dependent Domains

206

Recently Kalyabin has shown that if Q C R2 is a convex domain with d = diam (0) < 1 and if E : WP '(Q) -* WP '(R2) is an arbitrary extension operator, then QI-1/Pd(2-P)/P

inf IIEII

for p E (1, 2),

ISZI-1/21/2 (log(2/d)) IQI-1/P

for p = 2,

for p > 2,

where IQI is the area of 92 and the constants in this relation depend only on p (private communication). The following results on minimal norms of extension operators have been obtained by Burenkov and Gorbunov [38]. Let (a, b) be a finite interval in

R1 and E an arbitrary bounded extension operator: WP (a, b) -a Wp(R1), 1 < p < oo, l = 1, 2, .... Then there are absolute constants c1i cz > 0 such that ci (1 + 11(b - a)1-1-1/P) < inf IIEII < cz (1 + l1(b - a)1-1-1/P).

Let Q C R" be either a bounded domain in C°'1 or a special Lipschitz domain (cf. 1.6.2). If E is an arbitrary extension operator: Wp(S2) --+ W1 (RI), then there are constants c3(1), c4 (S2) > 0 such that c3(S2)1l1 < inf IIEII < c4(Q)1l1

c3(S1)1 < inf IIEII < c4(Q)1

for bounded S2;

for special Lipschitz Q.

In case Q. is a thin n-dimensional cylinder of the form w x R1, the asymp-

totic behavior of the minimal norm of an extension operator: V2 (Q-) 4 VZ (R") as e -- 0 was obtained by Maz'ya [135]. We follow this paper in Sec. 3.3.

The material of Sec. 3.4-3.7 is borrowed from the work by Maz'ya and Poborchi [153]. Two-sided estimates for extension operators from some particular domains depending on small parameters were given in the survey by Maz'ya [137].

CHAPTER 4

BOUNDARY VALUES OF FUNCTIONS

WITH FIRST DERIVATIVES IN LP

ON PARAMETER DEPENDENT DOMAINS

Introduction Let S2 C R" be a domain in

C°,1 and let TWP (SZ) be the space of the traces

ul., of the functions u E WP (Q). The norm II IITWw(Q) is defined as the norm in the factor space WP(S2)/WP(S2), Wp'(SZ) being the closure of Co (Q) in WP (SZ). According to Gagliardo's theorem [69], TWP (0) = WP-1/p(aQ) for

p E (1, oo) and TW1 (Q) = L1(8S2). Here WP-1/p(8SI) is the space consisting of functions on 8SZ with finite norm IIfIILp(8fl) +

(

ff

dsy ds If(X) - f(y)IP Ix _ yln+P-2

8SZ x 8SZ

l

1/p

I

where dsy, dsy are the area elements on o9Q.

Let Q. be a domain in C°,1 depending on a small parameter e in such a way that the limit domain limE,o Q, is non-Lipschitz. Then the norms II

' IITW;(SZa)

and

II

II

' IITW1 (SZe)

and

II

IIWPI-'"D(on )' -

p E (1, oo),

IILl(OQ,)

need not be equivalent uniformly with respect to E. In this chapter we find an explicitly described E-dependent norm of a function on 3 such that this norm is equivalent to I II Tw1(c ,) uniformly in E for certain domains depending on parameters. Among them there are small domains and narrow (of width E) cylinders or their exteriors. Theorems on small domains and narrow cylinders will be applied in Chapter 7 to characterize the space TWP (0) for a domain with the vertex of a peak on the boundary. I

207

-

4. Boundary Values of Functions with First Derivatives ...

208

We now describe some results obtained in this chapter. Let SZ C Rn be a simply connected domain of class C°,1 with compact closure. For a small positive e put Q. = E Q. Then the trace norm II f II TWD (c,) turns out to be equivalent (uniformly in e) to the norm = a(e)II f II Ln(aQc)

(J )P,o

1/p

If

I f (x) - f (y) I'QE (x, y)dsxdsy1

I

+(\ 8Ste x 8c2

,

/

where dsx, dsy are the area elements of the surface a Q., a(e) = el/P and the weight function QE is defined by

r2-n-p if Q. (x, y) with

El-n

p E (1, oo),

if p = 1

r=Ix - yl.

The norm Ilf II TWP (Rn\sie) is also equivalent to (f )P,as1E where

min{e(1-P)/P, E(1-n)/P} if p # n, a(e) =

{

logel)(1-P)/P

l (EI

if

p = n,

Qf(x, y) =0 for p= 1 and QE(x, y) =r 2-n-p for p E (1, 00). Let Q,, be a narrow cylinder of the form Q. = u),, x R1 C Rn, where

n > 2, w = e w and w C

Rn-1 is

C°' 1. Then the trace norm IIf II TWl

0 (x. U) _

a bounded simply connected domain in is equivalent to (f )p,an, if a(e) = El/p,

r2-n-' (r/e) for p E (1, oo), e`-"X(r/e) for p = 1.

Here X is the characteristic function of the interval (0, 1). For the exterior of a narrow cylinder, the trace norm IIfIITwD(Rn\?j,) is

equivalent to (f )P,an, with a(e)

min{E(1-P)/P

I el

E(2-n)/P} if p :A n - 1,

(1-P)/P logel)(

if p = n - 1

4.1. Traces on Small and Large Components of a Boundary

209

and

0

if p = 1, r2-n-p

Q. (X, y) _

if p E (1, n - 1),

r2-n-P + e2(2-n)r-1 (log(1 +

I

r2-n-p

+

e2(2-n)rn-2-p

r/e))-p

if p = n - 1,

if p E (n - 1, oo).

In all cases mentioned above an extension operator: TWP(52E) -4 WP (S2E) (or TWP(Rn \ 52E) -4 WP (Rn \ 52E)) exists with norm uniformly bounded in E. This operator is linear for p > 1 and nonlinear for p = 1.

4.1. Traces on Small and Large Components of a Boundary .4.1.1. Gagliardo's Theorem and its Consequences

Let 52 C Rn be a domain whose boundary can be locally represented as the graph of a uniformly Lipschitz function. Let TWP (52) denote the set of the traces Ulan for u c WP(52) (cf. Exercises 1.12-1.14). The space TWP(Sl) is endowed with the norm IIfIITw;(si) =inf{Ilullwp(n) : u E WP(52), Ulan = f}.

(1)

In a similar way we can define the space TLp(5l) and introduce the seminorm Ill IITLp(si) = inf{IIVutlLp(n) : u E Lp(s2), ulan = f }.

(2)

Let S be a measurable subset of 852. For a function f defined on S, we set dsxds

[f]P,Sff If(x)-f(y)IPIx-yln+P-2)

1/P

pE(1,oo),

(3)

SXS

where dsx, dsy are the area elements on S. The space WP functions f E Lp(S) having the finite norm

1P (S) consists of

IIf II wn-lip(s) = IIf II Lp(S) + (f]p,S, p E (1, no).

(4)

4. Boundary Values of Functions with First Derivatives ...

210

Here LP(S) is the space of functions defined on S and pth-summable with respect to the area. The following result is due to Gagliardo [69]. Theorem. Let 1 C R" be a bounded simply connected domain of class Co,' If S = 852, then the spaces WP'-'IP(S),

TWP (52)

and

TW1(52)

and L1(S)

p E (1, oo),

coincide with equivalence of norms. Furthermore, there are bounded extension operators

E: WP-'IP(S)-4 WP(R"),

1
and

E : L1(S) -> Wi(R"). The operator E is linear for p E (1, oo) and nonlinear for p = 1.

1

We now make a simple observation on extension operators with dilation.

Remark. Let S and E be as in Theorem. Given e > 0, put S. = e S. Then the formula

f -a EE f = (E(f o

o -D-1

with -I)x = ex, x E R", gives extension operators

EE : WP-'IP(SE) -* W(R) and

such that e-'IIEEfIILP(Rn) + IIVEEfIIL,(Rn)

< c(n, p, S)

\e(1-P)IPII

p E (1, 00),

(5)

e-lIIEEf IIL1(R^) + IIVEef IIL,(Rn) < c(n, S)IIf II L,(sc).

(6)

f II L"(S-) + [f]P,s.)

,

and

l(7)

We now state the "Poincare inequality" for functions defined on surfaces. Let S be a measurable subset of the boundary of a domain in C°" with positive area ISI. For f E L1(S), we introduce the seminorm

[f]l,s=ISI-1

SxS

4.1. 'IYaces on Small and Large Components of a Boundary

211

Lemma. If f c LP(S), 1 < p < oo, then IIf - f UL,(S) <_ [f]1,S, S)"+P-21S1-1[f]P,s,

IIf - f Ilip(s) < (diam

p > 1,

where f is the mean value off on S: 1 =

ISI-1 f f(x)dsx.

s

The proof of this lemma is easily carried out with the aid of Holder's inequality. Combining this lemma with (5) and (6), we arrive at the following assertion. Corollary. Let S be as in the above theorem and let S. = eS, e > 0. Suppose that EE : WP-11' (SE) - WP (R"), P E (1, oo), is an extension operator subject to (5) or

E. : L1(S.) - Wi (R") is an extension operator satisfying (6). Then e-1IIEc(f - f)IILp(Rn) + IIVEE(f - f)II L,(Rn) < e(n, S,p)[f]P,s,,

where 11, f EL1(S.)forp=1, f isthe mean value off on SE and

the seminorm given by (3) or (7).

.4.1.2. The Interior of a Small and Large Domain

Let SZ C R" (n > 2) be a bounded simply connected domain in C°". For e > 0 let Q = e Q. In the present subsection we indicate an e-dependent norm in the space WP-11P(a ), which is equivalent to the norm II' IITwp(n,) uniformly in E. The symbols c, co, c1, ... designate positive constants depending only on n, p,1 Q. The equivalence of positive quantities (denoted a - b) is meant in the sense that their ratios are bounded above and below by such constants.

4. Boundary Values of Functions with First Derivatives ...

212

Theorem. If p E [1, oo) and E E (0, 1), then the following relations hold

IIfIITWP(n.)-E"'IIfIIL,(a )+[f]p,8I I

where

,

I f IITL - (nE) - [f]P,BnE,

(1) (2)

is the seminorm given by (4.1.1/3) or (4.1.1/7).

Proof. Let u E Wp (Q,), u1en, = f. It will suffice to establish the estimate [f]P,an, < c

(3)

for the case E = 1 and 521 = SZ because the general case follows by using a similarity transformation. Consider the mean value f of f on 852. An application of Theorem 4.1.1 and Theorem 1.5.4 gives [f]p,an = [f - f]P,an 5 c IIu - JII ww(n) < c1IIVuIIL,(n)

Thus, (3) is true. The inequality E'/PII f IIL,(en,)

C (IIuIILp(na) + EII VuIIL,(n.))

(4)

follows from the estimate IIVIILp(an) <_ CIIVIIWi(n)

by contraction. A combination of (3) and (4) yields IIfIITL,(n,) 2 c[f]p,8Q,, IIf II TWP (n,) 2 C (El/PIIfIIL,(a

) + [f ]p,ocE)

To prove the reverse inequalities, one should construct extension operators E : WP-'IP(852E) -- WP (cu) for p > 1 or E : L1(OQ,) -4 Wi (QE) satisfying IIVEfIILp(n,) C IIEf

S El/PIIf IIL,(en.) + [f

(5) (6)

4.1. Traces on Small and Large Components of a Boundary

213

We introduce bounded extension operators WP (Rn) for p > 1 or EE : L1(812E)

E£ : Wp-'/P (aQE)

Wi (R").

subject to (4.1.1/5) or to (4.1.1/6), where S = 812, SE = a12,, (see Remark 4.1.1). Let f be the mean value off on 812E and let

Ef=f+EE(f-f). We claim that E is the required extension operator. Indeed, (5) follows from Corollary 4.1.1, and (6) follows from the same Corollary and the estimate

if

IPmesn(Qe) < CeIIf II L,(0Q,)-

This concludes the proof of relations (1) and (2).

1

Remark 1. Let 0 be the same as in Theorem. If o > 1 and )Q = e I, then by using a smooth partition of unity subordinate to a covering of 812,, by a finite collection of open balls, we can establish the relation IIfIIW;-'1p(an.) ifP> 1, IIfII Twp(nQ) _

(7)

IIf IIL1(aS2,) ifP =1.

Remark 2. Relations analogous to (1), (2) and (7) can be written for any connected component of the boundary of SZ,, = ell in case SZ E C°"1 is bounded

and multiply connected. For example, if Se is such a component, then

inf{IIuIIwi(n,) : uls, = fl min{el"P,1}IIfIILp(S,,) + [fJP,so, e > 0, p E [1, 00)

These relations can be verified in the same way as (1) and (7).

4.1.3. The Exterior of a Small Domain Let 12 and G be bounded simply connected domains in Rn, n > 2. Suppose that these domains are in the class C°'1 and that G contains the origin. Let e, a be positive parameters and let 0, = e 12, Go = g G, 12E C G,,. The case e = oo is also taken into consideration and then Goo = Rn. In the spaces

4. Boundary Values of Functions with First Derivatives ...

214

Wp-1/p(8S2E), Wp-1/p(8Ge) of the traces of the functions in WP (Ge \ 52e) we construct e, e-dependent norms uniformly equivalent to the factor-norms

(f)p,ac. =

uIas4 = f},

(1)

(f)p,aG,, = inf{IIu1Iw;(G,\nE) nlaGQ = f}.

(2)

In what follows c, co, cl,... are positive constants depending only on n, p, St, G.

By definition a '- b if cl < a/b < c2. Theorem. Let e E (0, 1/2). If fi, C G. and dist (SZef 8Ge) > c0E, then (f)p,OG0 - min{e1/p, 1}IIf II Lp(aGQ) + [f]P,aGQ, e < 00,

(3)

(f)p,aj2e ,,, a(E, 011f IIL,(aG,) + [f]p,ooE,

(4)

are given by (1), (2), is the seminorm defined by (4.1.1/3) or (4.1.1/7) and a(e, e) is determined by where p E (1, 00),

(1-n)/P min{en/P, En/P-1} if p < n, E(1-n)/p min{e, I logej(1-P)/P} C

a(E, e) =

if p = n,

IE (1-n)/p min{en/P, 1}

if p > n.

Proof. By Theorem 4.1.3, there exists a linear extension operator:

with norm uniformly bounded in E, e. Hence (f)p,aG, - IIfIITW,(GQ),

and (3) is a consequence of Theorem 4.1.2 (cf. also Remark 4.1.2/1). We now turn to (4). Fix positive numbers A, d such that SZ c BA, G 1 Bd and consider several cases. 1) We begin with the case Me > de. Note that then E e since K!, C GO implies e < c e. Under these conditions, relation (4) follows from Theorem 4.1.2 and Remark 4.1.2/2 (here the role of the domain SZ in Theorem 4.1.2 is played

by the domain Gee_' \ M.

4.1. Traces on Small and Large Components of a Boundary

215

To facilitate the remaining argument, we introduce a bounded domain g c R' of class C°'1 independent of e, p and satisfying fZ C g C G,-1 (this domain exists because dist (0, 8GeE-1) > co). Put gE = e g. 2) Let 2)e > 1. Then there is a linear extension operator: W1

-

- Wp(R")

whose norm is uniformly bounded in a (see e.g. Lemma 4.1.2/1). In this case (f)P,ac - inf{IIulIWD(9C\?jC) : ul a.E =

f},

and (4) is again a consequence of Theorem 4.1.2. 3) Suppose that tae < min{1, gd}. Let

uEWP(Go\1 ),

=f

The inequality [f JP,ac. <_ C IIuIIWy (GQ\ L)

follows from the estimate 1fJP,8nc < CIIVUIIL,(gc\fi,)1

which is provable in the same way as (4.1.2/3). Let us verify the inequality a(e, e) II f II Lp(80E) _< C IIUII W; (G,\?i,).

(5)

Note that ne C BaE C B2Ae C Bde C Gg.

From the estimate C IIVIILp(af) <_ IIVIILp(aBA) + IVvMILp(BA\i ), v E WP (Ba \ Q),

by means of contraction we obtain cIIf

IILp(8s

) <_

e1-'IpIIVUIILn(Ba,\n,)

+

Now, since a(e, Lo)e1-1/P < c, (5) is reduced to the estimate a(e, 0 IItIILP(aBA,) _< C II uIIWW (Bde\B)A

(6)

4. Boundary Values of Functions with First Derivatives ...

216

We may assume without restriction of generality that d = A = 1. So let 2E < min{ 1, Q}, U E Wp (Be \ BE) and let u(x) = 0 for IxI > 1 if Q > 1. Passing to spherical coordinates x = (r, 0), one can write

Q

e

e

2

u(e,0) -

f /2

u(r, O)dr

<

Iur(r,B)Idr, 0 E Si-1, E

whence

re

CIu(E,O)IP < Q-n]Iu(r,0)IPr''-'dr, e/2

r

+J

drP

(P-n)/(P-1)

elur(r,0)IPri-1dr,

\`

E

1

r)

(if p = 1, the last factor should be replaced by el-n). Integrating over the sphere Sn-1, we arrive at Cel-nI uIILn(8Be) <- Q-nIIuIILP(Be\Bo) A

+

= max 11, eP-n } for p # n and A = I log a IP-1 for p = n. Thus, min{Q"/P,A-1/P}IIUIILP(8BE)

E(1-n)/P

CIIuIIWPI(BPB,)

Therefore, estimates (6) and (5) hold. It is shown that c (f )P,ao < a(e, Q) II f II

[f ]P,ao .

To validate the reverse inequality in the case 3), one should construct

bounded extension operators E : Wp-1/P(BSZE) -+ Wp (Ge \ QE) for p > 1 or E : L1(ac ) _+ Wi (Ge \ E) satisfying C IIEf II Wi(G,\?iE) < a(e, Q) II f II LP(8n,) + [f]P,ooE, 1 < P < oo.

First we define auxiliary extension operators Ei : Wp-1/P(C7QE) -1 WP (Ge \ E)f P E (1, oo), 1 < i < 4,

(7)

4.1. Traces on Small and Large Components of a Boundary

217

(or Es : L1(852E) -* Wi (G , \ S2E) for p = 1) such that


where 1 < i < 4, 1 < p < oo, a1 =

E(1-P)IP,

a2 = (E1-nen)1/p' a3 =

E(1-n)/P, a4 = (EI logEl)(1-P)/P (the last for p = n). By Remark 4.1.1, there exists an extension operator f i-+ Ef from 852E to

Rn subject to E-IIIEefIIL9(Rn) + IIVEef IIL,,(Rn)

< C (E(1-P)IP)IIf IILp(ac) + [f]P,ac ), 1 < p < 00.

Thus, we may put Si = E. Next, it is possible to set Elf = f + EE(f - f),

f being the mean value of f on 852,. The required estimate for E2 (if P < oo) follows from Corollary 4.1.1 and the inequality If IPmeSn(GQ)

Let 0 be a function in Co (B1), IIfOIIW, (Rn) <

Furthermore, (-f 0 + EE(f - 7)) lass. one may put

V)IBIIa

= 1. We have

CE(1-n)/PIIf

=f

ULp(8Q )

because 52E C Baf C B1/2. Hence

E3f =f'+EE(f -f). Let p = n and let 1

W(x) =

if x E BAE,

log Ix/AI/ loge if x E Ba \ BAe,

0 ifxER"\BA. It is easily verified that cp E WP (Rn) and II0IIwp(Rn) < c I

logEl(1-P)/P.

4. Boundary Values of Functions with First Derivatives .. .

218 Since

IfIII'pIIW;(R') + IIEE(f - f)IIW;(Rs) < c (EI logE1)(1-P)/PIIfII Lp(an2) +

we can set

E4f =Jco+EE(f -f). An operator E satisfying (7) can be defined as follows. In case p < n put

E = E1 for e" > E"-P and E = E2 otherwise. If p = n, we may choose E = E4 for Pr > I log E I 1-P and E = E2 otherwise.

Finally, when p > n, one can define

E = E3 for e > 1 and E = E2 for oo < 1. The theorem is proved.

0

For e = oo we obtain the following assertion.

Corollary. If E E (0,1/2), then IIf IITW, (Rn\si,) - a(E) IIf II Lp(asi.) + [f ]P,an

where

,

is the same as in Theorem and a(e) _

min{E(1-n)/P, E(1-P)/P} for p # n, (El logel)(1-P)/P for p = n.

We point out one more property of the traces of functions defined on R"\Q

Remark. Let SZE be as in Theorem. The following relation is valid IIfIITL'(Rn\jje) - [f]P,en,, p E [1,oc)

Indeed, let

uELP(R"\1E),

ulase=f.

1

.

4.2. On the 'IYace Space for a Narrow Cylinder

219

Fix r > 0 such that S2 C Br. The estimate [f]P,8Q- < C II VUIIL,(Br.\iie)

is provable in the same way as (4.1.2/3). Hence IIfIITLP(Rn\cl)

c[f],,cz

.

The reverse inequality is a consequence of the estimate IIVE2fIIL,(Rn) 5 C[f]P,8ns,

where f H E2 f is the extension from 8QE to Rn constructed in the above theorem.

4.2. On the Trace Space for a Narrow Cylinder Here we consider a narrow cylinder (of width e) and study the problem analogous to that in Sec. 4.1.2. To a function defined on the boundary of a cylinder, we correspond an explicitly described norm (or seminorm) which is equivalent to the factor-norm of the form (4.1.1/1) (or to the seminorm (4.1.1/2)) uniformly with respect to E. Results on a narrow cylinder are applied in Sec. 4.2.3 to describe the trace space TWP for an infinite funnel.

4.2.1. An Explicit Norm in the Race Space for a Narrow Cylinder

Let w C Rs-1 (n > 3) be a domain in CO,' with compact closure and connected boundary ry. For the simplicity of presentation, we assume in what follows that w C B(n-1) Let SZ denote the cylinder

52={x=(y,z)ER": YEW, zER1} and put F = 852. Given a positive parameter E, we equip Wp (52) with the E-dependent norm IIUIIWD(n,E) = E IIUIILD(n) + IIVuIIL,(n), p

which induces the factor-norm

Oun1en=f}

IIfIITW;(n,c)=inf{IIuIIWi(c

(2)

4. Boundary Values of Functions with First Derivatives ...

220

in the space of the traces ul8S2 of functions u E WP (S2).

In this subsection c, co, c1, ... designate positive constants depending only on n, p, w. By definition a - b if co < a/b < c1. The following theorem gives an explicitly described norm of a function on r equivalent to the norm (2) uniformly in e E (0, 1].

Theorem 1. The relations IIfIITLP(0) - IfIp,r,

(3)

IIf IITW, (n,e) - e IIf IILp(r) + If Ip,r

(4)

are valid, where p E [1, oo), e E (0,1],

If IP,r =

C

1/P dsds If(x)-f(OPIx_CIn P-2 , p> 1,

ff

(5)

{x,£Er:Ic-zl<1}

ff

If I1,r =

If (x)

-f

I dsxds£,

(6)

{x,£Er:IS-zj<1}

x = (y, z),

_ (77, () and dsx, dsC are the area elements on F.

Proof. Let u E LP(S2), ul r =

f

.

Put rk = ry x (k - 1, k + 2). Then

fk+lf7J r -- E {Er:IC-zl<1} °°

If

PP,

v

k=-00

If(x)-f(S)IP ds

Ix - IniP-2

CO

[f]P,rk, p E (1,00).0

k=-oo

is the seminorm defined by (4.1.1/3). Let 52k = c x (k - 1, k + 2). It follows from Theorem 4.1.2/1 that Here

[f1P

rk -

IIvuIILp(Ilk), k c Z.

Summing over k c Z, we arrive at If Ip,r < c IIVuIIL,,(fl)

(7)

4.2. On the Trace Space for a Narrow Cylinder

221

The case p = 1 is treated in a similar way so that (7) is true for p > 1. Now suppose u E WP (1), u I r = f . Since If IIL,(rk) 5 C IIUIIwp (S2k), k E Z, p > 1,

one has

IIfIIL,(r) 5 cIIulIwi(n) Thus, the following estimates hold If IP,r 5 C IIf IITLP(f ) and


To validate the reverse inequalities, we construct an extension E f of a function f from surface r into SZ such that IIVEfIIL,(n) 5 cIfIP,r,

(8)

IIEf II Wp (n,c) 5 C (c IIf IIL,(r) + If IP,r)

(9)

Construction of the extension f -> Ef. Let 0k = ((k - 1)/2, (k + 1)/2), k E Z. Consider a smooth partition of unity {µk}k _, for R' subordinate to the covering {.rk}. In addition, let Ak E Ca (Uk), Akµk = Pk, k E Z.

We can assume that dist (supp Ak, R1 \ o'k) > CO

and that Iµk(z)I, IA (z)I < c for all k c Z and z E R1. Suppose f E LP io,(r) be given with If IP,r < oo. Let f k denote the mean value of f on the surface Sk ='Y X Qk

Define fk(x) = Ak(z)(f(x) - fk), x = (y,z) E Sk.

By Theorem 4.1.1, there is an extension Ek fk of the function fk from Sk to

R" subject to IIEkfkIIWy(R") < C (IIfkIIL,(Sk) + [fk]P,Sk)

(10)

4. Boundary Values of Functions with First Derivatives ...

222

(if p = 1, the last term may be omitted). We claim that the required extension operator f i- E f can be determined by

(Ef)(x) = E fkµk(x) +

µk(z)(Ekfk)(x),

(11)

k=-oo

k=-oo

where x = (y, z) E Q. Indeed, the identity E f Ir = f follows from (11). Let v(x) denote the first sum on the right in (11). Clearly VIGk

= fk+1 + (fk - fk+1)Pk,

Gk = w x (k/2, (k + 1)/2),

whence

IIVVIIL,(sl) = > Ilk - fk+IIPIIVpkIIL,,(Gk) k

< C E IIf - fkIIL,(Sk) < Cl E[f}P,sk k

(12)

k

Here Lemma 4.1.1 has been used at the last step. Note that [f lp,sk


dsx k

If (x) - f (S) IP

J "Er:jc-zj<1}

dst

Ix - SIn+p -2'

p> 1,

and therefore the last sum in (12) does not exceed c If Ip r. Thus, (13)

II VvIILn(cl) _< c If IP,r.

The same reasoning gives (13) for p = 1 as well. Let w(x) be the second sum in (11). With the aid of (10) we obtain IIWIIWD(o) < C

(IIfkIIL,(Sk) + [fk] sk)

(14)

k

(the term [fk]p Sk can be omitted in case p = 1). If p > 1, then [fkl p,sk

where

I1=

f

IAk(z)IPdsx

Sk

- c (II + I2),

f

eIn+P-2

Sk

12= f kIf(S)-fkIPds C

f

Sk

dsx

IAk(z)-Ak(()IP Ix - CIn+P-2'

4.2. On the 'Trace Space for a Narrow Cylinder

x = (y, z),

223

_ (77, (). Since cI( - zI,

IAk(z) - Ak(()I

the last integral over Sk is uniformly bounded with respect to k E Z, ( E Sk. Hence and from Lemma 4.1.1 follows the estimate I2 < c [ f ]P It is also clear that Il < c [f ]P Sk . Therefore, Sk.

[fk]p,sk 5 c [f ]P,Sk.

Furthermore, by Lemma 4.1.1, I IA llLP(Sk)

C[f]P,Sk, P j 1.

Combining two last estimates and (13), (14), we arrive at IIwIIw, (si)

< C Elf]

P,Sk

(15)

< CI If IP,r, P ? 1.

k

Now (13) and (15) imply (8). Let

f E LP(r),

If IP,r < oo.

In this case inequality (9) is to be checked. According to (13) and (15), (9) follows from the chain IIVIILP(Sl) < Cl E IfkVP < C2 E III IILP(Sk) <- C k

IIfIILP(r)

k

The proof of Theorem 1 is complete.

I

Remark 1. Formula (11) makes sense for x E Rn. One can observe from the proof of estimates (8) and (9) that these estimates remain valid if f is replaced by a wider cylinder B(n-l) x Rl. Remark 2. If w E C°'1 is multiply connected and r a connected component of 852, then

inf{IIVuIILP(n):uIr= f} - IfIP,r and

inf { IIuiIwn (D,E) : uir = f j - e IIf IILP(r) + If IP,r.

4. Boundary Values of Functions with First Derivatives

224

These relations can be proved in the same way as (3), (4).

Remark 3. If a E (0, oo), then

ff

If IP,r -

9(x, )dsds,

(16)

{xEEr::C-zI
where

9(x, £) =

If(x) - f

I PI x

-

I2-n-P if P > 1,

f (x) - f (f) I if p = 1,

and the constants in equivalence relation (16) depend on n, p, w, a. To establish this assertion, one should repeat the argument of Theorem 1 leading to (3) with

rk={xEr z E ((k-1)a,(k+2)a)} and 0k = ((k - 1)a/2, (k + 1)a/2).

1

Consider a narrow cylinder 1 = e 0 and put F = BSZE. The following trace theorem is obtained from Theorem 1 by using a similarity transformation.

Theorem 2. If p E [1, oo) and e E (0, 1), then the relations (17)

IIlIITL ,(12) N If IP,r.,E, II f IITW;(1E) ^

are valid, where II

IfIP,rc,E

_

C

If

x = (y, z),

. IITWD(Qt ),

ff

II

eliPll f

(18)

If IP,r.,E

IITLp(n,) are defined by (4.1.1/1-2), dsxds

1/P

If(x)-f(f)IPIx-_In p-2

, p> 1,

(19)

{x,eEr.:IS-zI
ff

If (x) - f (f) I dsxds£,

_ (77, () and dsx, ds are the area elements on r,:.

(20)

4.2. On the 'Dace Space for a Narrow Cylinder

225

4.2.2. Equivalent Seminorms Lemma stated below suggests an equivalent "coordinate-free" writing for seminorms (4.2.1/5-6).

Lemma. Let f E Lp,10c(F), p E [1, oo). Then If IP,r ,.,

ff

g(x, C)dsdst,

(1)

where I Ip,r is given by (4.2.1/5-6) and g(x, C) is the same as in Remark 4.2.1/3.

Proof. By the definition of I Ip,r, the right part of (1) is not greater than the left part. Thus, according to Remark 4.2.1/3, relation (1) is a consequence of the estimate c

ff g(x,C)dsxds£

<

(2)

Jf

{x,(Er:lx-(I<1} where

a={(x,C):x,CEr, IC - zI < 1/6}, x = (y, z),

(77, (), y, 77 E ry = 8w, z, C E R'. We express the left side of (2) as the sum of two integrals, the first being taken over the set {(x, C) E o,

: ly - iI < 5/6}

and the second over the complementary set. In the first case Ix - CI < 1 and the corresponding integral does not exceed the right side of (2). In the second case g(x, C) " If (x) - f (011.

So (2) follows from the inequality which occurs if g(x, C) is replaced by h(x,C)=If(x)-f(C)I1

in (2). Let I'k = ry x ((k - 1)/6, (k + 2)/6), k=0 ,± 1. . . . . Then

ff h(x,C)dsxds£ or

< >

ff

k=-oo 00 rhk

h(x,C)dsxds£. k

(3)

4. Boundary Values of Functions with First Derivatives ...

226

To bound the integral on the right, we construct a covering of the surface y = Ow by a finite collection of open (n-1)-balls {B(i)} 1 centered at y and such that diam (B(3)) = 1/4. Put n BU) rki)

y(i)

= {x

E Fk : y E

y(j)}, j = 1, ... , N.

Clearly N

ff h(x,C)dsxdsC. is=1 rj') xr(>) k k

ff rk xrk

(4)

For each pair (i, j), 1 < i, j < N, there exists a chain of balls B('0),

..., B(+m)

such that

io = i, im = j and B('°) n B(`°+0 # 0, v = 0, ... , m - 1. We have

h(x, ) < 2p-1(h(x, t) + h(t, )) for arbitrary x E 1'k'0), t E I'k'1), C E rk2). Hence, the integral on the right of (4) is dominated by

r(+°) xr('1) k

h(t, )dstds(.

ff

h(x, t)dsydst + c

ff

c

r(u1) k

k

k

Applying such fictitious integration m-1 times more, we find that the general term of the sum in (4) is not greater than m-1 -o

rk'm xrk

0 and diam B('') = 1/4, it follows that Ix - 61 < 1 for x E Fk`"), 6 E r(i-}1). Therefore Since B('°) n B(`v}1)

ff h(x,C)dsxdsf < c f dsx

rk') xrk')

rk

f

{Er:lx-I<1}

4.2. On the Trace Space for a Narrow Cylinder

227

A combination of the last inequality with (3), (4) gives fr

h(x£)dsds< c

ff

h(x,C)dsxd%

IX-C1<1

or

and (2) follows.

1

Corollary. If the integrals in definitions (4.2.1/19) and (4.2.1/20) are taken over the set

{x,eE1'E:Ix -Cl<e}, then relations (4.2.1/17) and (4.2.1/18) remain true. 1 The last assertion enables us to state relations analogous to (4.2.1/17-18) for some other "narrow domains".

Example. Let xl + ix2 = oeio, x' _ (o - 1,x3, ... , xn), n > 3 and e > 0 a small parameter. Consider a narrow doughnut

DE= {xERn:0E(0,27r),x'/eEc}, (see Fig. 26). An explicit norm uniformly equivalent to II obtained in the following way.

'

II Twy (DE) can be

Fig. 26

Construct a covering of Dr by a finite set of open balls such that the intersection of each ball with DE can be mapped onto a subdbmain of a narrow

cylinder of width e with the aid of a bi-Lipschitzian map (cf. 1.7). Then we introduce a smooth partition of unity on Dr subordinate to this covering and make use of above Corollary. This results in I1(x) - f(S)1P

(ff

J1

L

IIflITWI'(D.)

\ 1/p

1-n ff L

' p>

4. Boundary Values of Functions with First Derivatives ...

228

where L= {(x, e) : x,

E 8DE : Ix-CI <e}.

4.2.3. Traces on the Boundary of an Infinite Funnel As an application of the above results, we study here the space of boundary values of functions defined on an infinite funnel described below. Let W be a positive uniformly Lipschitz function on [0, oo), cp(z) -+ 0 as

z -4 oo, and let w C Rn-1 (n > 3) be a simply connected domain in

C1,1

The infinite funnel D corresponding to cp and w (shown in Fig. 27) is

D = {x = (y, z) E Rn : z E (0, oo), y/cp(z) E w} .

Positive constants c, cl,... appearing in this subsection and the constants in equivalence relations depend only on n, p, w, W. We assume that cp(z) - W(s)

for Iz - (I < 1. Theorem 1 stated below gives a description of the space TWp (D) for p E (1, oo). z t i

.11

y1 .r

11

yi

Fig. 27

Theorem 1. If p E (1, oo), the following relation holds 1/p

II f II TWy (D) - [f ]p,o + (

fS

I f (x) I pco(z)dsx/

dsxds

+

If(x) - f(SC)Ip Ix - In P-2) S

1/p ,

(1)

4.2. On the Trace Space for a Narrow Cylinder

229

is the seminorm given by (4.1.1/3), where o = {x E aD : z < 2}, S = {x E aD : z > 0}, x = (y, z), = (77,C) and M(z,() = max{cp(z), gyp(()}. Proof. Let (f )p,$ denote the sum of two last summands on the right side of (1). First we establish the relation II.fIITW,(D) - (f)p,S

(2)

for functions f defined on aD and satisfying f (x) = 0 for z < 1. Introduce a smooth partition of unity {µk}' 1 for [1, oo) subordinate to the covering by the intervals (k - 1, k + 1), k = 1, 2, .... Let {Ak} Oko= 1 be a set

of functions subject to

AkECo (k-1,k+l),.\kµk=pk for k>1. One may assume that 0 < )'k, µk < 1, Iak I + Ivk I < c. We now check the relation 00

(f)p,s - 1:04f)p,S+

(3)

k=1

where f (x) = 0 for z < 1. Indeed, it is obvious that

f

S

If(x)IPv(z)dsx -

f

k>1 S

Next, let if }p,$ denote the seminorm defined by the last term on the right in (1). Then { f }p ,S < C k>1

}P,S

because

I f (x) - f (S)IP < C E IIk(z)f (x) -1k(C)f (S)IP. k>1

Hence

(f)p,S < C E(Akf)p $ k>1

4. Boundary Values of Functions with First Derivatives ...

230

For the proof of the opposite inequality, we observe that

E{µkf}P,S < CE ff µk(z) - ttk(()IPIf(x)IPIx

d Clds£n+P-2

k>1 H

k>1

f

+cE :

I d(IdsC

n+P-2

H

k>1

with H = {(x, ()

S

x,( E S, I( - zI < M(z,()}. Since w(z) - gyp(() for

(x, () E H and

EIp:(z) -µt(()IP


i>1

the former of the last two sums over k > 1 does not exceed

d-2

{,ES:JS-z1
f If (x) I Pdsx

s

Ix - 61n

S

which is dominated by c

f If (x) I Pcp(z)dsx.

Because Ei>1 µi(()P < 1, the latter sum with repect to k is not greater than { f }P S. Relation (3) is established. P,

Let f E TWP (D) and u c WP (D), UI8D = f . We have CIIUIIWD(D) >_ E IlµkUIIwi(D) k>1

The support of tku is contained in the set Dk = {x E D : z E (k - 1, k + 1)}. Consider the mapping x H Fkx = X = (s, t),

s= y/W(z), t = z/Wk

with c°k = W(k). Then FkDk is a subdomain of the cylinder I = w x R'. The change of variable yields Co

1: k-PII(Iiku) IIUIIWi(D) > C kL=1 oFk 1II w;(O,wk),

4.2. On the Trace Space for a Narrow Cylinder

whence

231

00

IIfUITWp(D) J C E'Pk-PII(1kf)°Fk IIITWp(f1,wk)

(4)

k=1

(we recall that 11 - II Wp (sl,,k) and 11 II Twp (s1,pk) are introduced by (4.2.1/1-2)). -

By Theorem 4.2.1/1 (cf. also Remark 4.2.1/3), for each k > 1 Wk-PII(pkf)

o Fk 11ITWp(Sl,Wk) ti (likf)p,S

(5)

This along with (3) implies IIfIITWp(D) 2 c(f)P,S

Let us verify the reverse inequality. Suppose (f)p,s < oo and f (x) = 0 for z < 1. According to Theorem 4.2.1/1 and in view of (5), for every k > 1 there exists a function vk E WP (1) satisfying vk I0 = (µk f) o Fk 1 and CwkP-'

IIVkIIWp(n,Wk) <

'

(µkf)P,S.

The support of the function wk = (.\k o Fk 1)vk is contained in FkDk. In addition, wk I aj

= (µk f) o Fk 1 and IIWkIIw;(c,ck) 5 CIIVkI wp(0,,ok)

Putting uk = wk o Fk, we obtain that supp Uk C Dk, uk 18D = likf and that the estimate IIukIIW1(D) < C(/Lkf)P,S

is valid. Let u = r1k>1 uk. Then u18D = f and 00

o0

IIuIIWv(D)
Hence and from (3) follows the inequality IIfIITwp(D) <_ c(f)P,S

and, consequently, relation (2).

k=1

4. Boundary Values of Functions with First Derivatives ...

232

To conclude the proof of the theorem, one should derive (1) from (2). This can be made with the aid of a cut-off function 0 E C°°([0, oo)) such that

05 zP<1, Indeed, for any f E

I[O,1]=1, VI[3/2oo)=0.

we have

(f)P,s+[f]P,o
(6)

If f E TWP (D), then by (2) and Theorem 4.1.1, the right part of (6) does not exceed

CII(1-O)fIITWw(D)+CIIfIITW,(D) Hence (f )P,S + [f]p,, < C II f II TWD (D)

To verify the opposite inequality, suppose that (f )P,s + [f ]P,o < oo. It follows from (2) that (1 - V)) f E TWP (D) and II (1- /)f II TWP (D) <_ C ((1- Y')f )P,S.

Furthermore, by Theorem 4.1.1 IIbfIITWp(D) <
Note that the norm on the right is not greater than c ([f], P Q + 11f II L5(snv)) Thus, f E TWP (D) and IIfIITW,(D) <- IIOATW,(D) + II(1-1G)fIITW;(D)

< C ([f]P,a + (f)P,S)

which completes the proof of the theorem.

Remark 1. The surface o in Theorem may be replaced by the surface {x E 8D : z < a}, a E (0, oo). In this case the constants in (1) generally depend on a.

Remark 2. With the aid of Lemma 4.2.1, one obtains that relation (1) remains valid if the integration in the last term of its right part is taken over the set {x, E S : I x - I < M(z, ()). I

4.3. Inequalities for Functions Defined on a Cylindrical Surface

233

The following assertion is provable similarly to Theorem 1 (and even somewhat simpler).

Theorem 2. With the notation introduced in Theorem 1, we have IIf IITW (D) - Ill II Ll(8D\S) +

f

S

dsxdst If W - AO I M(z, C)n-1'

ff

+

w(z)I If (x)I dsx

{x,CES:IS-zI <M(z,()}

4.3. Inequalities for Functions Defined on a Cylindrical Surface In this section we prove several auxiliary assertions concerning functions defined on the boundary of the cylinder

SZ=wxR'cR', n>2, described in Sec. 4.2.1. These assertions will be used to study boundary values of functions defined in the exterior of the cylinder. We preserve the notation and assumptions introduced in Sec. 4.2.1. Let

G=B (n-1)

x R1 C Rn

B(in-1) be a circular cylinder with boundary 8G = S. We assume that 0 C and therefore 0 c G (see Fig. 28). We begin with the following assertion.

AZ

y: i Fig. 28

4. Boundary Values of Functions with First Derivatives ...

234

Lemma 1. Let T be a nonnegative function in L1(R') and let u E LP(G\S2). Then

f

R

< IIouIILp(G\!i) + fR' II ohUIl Lp(S)`y(h)dh,

(1)

where p E [1, oo), c = c(n, p,'y, `y) > 0, (Ohu)(y, z) = u(y,z + h) - u(y,z). Inequality (1) remains valid if r and S replace one another. Proof. Let B = B(In-1) From the well known inequality C IIVIILp(7)

IIVIILp(OB) + IIVvIILp(B\U), v E LP(B

we easily derive the estimate II (ohu)(., z) II Lp(8B)

C1I(AhU)(-,Z)jIP,P(,y) `-

+ II(ou)(-, z + h)II Lp(B\U) + 1I(Vu)(', Z) II Lp(B\U)

for any h E R' and almost all z E R1. Integration with respect to z E R1 yields c IIAhUIILp(r) <- IIDhUIILP(S) + IIVUIILp(G\ii)

Multiplying the last inequality by '(h) and integrating with respect to h c Rl, we arrive at (1). In a similar way one obtains the inequality which occurs if 1 r and S replace one another in (1).

Lemma 2. Let f E Lp,ioc(I'), p E [1, oo) and T a nonnegative function in L1(1, co). Then ff IIf(y,-) - f(77,-)IILp(Rl)d'Yvd7+? < clflp,r

(2)

7x7

and

ff

If (x)-f(C)IpW(IC-zl)dsxdsC+If lp,r

{x,CEr:IS-zI>1}

f 1

00

Ilohf Ilip(r)II'(h)dh + If Ip,r,

(3)

4.3. Inequalities for Functions Defined on a Cylindrical Surface

235

where x = (y, z), C = (q, (), (Oh f)(x) = f (y, z + h) - f (x) and I ' Ip,r is the seminorm given by (4.2.1/5-6). The constants in equivalence relation (3) may depend on IQ.

Proof. By Theorem 4.2.1/1, it is possible to extend f from r into Q in such a way that the extension u satisfies IH VuIIL,(a)

c f Ip,r.

Applying (4.1.2/2) (with e = 1) to a section z = const, we find that for a.e.

zER1 ff I f (y, z) - .f (rl, z)Ipd-yyd'y,i : c II (Vu)(', z)IIin(s)' ryxry

Now integration over z E R1 yields (2). To verify (3), note that If (x) - .f (015 If (y, z) - f (y, 01 + If (y, 0) - .f (77, 01. Therefore, the integral on the left side of (3) is not greater than

c f dz Rl

f dyy f 7

+c

IAh.f(x)I'`W(IhI)dh

IhI>1

IIWIILl(1,00) ff II f (y, ') - f (77,')II

Lv(Rl)dyydyn.

7X7

An application of (2) to the last integral shows that the left part of (3) does not exceed a constant times the right part of (3). To validate the reverse inequality, we first use the estimate

CIAh.f(x)Ip -< f If(y,z+h) - f(rl,z+h)Ipdy,7 +

f

If(ri,z+h) -.f(y,z)Ipd-y,,.

7

Put ( = z + h, C

_ (q, (). The last inequality implies that

f

oo LP(r)T (h) dh

II Ahf II

1

IIWIIL,(1,00) ff IIf(y,')-!(ij,')IILP(Rl)d-yydryxry

+

ff

{x,{Er:C-z>1}

If (x)

- f(e)I''( ( - z)ds.,dst.

4. Boundary Values of Functions with First Derivatives ...

236

It remains to apply (2) to conclude the proof of Lemma 2.

1

Let f be a function defined on the surface S = S"-2 x R' and let f (z) z) on Sr-2 with z E R1.

denote the mean value of f

Lemma 3. For f E Lp loc.(S), p c (1, oo), the following estimates hold

ff

dsds

I f ('z) - 7 (

)lp

X - CIn P-2 G C f I p,S'

(4)

{x,£ES:1(-zI<1}

f dl f If(z+h)-f(z)Ipdz
0

hp

(5)

Rl

Here x = (y, z), e = (77, () and I - Ip,s is the seminorm given by (4.2.1/5).

Proof. We consider 7 to be a function defined on S (and depending only on z). Then the left side of (4) is equal to If lP,s Clearly _

1

If IP,s < c J 1 Ilohf IILp(R3)dh f

de

f

do,

n_2 (IhI + Ia -

9I)n+p-2

with

ohf (z) = 7(z + h) - 7(z). By means of the change of variable a - B = I hIQ, the last integral over can be dominated by c I h I -p. Thus

IfIP,S
Sn-2

IlAhfllL,(RI)IhI-pdh,

1

and inequality (4) follows from (5). To verify (5), we observe that

f1 00

c

f

k=-oo sn Z

dh

fi

17(z + h) - f (z) I pdz

dO f dz k-1

f 0

l f (e, z + h) - f (0, z) I" .

(6)

4.3. Inequalities for Functions Defined on a Cylindrical Surface

237

Let us bound the general term of the last sum. By Theorem 4.2.1/1 (applied

to the cylinder G = B(i-1) X R1 and the surface S = 8G), there exists an extension u E LP(G) of the function f satisfying IIVuIIL,(G) <- cIfIp,S

Let

Gk={(y,z)ER":zE(k-1,k+1), 1/2
k = 0, ±l, ... , O E Sn-2.

z

r

Fig. 29

The set IIke) can be considered as a two-dimensional section of Gk by the hyperplane 0 = const. We introduce two variables r = IyI E (1/2, 1) and z E (k - 1, k + 1) in this section (see Fig. 29). Since u E WP (IIke)) for almost all 0 E Sn-2, Theorem 4.1.1 leads to the inequality k+1

k +1

fk -1

L -1

< c Jk+l dz k-1

dzd If (0, z) - f (0, C) I p

f

1

1/2

I(- ZIP

(Iurlp+Iuzlp)dr.

Integration of this inequality with respect to 0 E Sn-2 shows that the general term of the sum in (6) is dominated by c II VuIIi,(Gk). Therefore, the right part of (6) does not exceed c IIVuIILp(G) which is not greater than c if IP'S' This completes the proof of Lemma 3. 1

4. Boundary Values of Functions with First Derivatives ...

238

We conclude this section with two assertions concerning the domain w(e) _ is defined by analogy with (4.2.1/1). The next lemma follows from Corollary 4.1.3 by dilation.

Ri-1 \ 0. Given E E (0, 1/2), the norm II

Lemma 4. For any u E Wp (()(e)) the estimate a(e) IIuIIL,(7) < cIIUIIwp((,W,E)

is valid, where a(e) = I log

eI1-P if p

a(E) = min{ 1,

(7)

= n - 1 and

ep+1-n}

if p > 1, p> n - 1.

1

Passing to the limit as E -* 0 in (7), we arrive at another result.

Corollary. If u E WP (, (e)) and p E [1, n - 1), then IIUIILn(7) <- CIIVuIILn(w(1)).

4.4. A Norm in the Space TWp for the Exterior of an n-Dimensional Cylinder, p < n-1 We preserve the notation used in Sec. 4.2.1, 4.3. Let Q(e) = Rn\Sl be the exterior of a multi-dimensional cylinder Q described in Sec. 4.2.1. In what follows e is a small positive parameter. The norms II' II Twp (sl(e),E) are introduced in the same way as in (4.2.1/1-2). For functions defined on r = 8SZ we construct an explicitly described norm which is equivalent to II' II Twn (sl(-),E)

uniformly in e. Hence the theorems on traces of the functions in Wp

are

, = e Q. We begin with the following

obtained, where Q.(e) = Rn \ S2E and lemma.

Lemma. Let u E LP1 (RI), p < n. Then there exist a constant u and a sequence {uk} c Co (Rn) such that Uk -* u - u in LP(Rn).

Proof. Let (r, 0) be the spherical coordinates of a point x E R'. We first establish that there exists lim,.,, u(r, 0) for almost all 0 E Sn-1 Indeed, let 0 < r < R < oo. Then for a.e. 0 E Sn-1 " Iu(R,0)-u(r,0)IP=I f up(P,O)dp R r


Iu(P,0)IPpn-ldp.

4.4. A Norm in the Space TWP for the Exterior ...

239

Since the last integral is finite for a.e. 0, it follows that exists for the same 0 E Sn-1. In addition, we have

u(r, 0) = f (0)

IIf - u(r,')IIL,,(Sn-1) < Cr1-n/PIIouIIL,,(Rn),

(1)

and the inclusion u(r, ) E Lp(Sn-1) implies that f c Lp(Sn-1). Let us show that f (0) = const a.e. on Sn-1. To this end, we consider arbitrary measurable subsets U1, 0'2 of Sn-1 with positive measures and put u1(r) to be the mean value of the function Sn-1E) 0 H u(r, 0)

ono, i=1,2, r>0. LetQr={xERn:r
1U1(r) - U2(r)I <

cr(P-n)/PIIVul1LP(:r)

whence

U, (r) - 112(r) -+ 0 as r -+ oo.

Furthermore, (1) gives limr.,,. ui(r) = f;, where f; is the mean value off on o , i = 1, 2. Thus, f 1 = f 2, and f (0) = const by the arbitrariness of U1, 02 Let u = f (0), 1 1 E C°°([0, oo)), '1 [O 1] = 1, rll [2 oo) = 0.

Put k = 1, 2, ... , x E Rn.

Vk(X) = (u(x) -

Then Vk -+ u - zl in Lp,l°°(R') and also C IIV(vk - u)IILP

(Rn) <-

11 VU I ILP(Rn\Bk) + k-'IIu - uII

(2)

The first term on the right side of (2) tends to zero as k --> oo, whereas the second does not exceed 1/p

(fSn-1 dO J

I u(r, 0) - uI Prn-1-Pdr)P

(3)

k

Applying Hardy's inequality (1.1.2/7), we dominate quantity (3) by the expression C IIVUIILP(Rn\Bk) Thus

IIV(vk - u)IILP(Rn) -4 0.

4. Boundary Values of Functions with First Derivatives ...

240

Let Uk denote a mollification of vk with sufficiently small radius of mollification

so that IlUk - VkllW1(R^) < 1/k.

Then the sequence {uk} satisfies the conclusion of the lemma.

1

Let Il = w x R1 C R" (n > 3) be the cylinder described in Sec. 4.2.1 and S2(e) = R" \ 1 . Below we assume e E (0,1/2). The following theorem gives an explicit norm equivalent to II

.

ITW, (s1W,e) uniformly in E.

Theorem 1. The space TLi(Q(e)) can be represented as the direct sum L1(I')4.R1. Moreover, if f = g + A, g c L1(I'), A = const, then IIfIITL1(o(°)) ^- 1IgIILI(r)

Furthermore, the following relation holds IIfIITW (1l(-),e) - IIfIILI(r)

Proof. Let u E Li(I (e)), u I r = f. We now check that there is a constant A E R1 satisfying Ilf - AIILI(r) -< C IIVUIHLI(cl(r))

(4)

An application of Theorem 4.2.1/1 (see also Remark 4.2.1/2) to the cylinder

D = (Bii-1) \ 0) x R1 and the surface I' gives If I1,r 5 C IIDuIIL1(D),

I1,r is the seminorm (4.2.1/6). By the same theorem, there is an extension of f from r into S2 (this extension is relabeled as u) subject to where

I

5 c If Ii,r.

Thus, one may assume that u E L', (R') and that IIVUIILl(Rn) <_ C IIVUIIL1(f2(-)).

According to above Lemma, there exist a constant A E R1 and a sequence {uk}k>1 C Co (Rn) such that Uk - u - A in L1,lo,(Rn) and IIV(uk -

u)IIL1(Rn) -+ 0.

4.4. A Norm in the Space TW1 for the Exterior ...

241

Let w(e) = R"-1 \ w, y = 8w. Then by Corollary 4.3 Iluk(',z)IIL1(7)
Integrating with respect to z E R1, we arrive at IIukIIL1(r) -< C

k =1, 2, .. .

The replacement Uk with Uk - uj does not affect the validity of the last inequality. Hence it follows that the sequence {uklr} is convergent in L1(I'). If g is its limit, then II9IIL,(r) <- C IIVUIIL, (o(-)).

u - A in L1(R"), {uklr} is convergent to f - A in L1,1oc(F) and therefore g = f - A. So estimate (4) holds. Now let u c Wi (1(e)), u I r = f. In this case Corollary 4.3 gives Since Uk

11K, z)IIL1(.) <- c II(Vu)(.,

z)IIL,(W(.))

for almost all z E R'. Integrating with respect to z E R', we obtain (4) with A = 0. Hence, the following inequalities are valid IIl1IL1(r) <- c II f IITwi (o(0,e),

(5)

If-AIIL1(r) <-CIIflITL;(c )),

(6)

the number A E R1 in (6) being uniquely determined by f. To prove the reverse inequalities, we consider the extension operator E defined by (4.2.1/11). Let cp E Co (Rs-1), (p(y) = 1 for IyI < 1, W(y) = 0 for IyI > 2. Put (7) (Ef) (x) _ W (y) (Ef) (x), x = (y, z) E R'.

It follows from Theorem 4.2.1/1 (see also Remark 4.2.1/1) that if f E L1(r),

then E f E Wi (R"), E f Ir = f and IIEf IIw; (R.^) <_ C IIf IIL1(r)

Therefore, the inequality opposite to (5) holds.

242

4. Boundary Values of Functions with First Derivatives ...

If f = g + A, g E L1(I'), A = const, the function u = A + Eg is in L' (R'). Clearly ulr = f and the estimate IIVuIILI(R-) < c II9IIL1(r)

is true. This establishes the inequality opposite to (6). The proof of Theorem 1 is concluded. 1

We pass to the case p > 1. Theorem 2 stated below presents an explicit norm equivalent to II IITwn(sz<<),e) uniformly in e.

Theorem 2. Let n > 3 and p E (1, n - 1). The space TLp(Q(e)) coincides with the direct sum W P -l/p(I')+Rl and if f = g+A, g E W, -1" (F), A E R1, then IIIIITLp(1(-)) - II9IIwp-lIP(r)-

Furthermore, the following relation holds IIATWp(S2(e),e)

-IIfIIw,-1IP(r)

Proof. Let U E Lp(SZ(e)), ulr = f. An application of Theorem 4.2.1/1 (cf. also Remark 4.2.1/2) to the cylinder (Bin-1) \0) x Rl and the surface I' yields IfIn,r < c IIVujILP(n(-)),

(8)

where I Ip r is the seminorm defined by (4.2.1/5). Arguing as in Theorem 1,

we can validate the existence of the constant A satisfying (9)

IIf - AIILP(r) <- c IIouhhLP(n(-)).

Now (8), (9) and the relation II II wp-lIP(r) - I IP,r + II

IIL,,(r) (recall that

the norm in the space WP-1/1(I') is given by (4.1.1/4) ) imply that IIf -A11WP 1/P(r) <-c

IfIITLD(W-))

Let u E Wp(Sl(e)), ulr = f. Corollary 4.3 yields 11P., z) II LP(-Y) < c 11 (Vu) (-, z) IILP(R1-1\U)

(10)

4.4. A Norm in the Space TWp for the Exterior ...

243

for almost all z E R1. Integrating with respect to z E R1, we obtain (9) with A = 0. This in conjunction with (8) gives IIf II W;-lin(t) < C IIf II TW, (S2(°),E)

The inequalities opposite to (10), (11) are justified with the aid of extension operator (7) in the same way as the inequalities opposite to (5), (6) in Theorem 1. It should be taken into account that the operator E defined by (7) is an extension operator: Wp-1/p(r) -* Wp (R") with

IIEfIIw,(R.^)
(12)

The last estimate follows from Theorem 4.2.1/1 and Remark 4.2.1/1. This concludes the proof of Theorem 2.

1

Remark 1. Formula (7) determines a linear extension operator

E: WP-1/p(r)-pWp(Rn) subject to (12) for all p E (1, oo).

Remark 2. Let f E Lp ioc(I'), p E [1, oo) and let there be a A E R1 such that f - A E Lp(r). It is easily seen that then A = lim f k, where fk is the mean value of f on the surface {x E r : Ixi < k}, k = 1, 2, .... Hence and from Theorems 1, 2 we obtain the following assertions.

A function f E L1,io,(r) is in TL'(SZ(e)) if and only if the finite limit A = lim f k exists and f - A E L1(I'). In addition, IIf IITL11(ci(°)) "

IIf - AIIL1(r)-

A function f E Lp,i0E(I ), p E (1, n - 1), is in TLP(S2(e)) if and only if the

above limit exists and f - A E W,-1/p(r). In this case ill IITL1(fi(-)) '

IIf -

Allwp-1/n(r).

1

Put SZEe) = Rn \ S2E where Q. = e SZ is a narrow cylinder. We now present

a norm equivalent to

11

-

uniformly with respect to e E (0, 1/2).

4. Boundary Values of Functions with First Derivatives ...

244

The following relations are obtained from Theorems 1 and 2 by a similarity transformation: IIf IITWi (S

IIf

Here F. = )

)) '

Ill IIL,(r.), E(1-P)IPIIf

IITWp1(1lce,)) -

hI Lp(r.) + I f l p,rc,£ p E (1, n - 1).

and I - IP,r,,e is the seminorm given by (4.2.1/19).

For n > 3, p E [1, n -1), the space TLp(Q,(e)) coincides with the direct sum TWP (S2(Ee))+Rl and if

f =g+A, gETWp(1 )), AER', then

IIfIITLy(S2,(°)) -

II9IITwp(c(-°)).

4.5. The Exterior of a Cylinder, p > n-1 The same problem as in the preceding section is studied here in the case

p > n - 1. We begin with an auxiliary assertion. Lemma. Let g E LP,IOe(R'), p E (1, co) and let K be a function in Co (1, 2)

such that f K(t)dt = 1. Put

D={x=(y,z)ERn:zER1, IyI>1}, n>3,

(1)

and r

(Hg) (x) =

z

J

K(t)g(z + (IyI

- 1)t)dt, x E D.

1

Suppose that the seminorm I(g), defined by

IMP = J

00

IIOh9IIL p( R1 )hP(1

A

+ h) 2-n

,

is finite, where (Oh9)(z) = g(z + h) - g(z). Then II (H9)(y, .) - 9II Lp(R1) - 0 as IyI -> 1 + 0

(2)

4.5. The Exterior of a Cylinder, p > n - 1

245

and

II VH9IILp(D) < c(n,p, K) I (g).

(3)

Proof. Put r = I yI, h = t(r - 1), v = Hg and note that for (y, z) E D v(y, z) - 9(z) = r

f

1 1

2(r-1)

K (r h 1) (Ah9)(z)dh.

r-1

Holder's inequality yields

v(y,z) - 9(z)IP < c (r -

1)P-1

/

2(r-1)

-

r

I (Ah9)(z)IPd 1

Therefore, the first conclusion of the lemma follows by integration with respect

tozER1. We now turn to the proof of (3). Clearly IVvI <- Ivrl +

Ivzl

r

c

1

f 2 I (Ah9)(z)Idt, 1

whence C

IIVVIILp(D) <

TO (r - 1)P

f

l

dz (I I 2(Ah9) (z)I dt

I

P

An application of Minkowski's inequality leads to IIovIILp(D) < C J12dt 1 fRl dz f

(I(Ai9)(T)I ndr)1/p.

After the change of variable r = 1 + h/t in the last integral one obtains (3) thus concluding the proof of the lemma. I Below we preserve the notation used in Sec. 4.4. The following theorem gives an explicitly described norm equivalent to the norm II ' II Twn (QW,,) uniformly in E E (0,1/2).

Theorem. Let p E (n - 1, oo), n > 3, E E (0,1/2). Then IIfIITWp(1l(°),e) ^'

E1+(1-n)/PIIfIILp(r)

+ [flp,r

4. Boundary Values of Functions with First Derivatives ...

246

1/p

dsxds£

ff

+

I f (x) - f (C) I P Ix

- IP+2-n

(4)

{x,{Er.Ix-fl>1}

where dsx, ds are the area elements on r and [-]p,r is the seminorm defined by (4.1.1/3). If the left side in (4) is replaced by IIfIITLP(u(.)) and the first term on the right is omitted, the resulting relation is also true.

Proof. First we establish the relation dsxds

ff

[f1p,r+

If(x)-f(S)IPIx-SIp+2-n

{x,fEr:Ix-cI>1}

,,, IfIP,r+{f}P p, r, P in which f E Lp,iac(I'),

I

(5)

Ip,r is the seminorm given by (4.2.1/5), 1/P

{ f }p,r =

(foo II Ahf II LP(r) hn-2-Pdh)

(6)

and (Ohf) (x) = f (y, z + h) - f (x), x = (y, z) Indeed, by Lemma 4.3/2, the right part of (5) is equivalent to

IfIP,r+

ff

aszlp+2-n,

(7)

{x,{Er:IS-zI>1}

where x= (y, z), C= (77,C). Since Ix-CI ' IC - zI for I(- zI > 1, we deduce the equivalence of (7) to the left part of (5). Thus, (4) is valid if and only if IIf IITwp(s1(O,E) - NIf lIP,r

with IIIf IIIp,r.=

E1+(1-n)/P

IIf IIL,,(r) + If IP,r + { f }p,r.

(8)

Let u E Lp(SZ(e)), ulr = f. Estimate (4.4/8) is verified in the same way as in Theorem 4.4/2. We now turn to the inequality {f}P,r !5 CIIVuIILP(SZ(e))

(9)

4.5. The Exterior of a Cylinder, p > n - 1

247

In view of Lemma 4.3/1, it is sufficient to consider the case of the circular cylinder S2 (e) = {(y, z) E Rn : z E R1, IyI > 1} and I' = Sn-2 x R1. Let y = (r, 0) be spherical coordinates in Rn-1. Then h

P

ur(r,0,z + h)dr

C I (Ahf) (x) 1P <_

fz+h

P

P

h

u((h,B,()d(

f,

ur(r, 0, z)dr

(10)

whence

{f }p,r < c

n_

d0

JRl

/'dh + J

dz

dh hp+2-n

h

hP+2-n

1

[foo

(fI 1

(L z+h Iuc (h,

d(fP

0, () I

(\

1P

Lr

(r, 0, z) I dr)

We make the change of variable ( = z + ht, t E (0, 1) in the integral over (z, z + h) and then use Holder's inequality. We also apply Hardy's inequality to the second summand in square brackets. This results in c

1PP,r <

dz

R'

u T 6 z P+ u T 9 z p rn-2dr.

dO

S°-a

1

Hence (9) follows. Now (8) and (9) imply

Iflp,r+{f}p,r
(11)

Let u E Wp (Q (e)), ulr = f. An application of Lemma 4.3/4 yields CEP+1-nlf(., z)II LP(.1)

< EPIIu(., z)II LP(W(`)) + IIVu(-, x)II LP(W(e))

for almost all z c R1. Integrating with respect to z E R1, one obtains 6p+1-nIIfIILP(r)

<- CIIUIIwp(Q(e),E)

(12)

It follows from (4.4/8), (9) and (12) that IMIp,r

< C IIf IITW (sl(e),E),

(13)

248

4. Boundary Values of Functions with First Derivatives ...

where HI

IIIp,r is the norm given by (8).

To verify the inequalities opposite to (11), (13), we construct extensions f H Lf E LP(52(e)), f H Wf E WP (SZ(e)) of a function f from surface r into S2(e) satisfying II VL f IILp(ci(-)) < c (If IP,r + { f }p,r) ,

(14)

IIW.fIIwp(ow) <_ c11fIIIP,r.

(15)

Construction of extension operators L, W. Let f E Lp,loc(r), If Ip,r + If }p,r < 00.

Let E f be defined by (4.2.1/11). We put u(x) = (E f) (x) for x c G, G = B('- 1) xR1. According to Theorem 4.2.1/1 (cf. also Remark 4.2.1/1), u I r =

f

and

(16)

II VtIILp(G) <_ c If IP,r.

In addition, if f E Lp(r), the inequality cIIuIIwp(G,6) <_ 6llfhp(r) +

Iflp,r

(17)

e1+(1-n)/p takes place with b = Put S = 8G, g = ul s. An application of Theorem 4.2.1/1 to the cylinder G and the surface S gives

191P,s < c1110uIILp(G) <_ c21f IP,r, 61191ILp(S) <_ C311UIIW;(G,6) <_ C4111fIIIP,r

It follows from Lemma 4.3/1 that {g}p,S < C (II Dull Lp(G) + { f }p,r)

,

and thus the inequalities 191 p,s +

{g}p,s < c (I f l p,r + { f }p,r), 1119NIp,s <_ c III f IIlp,r

(18)

(19)

are valid. Here and III Nlp,s denote the seminorm (6) and the norm (8) respectively for functions defined on S. Inequalities (16)-(19) show that the

4.5. The Exterior of a Cylinder, p > n - 1

249

construction of the required extensions f H Lf, f H W f from surface IF to S2(e) is reduced to the case

S(e)={x=(y,z)ERn:zER1, IyI>1}, P=Si-2xR'.

(20)

We introduce a linear extension operator E : Wp-11P(r) -+ WP (Rn)

satisfying (4.4/12) (see Remark 4.4/1). Let 1i be a function in C'([O, oo)) such that O(t) = 1 for t < 1/2, ?I(t) = 0 for t > 1. We shall show that under condition (20) L and W can be given by

(Lf)(x) = (HY) (x) + (E(f - f)) (x), (WI) (x) = VG(EI yI) (Hi) (x) + (E(f - Y)) (x), where x = (y, z) E S ), j (z) is the mean value of the function f z) on the sphere Sn-2 and H is defined by (2). Proof of the required properties of L and W. In view of (4.4/12), one has

cIIE(f -f)IIx.P(R-)

ff

:5 If -7IP,r+

IIf(0,

f(a,-)IILr(R1)dadO.

Sn-2XSn

According to Lemma 4.3/2 (applied to ry = Sn-2 and r = Sn-2 x R1), the integral over Sn-2 X Sn-2 does not exceed c If IP,r. Furthermore, Lemma 4.3/3

implies that If Ip,r < c If Ip,r and therefore

IIE(f-f)IIwp(Rn)
f

1

dh IlohfllL(R') hp(1 + h)2-n

IIAhfIILp(Rl) hP

f

00

II1 hfIILP(R1) hP}+2-n

(21)

4. Boundary Values of Functions with First Derivatives ...

250

By Lemma 4.3/3, the first integral on the right does not exceed c if Ip,r The second integral is dominated by c If }p r. Hence

IIVHfIIL,(cW)
(22)

Thus, the formulas defining L and W have sense at least for those f E Lp,ioc(F) that satisfy If Ip,r + If }p,r < oo. The identity L f Ir = W f Ir = f follows from

the lemma preceding the theorem so that L and W are extension operators. Note also that (14) is a consequence of (21) and (22). Turning to the proof of (15), we put v(x) = ?p (EIyI)(Hf)(x). Then f1/e

I

2dr

(ET)Ipr

1

and therefore (23)

EPIIVIIP

In addition, the estimate IV)'(EIyI) pdy

II DHf I L,(Q(`)) + EP IIf IILn(R')

C IIVvIILP(B(e))

is valid. The second summand on the right is not greater than the right side in (23). Hence and from (22), (23) we derive the inequality IIV IIwi (fi(-),e) < c !I f nlp,r

which along with (21) implies (15). This completes the proof.

I

Let 0, = E SZ be a narrow cylinder and Q,(') = R' \ fie its exterior. Corresponding results for Q,(') follow from above Theorem by means of a similarity transformation.

Corollary. Let p > n - 1, n > 3, E E (0,1/2), and let r, = 8S2,. Then IIf II TWy(Q2(')) iv

E(2-n)/pIIf

+

IIL,(r.) + [f]p,rc

(e22_m

ff {x,£Erc:jx-£I>e}

If (x)

-f

ds ds Ip

IxI

p+2-n /

4.6. An e-Dependent Norm in the Space TWp ...

251

where is the seminorm given by (4.1.1/3). This relation remains valid if we replace IIf on its left side and omit the first by IIf IITLpi(st.(`))

IITyvp1(QE'))

term on the right.

4.6. An e-Dependent Norm in the Space TWp for the Exterior of a Cylinder of Width e, p = n-1 This section deals with the same problem that was considered in Sec. 4.4-4.5 but for p = n - 1. We begin with Hardy type inequalities.

Lemma 1. Let 0 < a < b < oo, p E (1, oo). If u is an absolutely continuous function on (a, b) and u(b) = 0, then

f

b

Iu(t) dt < Ip

c(p)

I

b

(t) I10-1

I

(log(t/a))p dt.

If u(a) = 0, then dt

Ja

b

IU(t) Ip

b

< C(p)

t(log(t/a))" -

/a

I u (t) I Pt" 'dt.

Proof. The change of variable log(t/a) = x leads to Hardy's inequality (1.1.2/7). 1 In the following lemma we construct an extension of a function from the

boundary of the circular cylinder B(n-1) x R1 into its exterior when this 1 function depends only on one variable.

Lemma 2. Let g E Lp,iac (R1), p E (1, oo). Suppose dh IIAh9IILp(R1) (1 + h) (log(1 +

Cf

J

1/n

h))r)

(1)

is finite with Ohg(z) = g(z + h) - g(z). Let D C R" be defined by (4.5/1) and put 1

(Fg) (x) = 1 log IyI fRi

9(z + h)dh

(IyI + IhI)(log(Iyl + IhI))

for x = (y, z) E D. Then

II(F9)(y, )-9IILp(Rl) -- Oas IyI - 1+0,

2

(2)

4. Boundary Values of Functions with First Derivatives

252

...

and if p = n - 1, the following estimate holds IIVFgIIL,(D) <- c(p)1(g).

(3)

Proof. For r > 1, h E R1, let K(r, h) = log r/ (2(r + Ihl)(log(r + IhI))2)

(4)

.

Because f K(r, h)dh = 1, we have (Fg) (x) - g(z) = fl K(I yI , h)(Ahg)(z)dh, x = (y, z) E D.

(5)

Put for brevity r = I yI , v = Fg. An application of Minkowski's inequality and Holder's inequality gives IIv(y,') - 9IIL,(R1)

< IRl K(r, h)IIAhgIIL,(R1)dh dh

< c logr

IlohgllL,,(R1) (JR1

(1 + Ihl)(log(1 + I hI ))P) 1/n'

dh Jo TO (r + h) (log(r +

1/p

h))p')

'

where p' = p/(p - 1), r E (1,00). Since

IIA(-h)gIIL,(R') = IIAh9IIL,(R1),

the right side of the last inequality does not exceed c (log r)1/p 1(g). Hence the first conclusion of the lemma is established. Turning to the proof of (3) for p = n - 1, we note that IIVvIIL,(D) <- IIvzIIL,(D) +

IIVrIIL,(D)

and bound each term on the right. It follows from (2) that IIvrIIL,(D) C c

rn 2dr

dz

fW

100

fRl

K (r, h) (Ohg)(z)dhl p

4.6. An e-Dependent Norm in the Space TWy ...

253

Consequently,

IIVrIIi n(

n><- fR'

dr

c

dz

+c

Rl dz

IOh9(z)I + Io(-h)9(z)Idh (r + h)(log(r + h))2 )

r UM

J

TP-1dT

1

(TO

P

)P z)I Ah9(z)I + Io dh I(r + h)2 log(r + h)

(6)

Let us check the estimates

)P

IOh9(z)Idh

dzJ

I

r (fooo (r + h)(log(r + h))2 hgog(r

f dz J

rP-1dr (f"O (r +Ah)

+

< CIO)"

h)) P < c I (9)P

(7)

(8)

Indeed, the left side of (7) is not greater than

C

(fr-1

dr /'°° dz [J1 r(logr)P Rl

+

f

°°

00 dr

IOh9(z)Idh (1 + h) log(1 + h) ) P

I oh9(z) I dh

-1 (1 + h)(log(l + h))2)

r

P

].

An application of Lemma 1 to each summand in the square brackets yields (7). To prove (8), we denote the left side of (8) by J and find that

J< cJ

R'

/foo

+J

f

dz 111

1

00

dr (fr IOh9(z)Idh)P rp+l r00

r p-1 dr (J

log(1 + h) I Ah9(z) I dh

P

(1+h)2log(1+h)) }

Apply Holder's inequality to the integral over (0, r) and then use Hardy's inequality (1.1.2/7) to bound the second term in the curly braces. This results in

/ °° dr

J
dz fR1

1

r IOh9(z) I Pdh

°O

r2 Jo (log(1 + h))P + J1

IAh9(z) IPdh

h(log(1 + h))P

4. Boundary Values of Functions with First Derivatives ...

254

The right part of the last estimate is dominated by c I (g)P, and (8) follows. The same argument shows that (7), (8) remain true if IOh9(z) I is replaced by IO(_h)g(z)I. Thus, the right part of (6) does not exceed cI(g)P so that IIvrIILp(D) < CI(g)-

We now obtain an analogous estimate for IIvzllL,, (D) It is readily seen that vz

- fRl 8h K(r, h) (Oh9) (z)dh,

where K is given by (4). Hence cll< Co

fRI

dz f rP-ldrl f1 00

< R1

dz

I

\8hK(r,

h)) (Ohg)(z)dhlP

rp-1(logr)Pdr(f (1+log(r+Ihl))Iohg(z)Idh)P `\fB.' (r +

Ihl)2(log(r + Ihl))3

l

One can easily check that the right part of the last inequality is not greater than the right part of (6). Therefore IlvzllLp(D)
This completes the proof of Lemma 2. 1 Below we preserve the notation introduced in previous sections. The following theorem gives an explicitly defined norm equivalent to II . IITW (szce),e) uniformly in e for p = n - 1.

Theorem. Let p = n - 1 > 2 and let e C (0,1/2). Then III II TW,(SZ(°),e) '" I logel(1-P)/P II! IILp(r)

ff

( +\ rxr

I f (x) - f

dsxe2p-i) ds 1/P

(9)

where e = Ix - I, Q(e) = 1 + e2P-2(log(1 + e))-P and dsx, dse are the area elements on r. If we omit the first term on the right part of (9) and replace the left part by If IITLp(Q<<)), we obtain the correct relation

4.6. An e-Dependent Norm in the Space TWy ...

255

Proof. Note that if f E Lp,ioc(r), then 1- f WI PQ(e) de P 1£

If IP,r + (f )P,r.

(10)

rxr

Here I - Ip,r is the seminorm given by (4.2.1/5),

(f)p,r = (

f

1/P dh hf Ilip(r) h(log(1 + h))P) '

II

(Oh f) (x) = f (y, z + h) - f (x), x = (y, z). Relation (10) is justified with the aid of Lemma 4.3/2 by analogy with (4.5/5) in Theorem 4.5. In particular, (10) implies that the norm on the right part of (9) is uniformly equivalent to the norm logel(1-P)1PIIfIILp(r) (11) RIfbp,r = I + If IP,r + (f)p,r. Let u c LP(SZ(e)), ulr = f . Estimate (4.4/8) is verified in the same way as in Theorem 4.4/2. Let us establish the inequality (12)

(f)p,r < c IIVuIILp(n(-)).

By virtue of Lemma 4.3/1, it will suffice to consider the case (4.5/20). Passing

to spherical coordinates y = (r, 0) in R"-1 and taking into account (4.5/10), one obtains c (f )P,r <

f

do

sn

+f 1

(

dzS

JR1

l

f"Alf"' uS (h, 0, ()d(

h(logh)p(\ f

P

h

hIur(r,0,z)Idr)Pj. (13)

To bound the right part of (13), first we make the substitution c = z + ht, t E (0, 1), in the integral over (z, z + h) and employ Holder's inequality. Next we apply Lemma 1 to the second term in curly braces. Then

c (f )p,r < Is-. dO

fR'

dz

f

(I ur (r, 0, z) I P + I uz (r, 0, z) I P) rn-2dr.

1

Hence (12) follows. Now (12) along with (4.4/8) implies If Ip,r + (f)p,r < C IIf IITL;(cW).

(14)

4. Boundary Values of Functions with First Derivatives ...

256

If u E WP (I (e)), ulr = f, the estimate loge1(1-P/P)IIf

I

IILp(r) <_ c I lull

wi(siw,E)

(15)

is provable with the aid of Lemma 4.3/4 in the same way as (4.5/12) in Theorem 4.5. A combination of (12), (15), (4.4/8) and (11) gives (16)

MfNP,r < cII!IITwp(11W,E)

In order to establish the inequalities opposite to (14),(16), we should construct extensions

f H Lf E L'(Sl(e)), f H Wf E Wp (S2(e)

of a function f from surface r into p(e) satisfying the conditions IIVLf iILp(n(-)) < c (lf IP,r + (&,r), IIWf IIw; (o(o,e) < c

MIP,r.

(17) (18)

Repeating the proof of estimates (4.5/16-19), one obtains analogous estimates in the case under consideration. That is, if we put 8 = I logel(1-P)/P

in (4.5/18) and specify the with in (4.5/17), replace IIIP,s by (11) in (4.5/19), the resulting inequalities are valid. norms III . IlIP,r, This means that the construction of the desired extensions L f, W f is reduced (as in Theorem 4.5) to the case of a circular cylindrical surface, i.e. to (4.5/20). III

-

Let E : Wp-11P(r) -+ WP(R") be a linear extension operator subject to (4.4/12). Put

t (t)= f E

I

log(et)/loge if t E (1,6-11, 1

We claim that under condition (4.5/20) the required extension operators L, W can be introduced by

(Lf) (x) = (Ff) (x) + (E (f - f)) (x),

(19)

4.6. An --Dependent Norm in the Space TWP ...

257

(Wf)(x) = tE(Iy1)(Ff)(x) + (E(f - Y)) (x),

(20)

where x = (y, z) E 11(e), Y (z) is the mean value of f z) on the sphere S"-2 and F is the operator defined by (2). Proof of inequalities (17), (18). Let I(.) be the seminorm defined by (1). Then oo IIOhf IILp(R1) dh P dh. cI(f) P IILp(R1) hP + h(log(1 + h))P Jo

1

f

The first integral on the right is estimated by using Lemma 4.3/3, whereas the second is not greater than c (f )p r. Thus, I(f) <_ c (If IP,r + (f )P,r)

(21)

Now it follows from Lemma 2 and (4.5/21) that definitions (19), (20) have sense at least for f E Lp,loc(r),

If Ip,r + (f )p,r < 00,

and, furthermore, L f = W f = f on F. Lemma 2 and (21) also imply IIV(1)IiLp(n(-)) < c (If IP,r + (f)P,r)

(22)

This inequality in conjunction with (4.5/21) gives (17). To verify (18), suppose that I f NIP,r < oo. First we check the estimate IIVIIWp(OO,e) < cIIIfIIIP,r,

(23)

where v (x) = 'E (I y I) (Ff) (x)

Let K be given by (4). In view of (5) 1/ e

IIvII(Q())

IIfIIL(R1)

+

dz

J

J

1

1/E

GE(T)PTn 2dT

rn-2'be(r)Pdr (JR1 K(r, h) I1hf (z) Idh ) P

258

4. Boundary Values of Functions with First Derivatives

...

By applying Holder's inequality to the last integral, one obtains cl(EIloge)PIIviILP(D()) < IIfIIL,(R1) + EP

J

1/f (r log

r)P-l I

log(er) IPdr fRi

1

< CZ (IIf iiLP ,(r)

+

I log'51,

'I

II

Ly(Rl )dh

(r + Ih)(log(r + l hl ))P

MP)

Thus, the following estimate holds because of (21) E IIvIIL,(n(-)) < C

(24)

I1IfMIP,r

To bound the LP-norm of the gradient of v, we observe that this gradient exists and is in LP,ioc(SZ(e)) since

Vie E W.' (1,oo), 0 < O/ < 1, Ff E LP' (WO)

(the last inclusion holds by Lemma 2). Hence for almost all x E 12(e)

lVv(x)I < I,' (Iy1)I I (F7) (x) - f(z)I + IVG£(IyI)f(z)I + IV(FJ)(x)I, and

C IIVvII L'(S(`))

<

f1<1Y1<11EI'+GE(Iyi)IPdyfRl

j(FJ)(x)-f(z)IPdz

1/e

+ llf IIi,(R') f

IVGE(r)IPrP-'dr + IIVF7Ilio(0(`)).

(25)

The last term on the right is estimated by (22), whereas the second does not exceed c I logell-Pllf IIi,(r) By (5), the first term on the right part of (25) is not greater than c

f

R1

P

dz f 1/E

IVE(r)IPrP-1dr

fRi K(r, h)(h f)(z)dh

(26)

1

In turn, quantity (26) is dominated by the first summand on the right part of (6) with g = f and thus does not exceed c I (f )P (see inequality (7) in Lemma 2). Taking (21) into account, we arrive at IIVVIIL,(c2(-)) < C f INP,r-

Comments to Chapter 4

259

The last along with (24) implies (23). Now (18) follows from (23) and (4.5/21) which completes the proof of Theorem. I We now present a corresponding result for the exterior of a narrow cylinder. It is obtained from above Theorem by means of the mapping 11(e) D x Hex.

Corollary. For p = n - 1 > 2, the following relation holds IIfIITH,p(Q<<)) - (e logel)(1-P)/P IIfIIL,(rE)

+

if (x) - f

1 I pQ(e/E) de p 1£

rExr,

I

1/p

(27)

/

where e = Ix - l; 1, Q is the function introduced in the preceding theorem and the remaining notation is the same as in Corollary 4.5. If we replace the left and omit the first term on its right part, we obtain part of (27) by IIf IITL'

the correct relation.

Comments to Chapter 4 Theorem 4.1.1 with p = 2 has long history. In 1871 Prym [174] showed that a continuous function on the boundary of a disk cannot be generally extended to a harmonic function on the whole disk with finite Dirichlet integral. Prym's work was not noticed by his contemporaries, and 35 years later Hadamard made the same observation in his note [81]. Hadamard's counterexample (dif-

ferent from that of Prym) was included in many textbooks on calculus of variations and partial differential equations. In an article on Plateau's problem and minimal surfaces, written in 1931, J. Douglas [52] gave the formula 1

87r Jo

2i

27r

JO

I f (0)

- f ('p)

12

(sin[(9 - gyp)/2])

2

for the Dirichlet integral for the harmonic function on the unit disk whose boundary values coincide with f. This formula was also used in the article by Beurling [23] in his analysis of exceptional subsets of the circle. The above formula became a starting point for the theory of functions with "fractional derivatives" in 1950s. In the works by Aronszajn [12], and by Babich and Slobodetski [16] the boundary values of Sobolev functions were characterized

260

4. Traces of Functions with First Derivatives ...

for domains with smooth boundaries. Theorem 4.1.1 for p E [1, co) was proved by Gagliardo [69].

The contents of Sec. 4.1.2-4.1.3 are taken from the paper by Maz'ya and Poborchi [149]. Most of the results in Sec. 4.2-4.6 can be found in the authors' work [152]. Lemma 4.4 is a special case of a result due to Sobolev [191] (see also Uspenski [204]).

Part III Sobolev Spaces

for Domains with Cusps

CHAPTER 5

EXTENSION OF FUNCTIONS

TO THE EXTERIOR OF A DOMAIN WITH THE VERTEX OF A PEAK ON THE BOUNDARY

Introduction Simple examples show that domains 1 C Rn with peaks directed into the exterior of 0 do not belong to EVp for p < oo (cf. Definition 1.6.2). The same is true for planar domains with inner peaks for p > 1. In this chapter we determine that for each such domain there exists a continuous linear extension operator: VP(Q) -+ V1o(R") acting in V1 o(Rn), where a is a weight nonnegative function equal to zero only at the vertex of the peak. We give sharp conditions on the weight a such that the extension operator exists. In particular, these results imply that planar domains with inner peaks are in

EVi . We also find sharp conditions on the parameters p, q, 1, n and the parameters describing the cusp that ensure the existence of a linear continuous extension operator: VP (Q) -+ V9 (Rn) with q < p. We now describe results of the present chapter using the domain

1_{x=(X',xn)ERn: xnE(0,1), IxI <W(xn)}, n>2, as an example. Suppose cp is a function in C°1([0,1]) such that wl(°,1] > 0, W(O) = 0 and the function (0, 1] E) r " w(r)/r is nondecreasing. Assume that

cp'(r) -+ 0 as r -4 +0, W(2r) < const cp(r), r E (0,1/2], (these conditions are not the most general; they will be refined in theorems below). Let V1 Q (Rn) be the weighted Sobolev space with norm I

IIuII Vp o(Rn) _ E IIaVkuIIp,R^ k=0 263

5. Extension of Functions to the Exterior of a Domain ...

264

The following assertions are consequences of theorems proved in Sec. 5.2-5.4.

In order that there exist a linear continuous extension operator: VP (Q) -4 VP o(Rn), it is sufficient, and for radial and nondecreasing or necessary that the inequality

a(x) < const

(W(r)1r)min{1,(n-1)/pl'

1p # n - 1,

be valid in the vicinity of the origin, where r = Jxl. In the case lp = n - 1 some additional restrictions are imposed on cp (not excluding though power cusps). Here the inequality just mentioned should be replaced by log(cp(r)/r)I(1-P)/P

a(x) < const (cp(r)/r)1I

In Sec. 5.6 we show that the existence of a linear continuous extension operator:

Vp(S2)->V9(S2), 1
In (Q)

with

_

tQ

\ cp(t) /

n/(p-1) dt

t < 00

1/q-1/p=l()3 -1)/()3(n-1)+1) if lq
and

1/q-1/p=(n-1)(j3-1)/np if lq>n-1. In case lq = n - 1 the factor I log(cp(t)/t)17 should be included into the integrand of In (Q), where ry = (1 - 1/q)/(1/q - 1/p), Q = (np - q)/(q(n - 1)).

Let n = 2. If o, is a weight such that u(x) < const(cp(x2)/x2)1-1/P

for x E Q and small x2, then, according to Theorem 5.5.2, there is a linear extension operator: VP (R2 \ S2) - VP o(R2). The inequality just mentioned is necessary for the existence of the extension operator if a(x) depends only on x2 for x E S2 and is nondecreasing.

5.1. Integral Inequalities for Functions on Domains with Peaks

265

Consider the planar domain D = B1 f1 (R2 \ SZ). By Theorem 5.6.4, the existence of a bounded linear extension operator:

VP(D)-*Vq(R2), 1
n = 2, 1/q - 1/p = (l - 1/p)(13 - 1)/(0 + 1). In Sec. 5.7 we study extension from a perturbed peak of the form 1(E) _ SZ \ BE, where E is a small positive parameter. Clearly Q(E) is in EVP for all

1 < p < oo and l = 1, 2, .... Let E. denote an arbitrary linear continuous extension operator: VP '(Q(-)) -4 1(R'). Then 119,,11 tends to oo as E -+ +0 if p < oo. We determine the speed of "degeneration" of operators E. Namely,

infIIEEII -

E

E

(AP(E) /

if lp#n-1,

)min{1,(n-1)/pl

(EIW (E)

(log

1+1/P

if

lp=n-1.

W(E) )

The symbol - designates equivalence uniform with respect to e. In the planar case the following relation holds inf IIEEII -

(E&(E))1-11P,

where EE : VP((R2 \ SZ) U BE) -+ VP(R2) is an arbitrary extension operator.

5.1. Integral Inequalities for Functions on Domains with Peaks Here we prove some auxiliary assertions on functions defined on a domain with an outer peak. In Sec. 5.1.1 the Friedrichs inequality is established for functions with supports in the vicinity of the vertex of the peak. Hence it follows that the spaces L,(1), WP(SZ), VP '(Q) coincide for such domains. Sec. 5.1.2 deals with Hardy's inequalities for functions on domains with outer peaks. Results of Sec. 5.1 are used in the following sections. For brevity we write . IIP,p instead of II . hp(St) and II IIP,z,c instead Throughout Chapter 5 c, co, c1, ... denote positive constants of II ' depending only on n, p,1, SZ. The symbol a - b means that c1 < a/b < c2. If a - b, the quantities a, b are called equivalent or comparable. II

IIv;(sz).

5. Extension of Functions to the Exterior of a Domain ...

266

5.1.1. Friedrichs' Inequality for Functions on a Domain with Outer Peak We begin with the description of the vertex of an outer peak. Let S2 be a domain in R' (n > 2) with compact boundary 852. Assume that 0 E 852 and that 8SZ \ {O} is locally a Lipschitz graph surface (i.e., it can be locally represented as the graph of a uniformly Lipschitz function in some Cartesian coordinates). At 0 we locate the origin of the Cartesian coordinates x = (y, z), y E Ri-1, z E R1. Let cp be an increasing function in C°'1([0,1]) 0 as t - +0 and let w be a bounded domain in such that cp(0) = 0, cp'(t) RI-1 of class C°,1. Fig. 30 illustrates the following definition.

Definition. The point 0 is the vertex of a peak directed into the exterior of 1 if it has a neighborhood U such that u n SZ = {x = (y, z) : x E (0, 1), y/cp(z) E w}. For the simplicity of presentation, we will also assume that w C that W(t) < t for t E (0,1].

B(n-1)

and

?z

yi / Fig. 30

Lemma. Let 0 be the vertex of a peak directed into the exterior of a domain 1 C R. Suppose u E Lip(Q fl U) and u(y, z) = 0 in the vicinity of z = 1. The following inequality holds IILIIp,S2nU < e IIOIUIIP,0nU,

1 < p < 00,

where c is a positive constant independent of u.

5.1. Integral Inequalities for Functions on Domains with Peaks

267

Proof. It will suffice to consider the case l = 1. Here we have u(cp(z)77, z) _

-

f

1

8t

(u(co(t)rl, t)dt

for almost all 71 E w and almost all z E (0, 1). Hence the required inequality follows for p = oo. Let p < oo. Then l u(W (z)71, z) 1P < c f l (Du) (w(t)71, t) I Pdt. Z

This estimate implies that IIuIIP,Qnu = 101 p (z)n-'dz

f

l u(P(z)rl, z) I Pd i

,

< c f cp(z)"-ldz f dt f1 l (Vu) (cp(t)ij, t) I Pdi7. l o

Z

W

The right part of the last inequality does not exceed 1

cf

,p(t)"-ldt L l (Vu)

t) JPdq = c Il VullP,unu

a

T he result follows. I The assertion stated below is a direct consequence of Lemma just established (cf. Corollary 1.5.2).

Corollary. If S2 is a bounded domain with an outer peak, then Lp(I) _ WP (Q) = VP (]) for any 1 < p < oo, I = 1, 2, .... In particular, Theorem 1.5.4 holds.

5.1.2. Hardy's Inequalities in Domains with Outer Peaks We begin with known weighted Hardy's inequalities for functions on intervals in R1.

1/p

Lemma 1. Let -oo < a < b < oo, 1 < p < q < oo. In order that there exist a constant C independent off , such that

fb w(x)

1/q

f (t)dt dx )

(Lb v(x)f (x)l9dx

(1)

)

5. Extension of Functions to the Exterior of a Domain ...

268

it is necessary and sufficient that the quantity

'sup (f r I w(x)I' dx)1/q J )

B

b

I v(x)I

-P/(P-1)dx

be finite. Moreover, if C is the best constant in (1), then B < C < B (q/(q - 1))1-1/P q1/q

Ifp= 1 orq=oo, then C=B. Lemma 2. Let -co < a < b < oo, 1 < p < q < oo. In order that there exist a constant C independent of f, such that

l1/9
6

q

(fb

I

w (x)

f

o

I

( Ja

11/P Iv (x) f (x) IPdxI

(2)

/

it is necessary and sufficient that

B

u) (f rs

r

1/q

6

w(x)Igdx I I

1-1/P Iv(x)I-P/(P-1)dxl

(f.

< 00.

The best constant C in (2) satisfies the same inequalities as in Lemma 1. 1 The proof of these assertions can be found in the book by Maz'ya [136] (Sec. 1.3.1). Lemmas just stated will be used in a full generality in the following chapter, while in the present one we need only the special case p = q. Let 0 be the vertex of an outer peak on the boundary of a domain ) C R'. We now establish some inequalities of Hardy type for functions defined on

UnQ. Lemma 3. Let Q C R' be a domain with the vertex of an outer peak on the boundary and let U and cp be the neighborhood and the function from Definition 5.1.1. Suppose that the function (0, 1) E) z -a co(z)/z is nondecreasing. If G = U n SZ, u E LP(G), p > 1, and u(x) = 0 in the vicinity of z = 1, then the following inequalities hold IIz-`tIIP,G < c IIVl

IIP,G,

IIz-101uIIP,G < c II71V1uIIP,c,

lp < n,

(3)

lp 5 n - 1,

(4)

5.1. Integral Inequalities for Functions on Domains with Peaks IIz-1

269

lp = n - 1,

0'2uIIP,G < c II a2V iuII P,G,

(5)

where a1()[log(z/W(z))](l-P)/P

al (x) = (z/W(z))`,) =a2(xW

Proof. It will suffice to verify the estimate Ilz-kaullp,G < c

IIz1-kapuIIP,G,

1 < k < 1,

(6)

in the following cases:

1° lp < n, a(x) = 1; 2° lp < n-1, o, (x) = al(x); 3° lp = n-1, a(x) = 0'2(x). Then (3)-(5) follow from (6) by iteration.

Let X = (s, t), where s E Rn-1, t E Rl,

t=1

s = ylgc(z),

z

dT W(T)

The transformation x -* X maps G onto the cylinder s E w, t E (0, 00). Put v(X) = u(x). Since Ioxul ^ IVxvl/co(z(t)) and

I D(y, z)/D(s, t) = W(z(t))n,

(6) takes the form J ds f c

J

I v(s, t) I

ds

nz(t)-kPW(z(t))na(z(t))Pdt

I vxv l

Pz(t)-(k-1)PW(z(t))n-Pa(z(t))Pdt.

s

Here v(s, t) = 0 in a neighborhood oft = 0 (we sometimes write a(z) or a(z(t)) instead of a(x) to emphasize the dependence of a only on the variable z, t and the like). The last inequality follows from the one-dimensional inequality nz(t)-kPW(z(t))na(z(t))Pdt low

< c f' IJ (tI Pz(t)(1-1)P

o(z(t))n-Pa(z(t))Pdt.

(7)

5. Extension of Functions to the Exterior of a Domain ...

270

According to Lemma 2, (7) holds with c = pP(p - 1)1-Pco,

co= sup r>0fX

1

\ J0

f-W

z(t)(k-1)P/(P-1)W(z(t))(P-n)/(P-1)cr(z(t))P/(1-P)dt)P-11

r

The substitution t -* z yields co = sup{A(b) : 6 E (0, 1)} where 6

A(6) =

W(z)n-ldz X

(fl

(z)-P)h/(P-1)dz)P-1.

(z(k-1)PW(z)1-n0' J

If p = 1, the second factor should be replaced by ess sup {cp(z)1-n(a(z))-lzk-1 : z E (6, 1)}.

Consider e.g. the case 1° lp < n, a = 1. Since V(z)/z is nondecreasing, we have

< const.

In a similar way the estimate A(6) < const is verified in cases 2°, 3°. The proof of the lemma is complete. 1 If lp > n, inequaltiy (3) is generally not true. However, it may be valid for some particular cusps.

Example. Power cusp. Let cp(z) = cza, A > 1 and

G={ (y, z) E R' : z E (0,1), y/cp(z) E w}. The same reasoning as in Lemma 3 leads to inequality (6) with or = 1 and z\(n-1)-kpdz/

c- 6SUP

( J06

al^-1)

n zkn-1D

C

J

1

-z 1

n

The last supremum is finite for all 1 < k < l if and only if lp < .\(n - 1) + 1. In this case inequality (3) holds.

5.2. Outer Peak. Extension Operator: Vy((l) - Vp o(Rn), !p < n- 1

271

5.2. Outer Peak. Extension Operator: VI(n) -4 VP s(R"), Ip < n-1 First we define the weighted Sobolev space VP 0 (G).

Definition. Let G be a domain in Rn and 0 E G. Let r be a Lebesgue measurable nonnegative function on G and suppose that o is separated away from zero and bounded in the exterior of any neighborhood of the point O. A function u is said to be in V1 Q(G) if D°u E Lp,1oc(G \ {O}) for lal < l and the following norm is finite I

II fill V

(G) _ E IJUVkuiIP,G, 15 P!5 00. k=0

We state the principal result of this section.

Theorem. Let 0 be the vertex of an outer peak on the boundary of a domain

1 c Rn and letlp
J (W(Ixl)/IxI)' ifx E Rn \ S2, Ixl < 1,

1 at the remaining points x E Rn. Then there exists a linear continuous extension operator: VP (S2) -a V1 Q(Rn).

(ii) Let W (2t) < ccp(t), t E (0,1/2].

(1)

Suppose a is a weight function on Rn such that the restriction Rn \ SZ D X H u(x) depends only on Ixl for small Ixl and is nondecreasing. If there exists a bounded extension operator: VP(S2) -+ V1 Q(Rn), then the estimate 0'(X) < C (W(IxI)/Ixl)`, X E Rn \ SZ,

is true for all x sufficiently close to the vertex of the peak.

Proof. (i) First we construct the required extension of a function u E V1 (SZ) in the following particular case. Let S2 = {x = (y, z) E R" : z E (0, 1), y/cp(z) E W}

(2)

5. Extension of Functions to the Exterior of a Domain ...

272

(cf. Definition 5.1.1) and let supp u is contained in a sufficiently small neighborhood of the vertex 0 of the peak. A sequence {zk}k>0 is defined by

z0 E (0,1), Zk+1 + cp(zk+1) = zk, k > 0.

One can easily verify that zk \, 0, zk+lzk 1 -+ 1. Moreover,

(3)

(Pk+1(Pk 1 -41,

where Wk = W(zk). Indeed, Zk

1

Pk

Pk+1 -

1

Wk+1 zk+1

WP(t)dt-40

since cp'(t) -* 0 as t -> +0. Choosing zo to be sufficiently small, we can also obtain z0 < 2z2. Put

1k = {x = (y, z) : z E (zk+1, zk-1), y/W(z) E w}, k > 1.

(4)

Note that Stein's extension operator: VP(cpk1Qk) -+ VP (R") has norm uniformly bounded in k (see Theorem 1.6.2). By Lemma 3.1.2/1, to every k > 1 there corresponds a linear extension operator Ek : Vp (1k) -4 VP '(R') satisfying t

IHVSEkVIIP,R- < CEAk 9IIDivIIP,ok,

(5)

i=0

where vEV(f2k), 0<s<1. Let {'7k}k>1 be a smooth partition of unity for (0, zl] subordinate to the covering {(zk+l,zk_1)}k>1: rlk E Co' (zk+1, zk-1),

1 ?7k(,) (z) I < C Wk

S = 0, 1.... ,

(6)

00

E'nk(z) = 1, z E (0, zl].

(7)

k=1

We introduce cut-off functions l k E C0 00(W-1) such that

Sk(y) = 0 if

I

I > 2Wk-1, WY) = 1 if 1Y1 <
(8)

5.2. Outer Peak. Extension Operator: Vl(1) -+ VP,o(R°), lp < n - 1

273

for all k > 1 and all s > 0.

Let u E Vp (1) and u(y, z) = 0 for z > z°/2. In this case the required extension u H E(°)u E VP o(Rn) can be constructed as follows. Put Uk = ulnk and define cc

(E(°) U) (x)

(EkUk) (x), x = (y, z) E R".

=E

(9)

k=1

Then E(°)ul, = u. Note that for any b > 0 the set Rn \ B6 has a nonempty intersection with only a finite number of supports of the functions l;kr/k. Therefore, the derivatives D'E(°)u E Lp,loc(Rn \ {O})

exist for lal < 1. We now check the estimate l rVj(E(°)u)IIP,Rn < C IIVIUilp,ci 0 < j < 1.

One can assume without loss of generality that u(x) = (cp(Ixl)/Ixl)I for all x E B1. Let Gk = { (y, z) : z E (zk+1, zk), lyi < 2Wk-1 }, k > 1.

Then supp E(°)u C UOO 1Gk and QlGk - Uk = (cok/zk)I

Hence 00

Il0'oj(E(°)U)IIp,,,>.

o

11V3(E(0)U)IIp,Gk'

k=1

Next k+1

11Vj (E(°)U) IIP,Gk S :Iloj

lip

i=k

k+1

j

< c> EW;'IIo9(EiUi)IIP' i=k s=°

(12)

5. Extension of Functions to the Exterior of a Domain ...

274

In view of (5), the right side in (12) is dominated by k+1

l

CE

i=k s=0

and thus 1 1/c/

Olkllv9(f'I°)N)Ilp,ck

p,$Zk Unk}1

(13)

s=0

Summation over k > 1 in conjunction with (11) yields t

IkrVi (E(°)u) lip < c E

llz'-`V8uIIP,11

8=0

Reference to Lemma 5.1.2/3 concludes the proof of (10). We now turn to the general case. Let Q be a domain of a general form with the vertex of an outer peak on the boundary and let U be the neighborhood from Definition 5.1.1. Choose a number P E (0, zo/4] (z° has been specified above) such that Bee C U and introduce cut-off functions V) E Co (B2e), T E CO '(U) satisfying ?p I = 1, T?p = i,i. By Theorem 1.6.2, there exists a linear continuous extension operator

E:VP(QUB012) VP(R"). For arbitrary function u E VP (1), we set v = (1 - O)u and extend v to be zero on the set Be/2 \ Q. The required extension operator

E:VP(Sl)-4 Vpo(Rn) is defined by Eu = 7-E(°) (?iu) + Ev. The proof of statement (i) is complete. (ii) Let f E Co (0, 3), f (t) = 1 for t E (1, 2). For any small p > 0, we put UPIn\U = 0,

ue(x) = f (z/B), x = (y, z) E U n f2.

Here, as above, U is the neighborhood from Definition 5.1.1. Clearly ue E VP (11) and

IluellP,l,s2 < c P1-!pp(3P)r-1.

(14)

5.3. The Case 1p = n - 1

Let E

:

275

Vp(0) -+ VP 7(Rn) be a bounded extension operator. By the

monotonicity of o//2p

II (euP/ ('1 z) Ilp,l,Rn-l dz.

IIUeIIp,l,O > CQ(o)p J e

Since the space VP(Rn-1) is imbedded into Lq(Ri-1) for q = (n - 1)p/(n 1 - lp), the last integral is not less than C

f

2P

IInPz)II9,nzdz,

(15)

e

where Qz is the section of U f1 0 by the hyperplane z = const. Quantity (15) oW(g)'-1-'P and hence is comparable to Clol-IPW(3O)n-1 > IIuPIIp,I,1 >

C2Q(o)PoW(o)n-1-lP.

The result follows because of the relation W(3o) - W(o).

5.3. The Case lp = n-1 In this section we give sharp conditions on weight that ensure the existence of a linear continuous extension operator: VP(Q) -* V' P,a(Rn) for domains with outer peaks in the case lp = n - 1. The principal result will be given in Sec. 5.3.3, while Sec. 5.3.1-5.3.2 contain auxiliary assertions.

5.3.1. Positive Homogeneous Functions of Degree Zero as Multipliers in the Space VP Q(R')

Lemma stated below gives a description of a class of multipliers in VP o(Rn) in case o = 0 at the origin. This lemma will be used in Sec. 5.3.3.

Lemma. Let o be a positive function on Rn \ {O} and suppose that o-(x) depends only on xl for small Ixl and is nondecreasing. If ( is a positive homogeneous function of degree 0 in C°° (Rn \ {O}), then (is a multiplier in VP Q(Rn) for lp < n and the following estimate holds for any u E Vj,o(Rn) II(UIIVV o(Rn) < C IIUIIV,,o(Rn)

(1)

5. Extension of Functions to the Exterior of a Domain ...

276

Proof. Since I (V 8() (x) I < c l x I -8, it will suffice to verify the estimate II I xl -8QV1-gull P,R° < C IIoVIUIIP,R°, 15 S < 1

(2)

provided u has compact support in a small neighborhood of the origin. Let supp u C Be and the number e be so small that a(x) depends only on IxI and is nondecreasing for x E Be. Note that (2) is a consequence of the inequality (3)

11 lxl-eoVI-9uilp,BQ <_ a 11 Ixi1-s(YVI-,+luliP,BQ,

where 1 < s < 1. Let v denote any derivative of u of order 1 - s. We observe that (3) follows from the estimate

f f f

e Iv(r,O)IPQ(r)Prn-l-'Pdr

dB

s, -1

0

Q


fSn-I

dO

I(Vv)(r,O)IPrn-1-(a-1)PU(r)Pdr

0

and the last is a consequence of the one-dimensional inequality P

e

tµa(t)Pdt

fo <

Cf 0

l f (t)

I Pµ+PQ(t)Pdt

(4)

for p > -1. According to Lemma 5.1.2/1, inequality (4) is valid if and only if sup{A(b) : b E (0, e)} < oo, where 1

A(b) = f b tI`Q (t)Pdt l f e t-

v(t)

P

dt I P

o

(if p = 1, the second factor should be replaced by ess sup{(tµ+1Q(t))-1 : t E (b, p)}). By the monotonicity of o, we obtain that

A(6) < L6tt(L

cc

±tdt\P-1

t

D

t1f


Thus, (4) is valid and (1) follows. The proof of Lemma is complete.

5.3.TheCase lp=n-1

277

5.3.2. Lemma on Differentiation of a Cut-off Function

Let cp be the function appearing in Definition 5.1.1 and let {zk}k>o be the sequence introduced by (5.2/3). Fix a 0 > 0 and put (k(Y) = 1 + 1og(Iyl/c0k)

y E Rr-1 \ {0},

0log(cpk/zk)

(1)

where Wk = co(nk), k = 0,1, ... .

Lemma. If A E C°°(R1) and A has bounded derivatives of all orders, then for any k > 0 the following estimates hold (o9(A°(k))(y)1

cIlog(Wkxk1)1-11y1

s > 1, y E

Rn-1 \ {0},

(2)

1(V.(A ° (k))(y) - (V.(A ° (k+1))(y)I < C l yl -e ((Wk - cok+1)co k

1

+ cOkZk

1), s > 0, Wk < I yl < cPk BZk,

(3)

where c = c (zo, s, 0, A, cp).

Proof. Let a E Z'-' be a multi-index and et the unit basis vector along the

axis ysinR"-1, i=1,...,n-1. We have D'+e' (A (k)(y) = Dy (A'((k(y))a(klayi)

(;)DA'((k(Y))D-+e

k

log lyl,

(4)

Q
where µk = (01og60kzk 1))-1. If I a I = j in (4), then

oj+1(A ° (k)(y)l <- C I Ml L

lVr(ac

° (k)(y)l lylr

-1, IyI

r=0

Now (2) follows from the last estimate by induction. Turning to (3), first we consider the case s = 0 : I(A o (k)(y) - (A o (k+l)I <_ C I(k(y) - (k+1(y)l = C Ilik log(IyI

1) - I4+11og(IylVk+1)I

< c (1(µk - 1k+1)109(IYIWk 1)I + II4+110g(Wk+1Wk 1)1)

0.

5. Extension of Functions to the Exterior of a Domain

278

...

If Wk < IyI < cps 8z°, then I(µk - 1k+1) 1og(Iylck

1)15

1µk - µk+1I log(zkl

C I log(Wk+lWk lzkzk+l)I < C2 ((Wk

-

lzkzk+l

1)

- lI

1) Wk 1 + (Pkzk 1).

Furthermore, 1)I < C (Wk -'Pk+1)'Pk

I1k+1

1

Thus, (3) holds with s = 0. Let s > 1 and let IVi ((A ° Ck)(y) - (A ° (k+l)(y))I CIYI-j((Wk

<

- cPk+1)c°k1 + Wkzk 1)

(5)

for j < s-1 and Wk < 1Y1 < wk'-Bze. Fix a y E Z'-', Iryl = s. Then 'y = a+ei for some 1 < i < n - 1 and a E Z+ 1. Using (4), we obtain

1Db((A°(k)(y) - (A°Ck+l)(y))I c

IyI1131-81(µk - µk+l)D°(A' ° (k)(y)I 0
+C Iµk+1I E

Iyi1131-8ID13(A'

o (k)(y) - D" (A' ° (k+l)(y)I

(6)

13
It follows from (2) (with A replaces by A') that

I (µk - MAD' (A' - (k) (y) < C IyI-11311µk

- µk+1I

< c IyI-1131 ((Wk - Wk+1)Pk 1 + Wkzk 1), Q < a,

and the first sum on the right part of (6) is dominated by the right part of (3). To bound the general term of the second sum, we use the induction hypothesis (5) (with A replaced by A' and j replaced by 101). Then the second sum on the right part of (6) is also dominated by the right part of (3). The proof of Lemma is complete.

5.3. The Case Ip = n - 1

279

5.3.3. Extension Operator: VP '(Q) -> VP 7(Rn), Ip = n - 1 The following assertion is the principal result of section 5.3.

Theorem. Let S2 be a domain in Rn with the vertex of a peak directed into the exterior of SZ and let Ip = n - 1, p E (1, oo). (i) Suppose that for some 5 > 0 w(t + ap(t)) = w(t) [1 + O((cp(t)/t)a)] as t -4 +0

(1)

and that the function (0, 1] E) t y W(t)/t is nondecreasing. Define a weight function o on R" by o(x) _ 60(ixI)/IxI),I log

MIxI)/IxI)I1-1/p

(2)

if x c B1 \ SZ, o(x) = 1 otherwise. Then there exists a linear continuous extension operator: Vp(Sl) Vp',o(Rn).

(ii) Let cp satisfy (5.2/1) and let or be a weight function on R" such that the restriction R" \ S2 x -- o(x) depends only on IxI for small IxI and is nondecreasing. If there is a bounded extension operator: VI(Q) --> Vp,o(Rn), then I1-1/p o(x) < c MIxI)/IxI)'I log for all x E Rn \ ?I sufficiently close to the vertex of the peak.

Proof. Let A be a function in C' (RI) such that A(t) = 0 fort < 1/3, A(t) = 1 for t > 2/3. By {zk}k>o we mean the sequence defined by (5.2/3). Recall that Zk \ 0, Zk+1Zk1 - 1, Wk+1Wk 1 - 1, cpkzk 1 -+ 0, where Wk = W(zk). Fix a

0 E (0,min{1/2, 5/1, 1/l}). Let (k be introduced by (5.3.2/1) and put

Xk=A0(k Clearly Xk E Co (R-- 1), k > 0. The following inequalities are easily verified:

(k(Y) < 0 if

JYJ > Wk -BZk)

5. Extension of Functions to the Exterior of a Domain ...

280

(k(y) > 1+log(wk-1/Wk)/(0log(c,k/zk)) if Ill < Wk-1Choosing zo E (0, 1) to be sufficiently small, we obtain that for all k > 1 Xk(y) =

1 if IyI < k-1 and

Xk(y) =

1-0 0 if IyI > (P'1-0z0 k

-

One may also assume that 2cpk-1 < W1-0ze k for k > 1 and that z0 < 2Z2k We now turn to construction of the required extension operator VP (0) 9 U H Eu E VP Q(R").

The general case is reduced to the case when 1 has the form (5.2/2) and u(y, z) = 0 for z > zo/2 (see the end of the proof of Theorem 5.2 (i)). Let {lJk}k>1 be a partition of unity with properties (5.2/6-7) and {Sk}k>1 a sequence of functions in Co (R"-1) satisfying (5.2/8). We introduce the cells S2k by (5.2/4) and linear extension operators Ek : VP(clk) -4 VP(R"), k > 1, subject to (5.2/5). Let ;9k be the mean value of u on 11k. The operator u H Eu is defined by Eu = v + w, (3) where 00

I 9k71k(z)Xk(Y),

(4)

k=1 00

w(x)

1:

Uk))(x),

(5)

k=1

x = (y,z) E Rn and Uk = UInk. It follows from (3)-(5) that EuIn = u. Note also that the derivatives DaEu E Lp,loc(R" \ {O}) exist for Ial < 1. The following estimates should be verified IIo VjvIIp,Rn < C IIVIUUIp,n, 0 < i:5 1,

(6)

II7VjwIIp,R° < C IVIUIIp,n+ 0 < j < 1.

(7)

We may assume without loss of generality that o,(x) is defined by (2) for x E B1 and o(x) = 1 for IxJ > 1. Put Gk =

{x = (y, z) : z E (4+1, 4), y E R"-1, IyI <

,k-°zk

1,

k > 1.

5.3. The Case lp = n - 1

281

If x E Gk, then

JIxI -zkl = IIlylz+z2 - zk1(IxI+zk)-1 (IyI2 + zk - zk+l) Zk l < (Pk(iPkZk

1)1-29

+ 2c'k < 3cPk,

so W(IxI) -'Pk + O(Wk) and

o'(x) - Qk = (Wkzk l)'11og(wk/zk)l1-1/p, x E Gk.

(8)

Proof of estimate (6). Clearly k+1

v(x) _ iv'i77i(z)Xi(y) i=k

_ Uk+lXk+l(y) +71k(z)(ilk - uk+l)Xk(y) +uk+1r)k(z)(Xk(Y) - Xk+l(Y) ), x E Gk.

(9)

Let a = (al, ... , a,) E Z+, j al = j. We will distinguish two cases:

1)an=0 and

2)

1) First suppose that j = 0. Then (8), (9) imply II7VIIP Gk

< CoP(IukI' + <

mesn(Gk) I log Wklp-1(Iuklp +

lk+l p) k

where ,bk = Ok/Zk- Since IukI"co

(10)

<- C IlullP,slk,

it follows that Ilavll p,Gk -- C

Ilullp,clkUnk+1,

k > 1.

(11)

Let 0 < j = lal < 1, an = 0. In this case k+1

IIo,D°vllp,Gk < Co-k> IuiI HHOjxillp,Gk i=k

(12)

5. Extension of functions to the Exterior of a Domain ...

282

Lemma 5.3.2 in conjunction with Xk(y) = Xk+1(y) = 1, yj < Wk, yields c

IIV X:IIP,Gk < Ilog

1_sZS

k

IyI-lidy

k

i= k, k+ 1.

< C Ok I log')k 11-P

Now the right part of (12) is dominated by ccpkzk "(IikIP + I9k+1IP) Using (10), one arrives at IIJD°vllP,ck < c IIz-`uIIP SZkUS2k+1) k > 1.

(13)

Clearly, (13) also holds for a = 0 in view of (11).

2) Let a = (13, j - s), where Q E Z+ 1, IQI = s < j < 1. Identity (9) implies that c

'IDavI < Iuk - uk+1IID'XkI + Iuk+1IIDI(Xk - Xk+1)I

k

(14)

on Gk, k > 1. Applying Lemma 5.3.2, we find WS kP-iP


3

II o,D' Xk II PP,Gk < C

,Ok-9(n-1-sP)I

flyl
-P,

I y I -'Pdy BZk

4Gk - Wkxk

1

Hence WkP-iPIIoD1XkIIP,Gk

CzkP(i-1)Wk-P

k> 1,

(15)

because p - O(n - 1 - sp) > 0. It follows from the inequality (16)

IIu - UkUIP,Ok < COklIVUIIP,nk

that

(Pk-PI7dk

- Uk+1IP < C IIVuIIP !ftk, k = Qk U Qk+1

The last and (15) give WkP-7PIUk

- Uk+1IPIIJD'XkIIP,Gk < c IIZ1-'VullP

Let us bound the quantity (P

- Xk+1)IIP,Gk

k > 1.

(17)

5.3.TheCase lp=n-1

283

e estimate If Iyl < then Xk(y) = Xk+1(y) = 1, and if Wk < IyI < cpk1-e zk, (5.3.2/3) holds. Therefore

k,

s

Wk -j )p II o

<

CWkp-n+2o"k (I((Pk

-' wk+1)Wk 1IP + 10k)

Li<-°z I1I-5Pdy

9(n-1-sP)(Pkzk 'p < C kzk ip, Ok = (Pkzk 1.

IP-1


DR (Xk - Xk+1) IIP,Gk

Here we have used that Wk - Wk+1 <- CWky)k because of (1), and, furthermore,

min{p, bp} > O(n - 1 - sp). Applying (10), one obtains WkP-7Pluk+1l ll0D1(Xk - Xk+1)lIP,Gk <

ClIz-1UIIP,Qk+i.

(18)

Estimates (14), (17), (18) imply that in the case 2) the inequality IIoD'vllP,Gk <

IIz-`ullp

-

+Ilz' 'VullPAk

(19)

holds with k > 1 and SZk = 11k U SZk+l. By (13), the same inequality (19) is true in the case 1). Since the multiplicity of the covering {1 k}k>1 is finite, summation over k > 1 leads to c IIo,D°vllp,G <-

IIz-`U,llp,n

+

lizl-`Vullp,n,

where a E Z+, j al = j < 1, G = Uk 1Gk. Now (6) is a consequence of Lemma 5.1.2/3 and the inclusion supp v C G.

Proof of estimate (7). It is easily seen that supp w c G and 00

OrkllojwIIP,Gk, 0 < j < 1.

Ilo.ojwlIP

(20)

k=1

Furthermore, k+1

j

'Pi-'IloB-6i(Ui - Ui)Il

IIVjwIIP,Gk < CE i=k

s=0

.

5. Extension of Functions to the Exterior of a Domain ...

284

With the aid of (5.2/5) and (16) we dominate the right part by k+1

l

CE EWi-'IloaullP,S2i' i=k

s=1

Hence I

QkiIVjWIlP,Gk <_ c> Ilze-'V uiJ7,

,

7 =0,...,1.

(21)

s=1

Relation (20) and estimate (21) imply that !

IIUv,wIIP 5 c Ilza-losUIlP,O s=1

Reference to Lemma 5.1.2/3 concludes the proof of inequality (7) and, therefore, statement (i) of Theorem. (ii) Let {ue} be the family of functions introduced in the proof of statement (ii) of Theorem 5.2. We recall that for any small e > 0

ue E Cm(1), uQ(x) = 1, z E (e, 2e), x = (y, z) E s2 n u, and estimate (5.2/14) is valid (U is the neighborhood from Definition 5.1.1).

Let ( E C°°(R" \ {O}) be a positive homogeneous function of degree 0 satisfying ((x) = 1 on the cone 2Iyi < z and ((x) = 0 outside the cone Iyj < z. Let E : VV(Q) -+ VP,Q(R") be a bounded extension operator, where Q(x) depends only on IxI for small IxI and is nondecreasing. We put ve = t;Eue. In view of Lemma 5.3.1, the following inequality holds clllvellV, ,(R') < IIEueIlvp,,(Rn) <_ c2IIUQIIP,I,si

By (5.2/14) and the monotonicity of a 2e

a(e)P [

(22)

II ve(', z)IIP,I,Rn-=dz < c

e

Fix a z E (e, 2e). Since ve(y, z) = 1 for y E W(z)w, an application of Lemma 3.1.4 to the function

(log(z/W(z)))'

B(zn-1) 9

y H ve(y, z) yields

Alive(', z)IIP,B= +

z1-nllve(', z)II P,B, > c,

(23)

5.4, Outer Peak. Extension for lp > n - 1

285

where Vi is the gradient of order l in yl, ... , Yn-1. Note that ve(y, z) = 0 for Jyj > z and hence IIve(',z)IIp,B.

cz'IIVive(',z)IIp,B.

Combining the last and (23) gives II Vive(', Z) II P,B > c [ log (z/w(z))]

1-p

(24)

This estimate and (22) imply that a(p)p log (W(p)/p) l1-p < c

for any p > 0 sufficiently small. Since W(3p) - cp(p), the result follows.

5.4. Outer Peak. Extension for lp > n-1 In this section a linear bounded extension operator: Vp(1) -+ Vp,Q(Rn) is constructed for a domain S2 C Rn with an outer peak and a weight function a in the vicinity of the vertex of the peak. subject to a(x) It is shown that the last relation is sharp. (,(Ixl)/Ixl)(n-1)Ip

5.4.1. Extension from a Peak to a Circular Peak and to a Cone To facilitate the proof of the principal result of Sec. 5.4, we state two lemmas.

Lemma 1. Let Q be given by (5.2/2), where w and W are the same as in Definition 5.1.1, 0 C B(n-1) Let M > 1 and put G = {x = (y, z) E R" : lyl < Mcp(z), z E (0,1)}.

F o r 1 < p < oo and l = 1 , 2, ... , there exists a linear continuous extension operator F : V7 (SZ) -* V7 (G). F can be constructed to have the following property: if u E VP (SZ) and u(y, z) = 0 for z > p, then (Fu) (y, z) = 0 for z > 2p provided p > 0 is sufficiently small.

Proof. Positive constants c appearing below depend only on n, p, 1, M, Q. Consider a sequence {zk}k>o given by zo = 1, Zk+1 + cp(zk+1) = Zk, k > 0.

5. Extension of Functions to the Exterior of a Domain ...

286

It is readily checked that zk \ 0, zk+lzk 1 -+ 1,

cpk+ltpk 1

-* 1 (where

Wk = O(zk)). Let S2k be defined by (5.2/4) for k > 1 and SZo = {(y,z) E S2 : z E (z1izo)}.

For each k > 0, let Ek

VP (R") be a linear extension operator satisfying (5.2/5). We also introduce a partition of unity {'qk}k>1 subject to (5.2/6-7). Put : VP (f2k)

o(z)

11 - 771(x) if z E [z1i1], if z E [0, zl].

0

Then CO

rio E C°O([0, 1]),

E 77k (Z) = 1, z E (0,1]. k=0

Az

Fig. 31

Let us extend a function u E VP(S2) to the circular peak G (see Fig. 31). By Theorem 1.5.2, there exists a linear mapping u H Pk E PI-1 such that IIV8(u - Pk)Ilp,nk C cWk

'IJVzullp,j1k,

k > 0, s < 1.

(1)

Putuk=ul0k, x=(y,z)EGand 0

00

(Fu)(x) = >'ik(z)Pk(x) +>71k(z)(Ek(uk - Pk))(x) k=0

k=0

(2)

5.4. Outer Peak. Extension for ip > n - 1

287

We claim that u H Fu is the required extension operator. Indeed, the identity Ful n = u follows from (2). Furthermore, Fu E VP (G \ B6) for any 6 > 0 sufficiently small. We now check the inequality c (IIV,uIIP,a + IIVIuHHP,ci), j:5

IIV3FuIIP,G

(3)

To this end consider the cells Gk = { (y, z) E G : z E (zk+1, zk) }, k > 0,

and observe that FulGk = vk + wk, where Vk = Pk + 7)k+1 (Pk+1 - Pk), k+1

Wk = E 7)iei (ui - Pi) . i=k

According to (5.2/6), k+1 Iloiwkllp,Gk

i

< CE (Pi-'IlVsl'i(ui - Pi)IIP i=k

s=0

This in conjunction with (5.2/5) and (1) yields Cik = SZk U Stk+l, k > 0.

Iloiwkllp,Gk 5 C

To bound Iloivkllp,Gk, we note that

i Iloivkllp,Gk <- IIViPkIIP,Gh + C

0ek-iIlV-(Pk+1-Pk)lIP,Gk

8=0

and then use the estimate IIQIIP,Gk

C

IIQIIP,Gkns1,

Q E P(-1.

This results in

i CllVivkllp,Gk

5IlVIPkllp,Gkno+0Wk'Ilvs(Pk+1-Pk)IIP,Gknn.

s=0

(4)

5. Extension of Functions to the Exterior of a Domain ...

288

The right side of the last inequality is dominated by j

k+1

C E E Wi, -j II Vs (u - Pi)

II Vj

IIP,2i

s=0 i=k

Hence and from (1) follows the estimate IIVjvkIIP,Gk

C

(II07uiiP,S2k + IIvIUIIPAk), k > 0.

(5)

Since the multiplicity of the covering {Ok}k>o is finite, (4) and (5) imply (3).

We conclude the proof of Lemma 1 by the observation that if u(y, z) = 0 I for z > zk with some k > 2, then (Fu) (y, z) = 0 for z > zk_ 1. In the following lemma we extend functions from a peak to a cone described below.

Lemma 2. Let S2 be the same as in Lemma 1 and let cp satisfy (5.2/1). Put

H = {x = (y, z) E R" : z E (0,1), I yI < z} and (W(Ixl)/Ixl)(n-1)/P

Q(x) _ 1

if x E H n B1,

ifxEH\B1.

Then there exists a linear continuous extension operator: VP (S2) -+ Vp a (H),

where 11andlp>n-1. Proof. Let

M = sup {cp(4t)/cp(t) : t E (0,1/4]}.

This value M specifies the circular peak G introduced in Lemma 1. Let F : VP (Sl) -* VP (G) be the extension operator constructed in Lemma 1. We define a sequence {ek}k>o by ek = 2-keo, where eo E (0, 1) is chosen to be so small that the following conditions hold: sup {cp(z)/z : z E (0, eo]} < min {1/4, M-1}

and (Fu) (y, z) = 0 for z > eZ if u(y, z) = 0 for z > e3. It is clear that then

{xEG:z<po}CH and ek-1-ek+1>2w(ek-1), k>1.

5.4. Outer Peak. Extension for Ip > n - 1

289

We also introduce the cylinders Dk = { (y, z) E Rn : z E (Pk+1, Pk-1), III < 1, for which { (y, z) E SZ : z E (Pk+l, Pk-1) } C Dk C G;

the first inclusion holds because 0 C B(,n-1) (cf. (5.2/2)) and the second by the definition of M. Construction of the extension operator: VP '(Q) E) u H Eu E VP Q(H). It will suffice to restrict our consideration to the case u(y, z) = 0 for z > '03. The general case is reduced to the case just mentioned with the aid of a smooth cut-off function and Theorem 1.6.2. Let u E VP (1l), u(y, z) = 0 for z > P3 and put

uk = FuIDk, k > 1. By Lemma 3.7.2/1, there is a linear map uk H Pk E Pt-1 satisfying IIVs(uk-Pk)IIp,Dk

C0k 8IIVzukllp,Dk, k>1, S<1.

(6)

According to Lemma 3.7.2/3, to each k > 1 there corresponds a linear extension operator Vp(Dk) D f y Ekf E Vpl(R.n) such that t C(Pk/Pk)(n-1)Ip

Pk

IIVsEkfIIp,Rn <

sIIVifIIp,Dk, 9 = 0,

1,

(7)

i=0

where Wk = co(Pk). Let {tik}k>1 be a smooth partition of unity for (0, P1] subordinate to the covering {(Pk+1, Pk-1)}, that is: A E CO (Pk+l, Pk-1), I/ks) (t) I < cPk s, s > 0, t c Rl,

(8)

00

E Ilk(t) = 1, k=1

t E (0, Q1]

(9)

290

5. Extension of functions to the Exterior of a Domain

...

We put

(Eu) (x) = > µk(z)Pk(x) + E lzk(z) (Ek(uk - Pk)) (x) k=1

(10)

k=1

for x = (y, z) E H and check that Eu is the required extension of u. The equality EuIn = u follows from the definition of E. Furthermore, (10) implies that Eu E VP (H \ W6) for any small 6 > 0. We now turn to the proof of the estimate (11) IIoV EuIIP,H
Let v(x) be the first sum on the right part of (10) and let Hk = {x E H : z E (Pk+1, Pk)}, Qk = (W(ek)/Ok)(' -1)/P

k > 1.

Then 00

IlkVjvIlp,H -

EO'kIIVjvIlp,Hk'

(12)

k=1

Because

= Pk+1 +µk(Pk - Pk+l),

VIHk

we have

IIVjvIIP,Hk 5 IIVjPk+1IIP,Hk

+CLek j IIV1(Pk-Pk+1)IIP,Hk i=0

An application of Lemma 3.7.2/2 gives (cf. Fig. 32)

1-1-j PkHHVi+jPk+1IIP,HknDk+i

IIVjvIIP,Hk 5 CUk 1 i=0

eks-j IlVi+s (Pk - Pk+1) IIP,HknDk+l .

+Cak i=0

(13)

s=0

The right part of the last inequality can be dominated with the aid of (6): I I Vi+jPk+1 I IP,HknDk+1

5.4. Outer Peak. Extension for 1p > n - 1

291

< IIDi+juk+1IIP,Dk+i +Cek

i-j11V1uk+111p,Dk+1, k+1

Iloi+9 (Pk - Pk+1) IIP,HknDk+1 < E II Vi+e (uv - Pv) II P,D,. v=k k+1

CE ev i-91IViuvIlp,D,. v=k

yi J.

Fig. 32

Hence and from (13) follows the estimate 7kllojvllp,Hk

C IIFuIIP,1,DkUDk}1,

(14)

where k > 1. Now (14) and (12) yield IJUVjVIIP,H
According to Lemma 1 and Lemma 5.1.1, the last implies Il7'jvllp,H < C 11VIU1IP,a, 0 < j < 1.

(15)

Let w(x) be the second sum on the right in (10). If j < 1, then k+1

j

IlojwIIP,Hk
ea'IIViE,(ue-P.)IIP

(16)

5. Extension of Functions to the Exterior of a Domain ...

292

In view of (7), the right part of the last inequality is not greater than k+1

I

CO"k 1

,r-j IIVr(us - Ps)IIP,D,

(17)

s=k r=0

1-jIIFuII

which is dominated by cu-1 p,DkUDk+1 with the aid of (6). This k ek means that if we replace v by w in (14), the resulting estimate is true. Note also that relation (12) holds where v is replaced by w. Now the inequality IloViwllp,H < C IloiullP,n, 0 < j < 1,

can be deduced by the same argument as (15) has been deduced from (12), (14). Estimate (11) is established and the proof of Lemma 2 is complete. 5.4.2. Extension Operator: VP (0) -+ VP Q(R') for lp > n - 1 In this subsection we prove the principal result of Sec. 5.4.

Theorem. Let lp > n - 1, 1 < p < oo, and let S2 be a domain in R" with the vertex of an outer peak on the boundary. Assume that condition (5.2/1) holds.

(i) If we put cr(x) _

(w(Ixl)/IxI)(n-1)IP

for x E Rn \ SZ, IxI < 1

and o(x) = 1 at the remaining points of Rn, then there is a linear bounded extension operator: VP (Q) -4 Vp Q (Rn).

(ii) Let or be a weight function on Rn and suppose that the restriction Rn\1l E)

x H o(x) depends only on IxI for small IxI and is nondecreasing. If there exists a bounded extension operator: VI(Q) VPa(R") and the domain w from Definition 5.1.1 contains the point y = 0, then o(x) < C

(w(Ixl)/Ixl)(n-1)Ip

for all x E Rn \ Sz sufficiently close to the vertex of the peak.

Proof. (i) The case of a general domain is reduced to the case when 1 has the form (5.2/2) in the same way as in Theorem 5.2 (i). By Lemma 5.4.1/2,

5.4. Outer Peak. Extension for Ip > n - 1

293

it is sufficient to construct a linear extension operator: VP,Q(H) - VP Q(Rn), where H is the cone introduced in that lemma and the weight is given by (,(Ixl)/IxI)(n-1)/P

r

if IxI < 1, if IxI > 1.

Il l 1

To define an extension

V, (H) E) uHEuEVP,o(R"), A or we can assume that supp u is contained in a small neighborhood of the origin. The general case is then provable with the aid of a smooth cut-off function and Theorem 1.6.2.

Let ek = 2-k, k > 0, and let u E VPo(H), u(x) = 0 for IxI > 92. Put

H(k)={xEH:Qk+1 1. By Theorem 1.5.2, there is a linear map u H Pk E Pi-1 such that

llvs(u -

<_ C Qk 9llV1uIIp,H(k),

Pk)IIp,H(k)

k > 1, s < 1.

(1)

According to Lemma 3.1.2/1, to each k > 1 there corresponds a linear extension operator D f H SO E Vp(Rn) VP(H(k))

subject to t

9IIVifllp,H(k), S < 1.

Iloe(Ekf)IIP,R- <

(2)

i=0

Let {1k}k>1 be a family of functions satisfying (5.4.1/8-9). We put Uk = UJH(k), 00

µk(Ixl)Pk(x), x E Rn,

v(x) _

(3)

k=1

w(x) = Eµk(Ixl)(Ek(Uk - Pk))(x), x E Rn,

(4)

k=1

Eu = v + w.

(5)

5. Extension of Functions to the Exterior of a Domain

294

...

It follows from (3)-(5) that Eul H = u and that Eu E VP (Rn \ Ba) for any 5 > 0. We now check the estimate Ilo V j (Eu) Ilp,R- < C IIUII V;,,(H), 0-5j51-

(6)

Proof of inequality (6). Consider the annuluses

Ak={xER'j:Ok+l1 and note that v(x) = Pk(x) +µk+1(IxI)(Pk+1(x) - Pk(x) 1, X E Ak. Hence IITVjvllp,Ak <- C

7kllVjPkllp,Ak

9

[0k '11v8 (Pk+1 - Pk) IIP,Ak, k > 1,

+CO'k

(7)

8=0

(n-1)/p

where Qk =

Since

IIQIIp,Ak <- c IIQIIp,H(k)

for

QEPP_1,

we have c II V,Pk ilp,Ak <- II ViUIIp,H(k) + lo; (u - Pk) II p,H(k), i <<- 1 - 1, k+1 C

IIv,

(Pk+l - Pk) IIP,Ak <- E 11V- (u - Pi) II p,H(.) ,

S < 1-1.

i=k

A combination of these inequalities with (1) and (7) yields k+1

IkYV3vllp,Ak <- IIGVjuIIp,H(k) + E Ilo-o(uIIp,H(.), k > 1.

(8)

i=k

Because suppv C B0i, (8) implies IlcTo,vllp,R° < C IIuIIV,,o(H), 0 < j < 1.

(9)

5.4. Outer Peak. Extension for !p > n - 1

295

An analogous estimate holds for the function defined by (4). Indeed, if

k>1and0<j
k+1

j

IloVjwlIP,Ak
JIV9Ei(ui-Pi)IIP s=0

Applying (1) and (2), we arrive at k+1

ikrVjwllp,Ak

CE Ik YVzuilp,HCNN i=k

It follows from the last inequality and the inclusion supp w c Bei that IJUVjwllv,R.n < c 110 VIUi1P,H.

This estimate and (9) give (6), and thus Eu is the desired extension of u. The proof of statement (i) is concluded. (ii) Consider a function f such that f E C0 (1, 2), f > 0, f 1(4/3,5/3) = 1.

Let U be the neighborhood from Definition 5.1.1. For any small p > 0, we put uel Q\u = 0, ue(x) = f (z/,o), x = (y, z) E U n Q.

Then ue E COO (0) fl Vp (0). Let E : VP (0) - Vp o (Rn) be a bounded extension

operator where o(x) depends only on IxI for small IxI and is nondecreasing. We claim that IIVIEuQIIP,nQ >-

Ce-i+n/P

(10)

with

Hp ={(y,z)ERn:zE(P/2,2P),yERn-1}. The proof of this fact is rather cumbersome. Note that the function (Eue) (y, ) is in Vp (Rl) for almost ally E Rn-1. By

Theorem 1.5.2, for the same y c Rn-1, there is a polynomial P(y,, ) E Pi 11 satisfying 2e

II (Eue) (y, ') - P(y, ') 11%(@/2,2e) < C

eiP

JB/2

a` (Eue)

8zi

(y, z) Pdz.

(11)

5. Extension of Functions to the Exterior of a Domain ...

296

We can assume that 1-1

P(y,z) _ > ak(y)zk,

(12)

k=0

where e

ak(y) = e-1-k f bk(t/e)(Eue)(y,t)dt

(13)

p/2

and {zik}j o are standard functions in Cow (1/2,1) (see Theorem 1.5.2). Put ve(y) = P-1

f

2

((Eue)(y, z) - P(y, z))dz, y E Rri-1.

(14)

e

Clearly (Eue) (0, z) = ue(0, z), z E (P/2, 2P)

If y = 0, then all coefficients (13) equal zero. Hence 2p

2

ve(0) = 0-1 f ue(0,z)dz = f f(z)dz > 0. e

1

From Sobolev's imbedding

VI(B(n-1)) e P

C

C(B(n-1)) e

nL o

o(B(n-1))

e

we obtain C

0(n-1)/PIIVQIIoo,BQ

5 IIVPIIP,B, + QIIIVIVpHIP,BQ,

whence C C e(1-n)/PIIVPIIRRn-1

+

LO'-(n-1)/PIIDIVeIIRRn-1.

15)

Let us bound the right side in (15). If Ial = 1, then 2e

D°v p(y) = P-1 f Dy ((Eue) (y, z) - P(y, z)) dz. e

It follows from (13) that I D'ak (y) I < C e-k-1/P II Dy (Eue) (y, -) II Ln(e/2,e)

(16)

5.5. Inner Peaks

297

and, consequently, I Dy P(y, z) I <_ c e 1I" II Dy (Eue) (y, ') II L,,(el2,e), z E (e/2, 2B).

A combination of the last estimate with (16) yields IIVzvoIIP,R^-1 < Ce-1IPIIVzEuellp,n".

To bound

(17)

IIveIIP,R^-1, we apply Holder's inequality to the integral on the right

in (14) and then use (11). This results in IIVpIIP,R^-1 < Ce1-1/PIIVjEu,IIP,nQ

(18)

Now (15), (17) and (18) give (10).

The remaining part of the proof of statement (ii) is not difficult. By the monotonicity of o,, (10) implies !+n/PQ(e/2), IIUVLEueIIP,nQ > CC

whereas

IIaVICU"llP,R° < c IIueIIRi,n C C1e11P-1,p(2L0)(n-1)/P

Since W(2p) - W(e), the inequality U(e) 5 C

is valid. The proof of Theorem is complete.

I

Remark. One can observe that the proofs of Lemma 5.4.1/2 and the preceding theorem remain true for p = oo. So that domains satisfying the assumptions of the above theorem are in EVE for all l = 1, 2, .... However, this result is well known for a wider class of domains. See Whitney [215], [216] and the comments to the present chapter.

5.5. Inner Peaks 5.5.1. A Multi-Dimensional Domain

Let SZ be a domain in Rn(n > 3) with compact boundary 851. Suppose that 0 E 8SZ and that 8SZ \ {0} is locally the graph of a uniformly Lipschitz

5. Extension of Functions to the Exterior of a Domain ...

298

function. At 0 we locate the origin of the Cartesian coordinates x = (y, z), y E

Rn-1,

z E R'. Let cp and w have the same sense as in Definition 5.1.1.

Furthermore, we assume that w is simply connected.

Definition. The point 0 is the vertex of a peak directed into SZ if this point is the vertex of a peak directed into the exterior of R" \ Q. That is, the point 0 has a neighborhood U such that U\ 52 = {x : z E (0,1), y/cp(z) E w}. I It turns out that multi-dimensional domains with inner peaks are "better" than those with outer peaks.

Theorem. A domain in R", n > 3, having the vertex of an inner peak on the boundary is in EVp for any l < p < oo and l = 1, 2, ... .

Proof. We can assume without loss of generality that w C Bin-11 Clearly the result will be established if we construct a linear continuous extension operator F : Vp (S2) --> VP (G), where

S2 = {x = (y, z) : z E (0,1), y/W (z) E

Bin-1)

\ w},

G = {x = (y, z) : z E (0,1), IyI < cp(z)},

and F has the following property: there is a number r E (0, 1), depending only on cp, such that if u E VP (52) with u(y, z) = 0 for z > r, then Fu = 0 in the vicinity of z = 1. We observe that the argument in Lemma 5.4.1 applies verbatim to define F by (5.4.1/2). One should only put Qk = {(y, z) : z E (zk+l, zk-1), y/cp(z) E Bin-11 \ w}, k > 1, in (5.4.1/1).

1

Remark. It is easily seen that if SZ C R", n > 3, has an inner peak, then Q satisfies the assumptions of Jones' extension theorem (see the comments to Sec. 1.6).

5.5.2. Planar Domains with Inner Peaks

First we describe the vertex of an inner peak on the boundary of a planar domain. Let SZ be a domain in Rz with compact boundary 852. Suppose that 0 E 852 and that 852 \ {O} is locally the graph of a uniformly Lipschitz

5.5. Inner Peaks

299

function. At 0 we locate the origin of the Cartesian coordinates (x, y). Let, furthermore, cp_, cP+ be functions in C° 1([0,1]) satisfying W± (0) = 0, cpt(t) --->

0 as t -+ +0 and W_ (t) < w+ (t) for t E (0,1] (see Fig. 33).

Definition. The point 0 is called the vertex of a peak directed into S2 if there is a neighborhood U of this point such that U S2 = {(x, y) : x E (0, 1), W- (X) < y < W+(X)I.

Fig. 33

We need two auxiliary assertions to prove the principal result.

Lemma 1. Let 0 be the vertex of an inner peak on the boundary of a planar domain 1. Put G= { (x, y) : x E (0,1), W- (x)
II={(x,y):xE (0, 1), yER'}. For any l = 1, 2.... there is a function E

CQ-1)

(II) n C°° (G)

such that z/'(x, y) = 1 for y > cp+(x), zO(x, y) = 0 for y < cp_ (x) and

y)

c `o+(x) - cP (x))-e, (x, y) E G, s = 0, ...,1.

Proof. Let d+ (x, y), d_ (x, y) denote the regularized distances (see, for example, Stein [194], Chap. VI, §2) from the point (x, y) to the sets F+

(x, y) : x E [0,1], y > co+(x) },

F_ = { (x, y) : x E [0, 1], y < co_ (x) }

5. Extension of Functions to the Exterior of a Domain ...

300

respectively. We claim that the required function can be defined by

-'=d`./(d++d` )Indeed, since d_ I F

(1)

= d+ I F+ = 0 and d f E C°° (G), it follows that V)IF_\{O}

0' V)IF+\{O} = 1

and 1/i E C°O(G). Using known estimates

1Vkd-IIc

cdl-k, IVkd+lIG

cd+ k, k=0,1,...

(see Stein [194]), one obtains by induction on l that

IVkdi

lIG <

cd!-k,

IVkd+IIc <- cd+ k, lVk(d+ +d' )-1IIG < cd-I-k,

where d = d+ + d- and k < 1. Thus V) C- C'- 1 (11) and c>dl+k-8d-l-k <

IVs0IIG <-

cld-', s < 1.

k=0

Furthermore, we have

d- (x, y) - y - w- (x),

d+(x, y) - w+(x) - y

if (x, y) E G, and hence d(x, y) - W+ (x) - w_ (x) on G. This completes the 1 proof of Lemma 1. The following lemma gives a necessary condition on a for the existence of a continuous extension operator: VP(Q) -+ Vp v (RZ).

Lemma 2. Let G be the same domain as in Lemma 1. Suppose that a is a positive bounded function on G which is separated away from zero in the exterior of any neighborhood of the origin and Q(x, y) depends only on x. If U E VP o(G),

aku/aykl y=w-(y)

0

for k = 0, ... ,1 - 1 and almost all x E (0, 1), then Il fO'(w+ -

W-)'/P-' 11p,(0,1) 5 C lI 7VlulIP,G,

(2)

5.5. Inner Peaks

301

where f (x) = u(x, w+ (x)).

Proof. An application of the Taylor formula to the function u(x, ) on the interval [cp_ (x), W+ (x)], x E (0,1), gives 1

u(x, cp+(x)) =

G+(x) a17I

(1 - 1)! w-(x) ay!

(x,

y) (W+(x) - y)

!

dy.

By Holder's inequaltiy,

f(x)I(+(x) -

(x))

1/p

!

1/P-1

(l

11), L(Z)

ayl

P

(x, y) dy

The last implies (2) with c = 1/(l - 1)!.

1

Theorem stated below presents sharp conditions on the weight Q for a linear bounded extension operator: VP '(Q) -+ Vp,o (R2) to exist.

Theorem. Let Q be a planar domain with the vertex of an inner peak on the boundary and let G be the same as in Lemma 1. co=cp+-cp-. If (i) Put o ,(x, y) = (co(x)l

x)!-1/P

for (x, y) E G

and a = 1 on R2 \ G, then there is a linear bounded extension operator: Vp (S2)

VP',Q (R2),

where 1 < p < oo, l > 1.

(ii) Let cp = cP+ - cp_ be nondecreasing and satisfy the condition W(2t) - co(t),

t E (0, 1/2]. Suppose that o, is a weight function on R2 and aIG depends only on x and is nondecreasing. If there exists a bounded extension operator: Vp (S2) Vp,. (R2), then o(x, y) < c(cp(x)/x)!-1/P

for all points (x, y) E G sufficiently close to the vertex of the peak.

Proof. (i) Let a E (0, 1] be a number satisfying W+ (x) < x for x E (0, a] and the set

U+={(x,y):0<x
is contained in the neighborhood U from the definition of the vertex of an inner peak. One can assume without loss of generality that a = 1. We consider several cases to extend a function u E VP (S2).

5. Extension of Functions to the Exterior of a Domain ...

302

1°. Suppose u E VP (S2), u = 0 in the exterior of U+. Extending u to be

zero on R2 \ (Q U U), we obtain u E VP (R2 \ G). Let

D+=R2\{(x,y):XE [0,1), y<
Put u+ = E+ (u I D+) and define

outside G,

ru

(3)

on G,

Ou+

Sl

where b is the function described in Lemma 1. We claim that Elu E Vp,Q(R2)

and that Ik'Vk(E1u)IIP,G
(4)

Let II be the strip introduced in Lemma 1. Note that for any small 6 > 0 the function given by (1) is in VV (II \ B6). Therefore 'pu+ E VP(II \ B6). Since 'pu+ = u on II \ G, it follows that Elu E VP' (R2 \ B6). We now turn to (4). Only the case k > 0 needs to be checked. By Lemma 1 k II

Ic.w9-kV8u+jjP,G.

k(E1u)IIP,G < CE

(5)

8=0

Let v denote a derivative of u+ of order s, s < k. Then v(x, ) is in Vp-8(R1) for almost all x c (0, 1) and, furthermore, v(x, t) = 0 for t > x. Hence 1 t

v(x, y) _ (l(- s)-

81-ev

8

1)! fYX

8t1-8

(x, t)(t - y)1-a-ldt, (x, y) E G.

(6)

An application of Holder's inequaltiy yields 1/p

x

I v(x, y) I < c x!

1/p

L

P

(Viu+) (x, t)

dt

)

(7)

5.5. Inner Peaks

303

Since y)'P(x)s-k

O '(X,

for (x, y) E G, (7) implies that

cp(x)-1/pxs-l+1/p

<-c 1

IIcyy,s-kVsu+Ilp,G < C flu+I1p,l,R2.

(8)

Clearly, the last estimate is also true for s = k. Now (5) and (8) imply (4). Thus, the required extension operator in the case 1° can be defined by (3). 2°. Let u E VP AI u 1 Q\U

= 0. Extending u to be zero outside U, we

obtain that u E VP (R2 \ ?7). put D = R2 \ { (x, y) : x E [0,1], Y E [gyp- (x), x] }

and w = uID. By Theorem 1.6.2, there exists a linear continuous extension operator F : VP (D) --> VP (R2). The function u - Fw satisfies the assumptions of 1°, and the desired extension u H E2u can be given by

E2u = Fw + E1(u - Fw). 3°. Consider the general case. Suppose B2i. C U and introduce a cutoff function 77 E Co (B2i.), 77IBr = 1. If u E V(S2), then ?7u satisfies the assumptions of 2°, whereas the function (1- y)u (extended by zero to B,. \ Q) is in VP '(DUB,). Thus, the required extension operator u H Eu can be defined by

Eu = E2(r1u) + E('') ((1 - y)u),

where E(') : V(1 U Br) -4 VP(R2) is a linear continuous extension operator. The proof of statement (i) of Theorem is concluded.

(ii) Let g, h be functions in C°°(Rl) such that supp 9 C (0, 3), 91(1,2) = 1, hl (-o0 2) = 1, hl (3 00) = 0.

For any small p > 0, we set 9(x/P)h(y/P) if x E (0, 3P), y E (W+ (x), 3P), ue(x,y) =

1 0 at the remaining points of Q.

5. Extension of Functions to the Exterior of a Domain ...

304

Clearly

ue E Cm (Q),

IIu0IIP,1,sl <

C,0-1+2/p.

Put f (x) = ue (x, co+(x)), X E (0,1).

If there exists a bounded extension operator: VP (52) -* V1 o (R"), then it follows from Lemma 2 that UV-!+l/Pf C1,0-1+2/P.

II

(9)

IIP,(0,1) < C II uell P,1,Sl <

On the other hand, f I (e 2p) = 1, and in view of the monotonicity of a and cp, we have 110'W-l+l/Pf IIP,(0,1) ? ca(Q)

V(2p)-1+1/Pe1/P

(10)

Inequalities (9) and (10) imply the desired estimate for the weight function u. The proof of Theorem is complete. I The theorem just proved in conjunction with Theorem 1.2.5 implies the following assertion. Corollary. A planar domain with the vertex of an inner peak on the boundary belongs to the class Evil-

5.6. Extension Operator: VP '(0) -* Vq (R"), q < p Let I be a bounded domain in R" with the vertex of an outer peak on 852. In this section we turn to the problem of the existence of a linear extension operator: VP (52) -+ V9 (R") with q < p. The following three cases are distinguished:

lq < n - 1, lq = n - 1 and lq > n - 1. Three theorems of the present section deal with each of these cases. Furthermore, we find conditions that ensure the existence of a linear continuous extension operator: VP '(Q) -+ VQ (R2), q < p, where SZ is a bounded planar domain with an inner peak. Positive constants c, cl,... appearing in this section depend only on p, q, 1, n, Q.

5.6.1. Outer Peak, the Case lq < n - 1 Here we prove the following assertion.

Theorem 1. Let 0 be the vertex of a peak directed into the exterior of a bounded domain SZ C R. Suppose that 1 < p < oo and that either

lq
5.6. Extension Operator: Vp(1) -+ V, (R"), q < p

305

(i) Let (0, 1] D t i-+ W(t)/t be a nondecreasing function. If

1/q- 1/p=l(/3- 1)/((n- 1)/3+1)

f

and

p

l \ W(t) /

J0

n/(P

1)

t

(1)

00,

(2)

<

then there is a linear continuous extension operator: VP (1)

Vq (Rn).

(ii) If there exists a bounded extension operator l; : VP (S2) -- V(R)

(3)

and cp satisfies (5.2/1), then (2) holds with ,8 defined by (1). Proof. (i) First we check that the space VP '(Q) is continuously imbedded into the weighted space Vq,T(S2), where

(1 ,r (x) {l

ifxEIl\U,

(z/cp(z))` if x= (y, z) E Q n u

and U is the neighborhood from Definition 5.1.1. Indeed, let it E Vp ([l) and let v = I V j u 1, 0< j< 1. Putting D = U n Q and applying Holder's inequality, we obtain 1/q-1/p

r(x)Pgl (P-q)dx)

II TV II q,D 5

D

c

(f'

II V I IP,D

1/9-1/P (Z/W(Z))lpq/(P-q)W(z)n-1dz

I

/

IIvIIP,D.

The last integral is equal to that in (2), whence IITVjUllq,n < C IjojUIlp,Q, 0 < j < 1,

and the continuity of the imbedding VP (1) C Vq T (SZ) is verified. Now the proof of statement (i) is reduced to the construction of a linear continuous extension operator: Vq,,r(c) -+ Vq(Rn).

5. Extension of Functions to the Exterior of a Domain ...

306

Let {zk}k>° be the sequence given by (5.2/3). Ifc has the form (5.2/2), u E VQ T(1l) and u(y, z) = 0 for z > z0/2, we define the extension E(°)u of u by (5.2/9). The same reasoning as in Theorem 5.2 (i) leads to the inequality which occurs if one replaces p with q in (5.2/13). This inequality implies that (we preserve the notation introduced in Theorem 5.2) zks-1)gllosuljq,!5k

IIVjE(°)uIIQ,G, < CUk q s=0

k > 1.

< C1 s=0

Here the relation Tl,% ' Qk 1 = (zk/W(zk))1 has been used. Since the mul-

tiplicity of the covering {(lk}k>1 is finite and suppE(°)u is contained in the closure of Uk>1Gk, the following estimate holds 1

IIo7E(°)uIIq,R- 5 C I1

z8_1TVsuIIq,S2,

j < 1.

S=0

In view of Lemma 5.1.2/3 and Theorem 1.2.5, we obtain E(°)u E VQ (Rn) with IIE(0)1IIq,1,Rn < C IITVIUj

q,S2.

Thus

V97"(0) E) uHEMU EVq(Rn) is the required extension. In the general case a bounded extension operator: V1 T(1) -> V9 (R") can be constructed in the same way as in Theorem 5.2 (i).

(ii) Let f be a function in Co (1, 2), f (z) = 1 for z E (4/3, 5/3). Put ,ok = 2-k, k > 0 and fix an integer N > 0. A function UN is defined on SZ by N

UNln\U = 0, UN(X) = Y1akf (z/Qk+1), x = (y, z) E o n U. k=0

Here {0k}k>0 is a sequence of positive numbers described below. We have C IIUN IIp,1,n >- II EuN II q,1,R^

(4)

5.6. Extension Operator: Vp (S2) -+ Vq (R"), q < p N

307

5ek+1/3 k+1/3

Cl E

II

/3

k=O

(SuN) (', z)II q,,

Rn-1dz.

(5)

By Sobolev's imbedding Vq (Rn 1) C L,.(Rn 1), r = (n - 1)q/(n - 1 - lq),

the general term of the sum in (5) is not less than 5ek+1/3 C

4ek+1/3

where Q. is the section of inU by the hyperplane z = const. The last integral is comparable to i9, and hence (5) implies that

Let

k > 0.

akpk

Then N w(ek)n-1-lPq/(P-q)

k+lpq/(p-q) <

C

,

k=O

where p < oo. The general term of the last sum is comparable to ek

(tl

w(t))lpgl(P-q),(t)n-1dt,

Jek+1

whence

f 1

(tlw(t))lP9I(P-9)W(t)n-1dt < c.

N}1

Passing to the limit as N -* oc yields (2). This completes the proof.

1

The proof of Theorem 1 contains the following assertion.

Theorem 2. Let SZ be the same as in Theorem 1 (i). Suppose lq < n-1, q > 1 or l = n - 1, q = 1. If T is a weight function defined on 11 by 7- (x) = 1 for

xEIl\B1and

T(x) = (IxI/W(Ixl))` for X E SZ n B1,

5. Extension of Functions to the Exterior of a Domain ...

308

then there is a linear continuous extension operator: Vq T(Q)

VQ (R").

Example. Power cusp. Let S2 = {x : z E (0, 1), jyj < cz-'I, A > 1.

(6)

If p > l and either lq < n-1, 1 < q < p or l = n - 1, q = 1, Theorem 1 says that the existence of a bounded extension operator acting as in (3) is equivalent to q-1 > p-1 + l(A - 1)/(1 + A(n - 1)).

5.6.2. Extension Operator: VP '(Q) -> V9 (R"), lq = n- 1 We state the following criterion for the required extension operator to exist.

Theorem 1. Let 0 be the vertex of a peak directed into the exterior of a bounded domain S2 C Rn and suppose lq = n - 1, q > 1, p E (q, oo). (i) Assume that (5.3.3/1) holds and that the function (0, 1] D t F+ W(t)lt is nondecreasing. If/3 is defined by (5.6.1/1), a = (1 - 1/q)/(1/q - l/p)

and

f1 (

n/(A-1)

3

P(t)

)

log

(1)

dt <

o0

(2)

then there is a linear bounded extension operator: Vp (1) -+ Vq (R").

(ii) Let cp satisfy (5.2/1). If there exists a bounded extension operator E VQ(R"), then (2) is valid, where a and /.l are defined by (1) and

VP(52)

(5.6.1/1), respectively.

Proof. (i) Let U be the neighborhood from Definition 5.1.1 and put -r(x) = (z&(z)) l [log (z/w(z))] -1+1/q x = (y, z) E St n U.

(3)

By Holder's inequality ll of HHq,inu 5 c IJf IIp,onu

(4)

5.6. Extension Operator: VP (1) -+ Vq (R" ), q < p

309

with f E L,, (S2 fl U) and

fl c=

\

(z)n_1dzq

(z)) n °

F .

I log (cp(z)/z) I D4 n

( z/V

(5)

We observe that (1)-(5) and (5.6.1/1) imply the continuity of the imbedding Vp (SZ) C VQ,,r(Sl), where the weight function T is defined by (3) in SZ fl U and

n\U = 1. Thus, statement (i) will follow if we construct a linear continuous extension operator: V9 T(Q) -4 Vq (R"). TI

Let {Zk}k>0 be the sequence given by (5.2/3) and let S2 have the form (5.2/2). Suppose that u E Vq (1) and that u(y, z) = 0 for z > z0/2. Consider the extension operator u -+ Eu defined by (5.3.3/3-5). The same argument as in Theorem 5.3.3 (with the same notation) leads to the inequalities which occur if we replace p with q in (5.3.3/19), (5.3.3/21). Since Uk " r(x)-1 for x E S2k and Qk - o(x) for x E Gk, these inequalities imply that IIV3vIIq,Gk C C (IITZ-lullq

1%

+ ll

(6)

I

IIVjWU ,Gk

CY:

ITZ'-lV8ull9

k > 1, j < 1.

(7)

s=1

Furthermore supp v, supp w C Uk>1Gk. Summing (6) and (7) over k > 1 gives IlV.ivU < C (IITZ-1uIIq,n +

IITZ1-10uIIq cz),

!

Ilo,wllq < cE

IITZ9-lVsullq

sz

a=1

Lemma 5.1.2/3 and Theorem 1.2.5 say that ,6u E V9 (R") and that IIEuHq,1,R° G C IITVIUIIq,o

Thus, u i--> Eu is the required extension operator in case S2 has the form (5.2/2) and supp u is contained in a small neighborhood of O. The general case is provable in the same way as in Theorem 5.2 (i).

(ii) Let {ur,}N>0 be the sequence given by (5.6.1/4). We introduce a smooth in R" \ {O} positive homogeneous function

of degree zero such that

5. Extension of Functions to the Exterior of a Domain ...

310

<(y, z) = 1 for I y I < z and C(y, z) = 0 outside the cone I yI < 2z. According to Lemma 5.3.1, the function VN = V UN is in VQ (R') and I IVNIIq,1,Rn < C IIetNIIq,I,R^, N=0,1,.... Therefore

N IIUNIIp,I,Si > CE J

1P5ek+1/3

k=p 4Pk+1 /3

(8)

IIvN(',z)IIg1,R^-ldz.

Note that VN (X) = ak if x E Q n u, z E (40k+1/3, 50k+1/3). B2z-l)

An application of Lemma 3.1.4 to the function

Y H ak lvN(y, z)

yields q-1

2z Clog

I1VIvN(',z)IIQ,B2 +zl-nIIVN(',z)I1y,Ba.

P(z)/

cak,

where Vi designates the gradient of order l in the variables Y1.... Ys-1. Since vN (y, z) = 0 for I y I > 2z, the Friedrichs inequality II VN(', z)IIq,B2. < cz1II VivN(', Z)11 ,,,B2=

holds and hence 1-q

cak [log

IIVivN(', Z) II q,B,. >_

z E (4Pk+1/3, 5Pk+1/3)

It follows from (8) and the last inequality that N

Cl

\

(E k=0

11/P

> II UNI T P,l,a

N

lPk)

> C2 (t ak

1/q

J

k=0

If we choose positive coefficients ak such that CIPk

ak (log (Pk/co(Pk))) k)))

1-q Pk,

k > 0,

5.6. Extension Operator: Vy (1) -> Vq (Rn), q < p

311

then

N

y

P(g-1)

n-1 (

n-q

Ok

Bk

q-p

\

to

W(ek) )

k=0

g

<

C.

)

W(9k)

The general term of the last sum is equivalent to to

ek

n/(0-1)

logW(t)t

where a and /3 are defined by (1) and (5.6.1/1) respectively. Therefore

/

1

p )n/(0-1)

I

LN+l

W(t)

log(t)t

° dt -
oo yields (2). Passing to the limit as N The following assertion is established in the proof of Theorem 1.

Theorem 2. Let Q be the same as in Theorem 1 (i). Suppose that lq n - 1, q > 1. If r is a weight function defined on S2 by T(x)

_

(IxI/W(IxI))' [log (IxI/w(Ixl))] -1+1/q

for x E SZ n B1,

for x E 1 \ B1i

1

then there is a linear bounded extension operator: /.(1Z)

Vq (Rn).

Example. Power cusp. Let Q be as in (5.6.1/6) and assume lq = n - 1, 1 < q < p. Theorem 1 says that a linear continuous extension operator acting as in (5.6.1/3) exists if and only if either one of the following conditions holds q-1 > P-1 + 1(A - 1)/(1 +,1(n - 1)) or

q-1 = p-1 + l(A - 1)/(1 + A(n - 1)) and 2q-1 < 1

+p-1.

5.6.3. The Case lq > n - 1 The following theorem gives sharp conditions for the existence of a continuous extension operator: VP (S2) -4 V9 (Rn), lq > n - 1.

5. Extension of Functions to the Exterior of a Domain ...

312

Theorem. Let 0 C R" be a bounded domain with the vertex of an outer peak on the boundary and let (5.2/1) hold. Suppose that p c (1, oo), q c [l, p)

and that lq > n - 1. (i) If (1)

np(1/q - l/p) = (n - 1)(Q - 1)

and inequality (5.6.1/2) is valid, then there is a linear continuous extension operator: VP '(Q) -a V.I (R").

(ii) If there exists a bounded extension operator acting as in (5.6.1/3) and the domain w from Definition 5.1.1 contains the point y = 0, then (5.6.1/2) is true with ,Q defined by (1).

Proof. (i) By Theorem 5.4.2, there is a linear continuous extension operator E : VP '(Q) -4 VP Q (Rn)

where a(x) = (cp(Ixl)/Ixl)(n-1)/p for Ixl < 1 and o,(x) = 1 for Ixl > 1. Since 52 has compact closure, one can assume that for all u E Vp (52), supp(Eu) is contained in some ball independent of u. We now prove the continuity of the same operator E as an operator: VP (52) -* V9 (R"). It will suffice to check the inequality (cf. also Theorem 1.2.5) IIVj5tIIq,B < C J1UVjEU p,Q, 0 < j < 1, u E Vp(S2).

Fix a multi-index a, lal < l and put v = D'Eu. By using Holder's inequality, we find 111q-11p (fU(x)Pg/(q-P)dx)

IIVIIq,B1 < II0'VIIP,Bi B,

The last integral is a constant times the integral in (5.6.1/2) so that IIVIIq,BI C IIQVIIp,BI, and statement (i) follows.

(ii) Let f E Co (1, 2), f ? 0, f l(4/3,5/3) = 1.

Put pi = 4-i, i > 0, fix an integer N > 0 and define the function uN on S2 by N

uN In\U

0, UN(x) _ E aif (2z/Pi), i=0

5.6. Extension Operator: V-1(f)) -> Vy (Rn), q < p

313

where x = (y, z) E SZ n U, U is the neighborhood from Definition 5.1.1 and {ai}i>o a sequence of positive numbers specified below. We shall check the estimate IIVleUNllq,11,

- ca9ei-lq, i=0,...,N,

(2)

with fIi = {(y, z) : y E Rn-1, Z E (Qi+1, Qi)}.

The proof of (2) will be similar to that of inequality (5.4.2/10) in Theorem 5.4.2 (ii). We have (EuN)(y,') E Vq(R1)

for almost all y E Rn-1. By Theorem 1.5.2, to each N > 0 there corresponds a set of functions {Pi}i'_0 such that Pi is defined on Rn, P1(y, ) is a polynomial of degree l - 1 in R' for almost all y E Rn-1, and the following inequalities hold I1(EuN)(y,') - Pi(y,')II Lq(e;+j,e;) < C(l, q) Lotlq J e;+ 1

at (EuN) azl (y, z)

q

dz, 0 < i < N.

(3)

One may assume that i-1

Pi(y,z)=EakW(y)z

k

,

i=0,...,N,

k=0

where (cf. Theorem 1.5.2)

ak (y ) = P

k-1

k ((2Q+1)-1t) (EuN) (y, t)dt

(4)

L+1

and {z/ik}k o are certain standard functions in C01(1/2, 1). For 0 < i < N, we put 1

vi(y)=

e,

2Qi+1 f2LO,+,

((EuN)(y,z)-Pt(y,z))dz, yERn-1.

(5)

Since (0, z) E SZ for z E (0,1), (EuN) (0, z) = UN (0, z) for almost all z c (0,1). In particular, (EUN) (0, z) = 0 if z E (Qi+1, 2ei+1), and all coefficients (4) equal zero for y = 0. Hence 2

e;

vi(0) = (2Qi+1)-1 f 2e,+1

UN (0, z)dz = ai f, f (z)dz. 1

5. Extension of Functions to the Exterior of a Domain ...

314

By Sobolev's imbedding V 9 (Ben-1)) C C(Ben-1)) f1 L,

(Ben-1)),

one obtains Ce(n-1)/gllvillo,BQ,

whence

< II'iIIq,B0, +esllolvillq,B,;,

(1-n)/q

l-(n-1)/q

c>i < Cei

Ilvillq,Rn-1 +Cei

Ilolvillq,Rn-1.

(6)

We now bound the right side in (6). Fix a multi-index 'y E Z+ 1, Iryl = 1. It follows from (5) that D7vi(y)I <_Cei

1f

e;

lei+1

(ID1(EuN)(Y,z)I +ID-Pi(y,z)I)dz.

(7)

An application of Holder's inequality to the integral in (4) yields k-1/9II Dy (EUN) (y, -) II

C Qi

I Dryak') (y) I

L,(e.+1,2e:+1)+

and so

DyPi(y,z)I

Cei 1/gjID7(EuN)(y,-)IIL9(e;+1,2ej+1)

(8)

for z E (ei+1, ei) Estimates (7), (8) imply that Ilolvillq,R'-1
1/gIJV1EUNllq,n:,

(9)

where Hi is the strip introduced in the line preceded by (2). To bound IIViIIq,R^-1, we first apply Holder's inequality to the integral in (5) and then make use of (3). This results in Ilvillq,R^-1 <

Cei-1/gIIVIEtNIIq,n;

(10)

From (6), (9), (10) follows (2).

The remaining argument is standard. Estimate (2) and the boundedness of the operator S : VP (S2) -a V9 (RI) imply N

ClEa%Li -Iq < IIVIEuNIIq,R^ _< C2IIUNIIP,l,il i=0

5.6. Extension Operator: 1/1(fl) -> Vy (R^), q < p

315

Hence N

aI

C3

N nq -i G IIUNIIp,l,S2 < C4 (EapP

i=0

-'PW(Pi)n-1)

q/p

i=0

for arbitrary positive coefficients {ai}i>o. We now choose ai to satisfy aqPn-`q = ap Pi 1-`p(P(Pj)ri-1, i > O.

Then N W(Pi)('i-1)q/(q-P)P(np-9)/(P-q) < C.

i=0

The general term of the last sum is equivalent to

f

ei+1 '

(ta/p(t))n/(P-1) dt

t

,

where /3 is defined by (1). Thus (tp/O(t))n/(A-1) at

1

< C.

Passing to the limit as N -4 oo gives (5.6.1/2). This completes the proof.

I

Example. Power cusp. Let 0 have the form (5.6.1/6) and let p > q > 1 with lq > n - 1. By the theorem just proved, the existence of a linear continuous extension operator acting as in (5.6.1/3) is equivalent to

q-1 > p-1(1 + (A - 1)(n - 1)/n). 5.6.4. Inner Peak, the Case n = 2 The following assertion gives a criterion for the existence of a linear continuous extension operator: VP '(Q) -- VQ (R2) for a planar domain described in Definition 5.5.2.

Theorem. Let 0 be the vertex of a peak directed into a planar bounded

domainI and let 1
5. Extension of Functions to the Exterior of a Domain ...

316

(i) If 1/q - 1/p = (1 - 1/p)(,3 - 1)/(a + 1)

(1)

and inequality (5.6.1/2) is valid with n = 2, then there is a linear continuous extension operator: VP (Q) -+ Vq (R2).

(ii) Let cp be nondecreasing and satisfy (5.2/1). If there exists a bounded extension operator .6 : VP (S2) -4 VQ (R2), then (5.6.1/2) holds with n = 2 and )3 defined by (1).

Proof. (i) By Theorem 5.5.2, there is a linear continuous extension operator E : Vp(Q) -4 VP o(R2),

where o is the weight function defined in statement (i) of that theorem. We can assume that supp(Eu) is contained in a disk independent of u E Vp (S2). Let v = D'Eu, J < 1. An application of Holder's inequality yields 1/q-11p IIV IIq,G

IIorVIIP,G (ff

y)Pg/(q-p) dxdy) a(x,

G

(we use the same notation as in Theorem 5.5.2). The last integral is equal to that in (5.6.1/2), and hence IIVIIq,R2
This implies that the operator E mentioned above is continuous as an operator: Vp (1l) -+ VI (R2). The proof of statement (i) is concluded.

(ii) We introduce functions f,g with the following properties: f E Co (1) 2), f(4/3,5/3) = 1, g E Cm(Rl), 9I(_. 1) = 1, 9I(2 oo) =

0.

Fix a number go E (0, 1) satisfying w+ (x) < x/2 for x E (0, go] and put Pk+1 = ek/2, k = 0,1, .... For each N > 0, uN is defined on SZ by N

E akf (x/ek+1)9(Y/Pk+1) if x E (0,1), y > cp+(x), UN (x, y) =

k=O

0

at the remaining points (x, y) E 11,

5.6. Extension Operator: 4 (S2) - Vy (R^), q < p

317

where {ak}k>o is a sequence of positive numbers specified below. An application of Lemma 5.5.2/2 gives 1

f I UN(x,W+(x))I gV(x)1-lgdx < C IIeUNIIq,l,R2 0

Hence N

ak'P(Pk)'

C1

10k

k=0 N 50k+1/3 < E Jf I UN(x, P+(x)Igw(x))1-lgdx

k=0 4ok+1/3

< C2IIEUNIIq,l,R21

(2)

Since

C3IIEUNIIq,I,Rs < IIkNIIp,I, l < C4 (E aL0k

(

we arrive at

N

(

1

11/P,

J

k=0

N

1/q C

1/P

C'Pk kok lp)

Let akekW(Pk)1-lg = C

00k

lp,

k 1 0.

Then N (Pk)P(1q-1)/(q-P)e2+p(lq-1)/(p-q) <

k=0

The general term of the last sum is comparable to

t t7' bQk+l(t) where ,Q is defined by (1). Therefore

2/-1) 1QN+i

Ut))

d < const,

C.

< pp,

5. Extension of Functions to the Exterior of a Domain ...

318

and (5.6.1/2) with n = 2 follows by passage to the limit as N

co.

Let p = oo. In this case we put ak = ek, k > 0 and observe that IIEtNIIq,I,R- < C IIUNIIoo,l,1l < C.

Inequality (2) implies N ek+1W(Pk)1-!q <

C.

(5)

k=0

If p = oo, the general term of the last sum is comparable to (3), and (4) follows

from (5). Thus, (5.6.1/2) is also true for p = oo. The proof of Theorem is complete.

I

Example. Power cusp, n = 2. Let Sl = B2 \ {(x, y) E R2 : x E [0,1], 0 < y < cxA}, .\>1. By above Theorem, a bounded extension operator: V() -* Vq '(R2 ) exists if and only if q < (A + 1)/(l(A - 1) + 2p-1).

5.7. Small Perturbations of Peaks in the Vicinity of the Vertex We have seen in previous sections that there is no bounded extension operator: VP '(Q) -* VP (R") when 1 < p < oo and Q is a domain with the vertex of an outer peak on the boundary. Suppose that for any sufficiently small e > 0 we are given a perturbed domain II(E) of class C°,1 such that OW is converging

to S2 as e -4 0 in a sense. In this situation the norm of a bounded extension operator: VP (SZ(e)) -+ VP (R") is expected to be growing as e -+ 0. We are concerned with two-sided estimates for the norm. Clearly, this "speed of degeneration" depends on the form of the perturbed domains Q('). `El. Here we illustrate this effect using a truncated peak. 5.7.1. Truncated Outer Peak. Extension Operators: Vp (Q(E))

VP (R")

In this subsection our consideration is restricted to the model domain defined by (5.2/2) (cf. Fig. 30 in Sec. 5.1.1) and to its perturbation of the form l (e) = {x = (y, z)

E R',

z E (e, 1), y/cp(z) E w}.

(1)

5.7. Small Perturbations of Peaks in the Vicinity of the Vertex

319

(see Fig. 34). Here w and cp have the same sense as in Definition 5.1.1 and satisfy the assumptions formulated at the beginning of Sec. 5.1.1. For the simplicity of presentation, we will assume that the following conditions hold: 0 C Bin-11, the function (0,1] E) z H cp(z)lz is nondecreasing and W(z) < z for z E (0, 1]. Furthermore, cp satisfies (5.2/1). In this section e E (0, 1/2) and c denotes various positive constants depending only on n, p, 1, cp, w. The relation a - b implies that a/b is bounded above and below by such constants. By E we mean an arbitrary bounded extension operator: VP (Q(E)) - VP (Rn) with 11(e) defined by (1). In each case lp < n-1, lp = n-1 or lp > n - 1 sharp estimates for IIEII are given. We now state three theorems concerning the cases just mentioned. AZ I

/

0

yn-1

/ /

Y1 `/ Fig. 34

Theorem 1. Let p E (1,oo) and lp < n- 1 or p = 1 and l =n-1. Then inf IIEII - (e/cp(e))1.

Theorem 2. Let lp = n - 1, p > 1. If (5.3.3/1) holds for some 6 > 0, then (e/cG(E))](1-P)IP

inf IIEII - (E/cG(e))' [log

Theorem 3. Let domain w (cf. (1)) contain the point y = 0. The following relation is valid in case lp > n - 1: inf IIEII -

(,/,(,))(n-l)/p.

5. Extension of Functions to the Exterior of a Domain ...

320

The proofs of these assertions do not require new ideas and can be mostly carried out by repeating the argument of corresponding theorems in Sec. 5.25.4. The distinction is that one should stop in a finite number of steps. Here we give a detailed proof of Theorem 1 leaving Theorems 2 and 3 to the reader.

Proof of Theorem 1. First we construct a linear operator E satisfying 11C11 <_ c (c/w(e))`.

As in Theorem 5.2 (i), the sequence {zk}k>o is introduced by (5.2/3). The number zo E (0, 1) can be chosen to be so small that zo < 2x2. We may assume without loss of generality that e < zo/2 and that ZN = e for some N > 2 (otherwise, if e E (zN+l, ZN)) we put ZN = e and find ZN+1 such that ZN+1 + p(zN+l) = e). Thus, a finite decreasing sequence {zk}N-+ol is constructed with the following properties: ZN = e and zk - zk+1 - wk for k < N where wk = cp(zk). Let 177k IN 1 be a collection of functions subject to (5.2/6) and to N

E 77k (Z) = 1, z E [E, zl]. k=1

Put 1Zk = {(Y, z) E 1IEI : z E (zk+l, zk_1) },

1 < k < N - 1,

aLlinear : Z E (E,zN_1)}. HN = {(y,z) E SZ(E)

To each 1 < k < N, we correspond

extension operator

Ck : VP(fk) -+ Vp(Rn)

satisfying (5.2/5). Also let {G}k 1 be cut-off functions in Co (Rn-1) with properties (5.2/8). As in Theorem 5.2 (i), it is sufficient to construct the required extension VP(S2(E)) E) U H Su E VV(Rn)

when u(y, z) = 0 for z > zo/2. If this is the case, we set Uk = uInk , N (Eu)(x) = E 71k(z)G (y)(Ekuk)(x), x = (y, z) E Rn, k=1

5.7. Small Perturbations of Peaks in the Vicinity of the Vertex

321

(cf. (5.2/9)). The identity Eul,(e) = u follows from the definition of Eu. We now turn to the estimate IIEuIIp,I,R°

< C (E/((E))`IIUIIp,t,l2(0 .

(2)

Consider the sets Gk = {(y, z) E R" : z E (zk+1, zk), I y I < 2cpk_1}, 1 < k < N.

The same argument that was used in Theorem 5.2 to prove (5.2/13) leads to the inequality I

IIVjEuIIP,ck < CQkpj:zk9-I)pllosuII'f1k, 1 < k < N,

(3)

8=0

Ok = 1k U Qk+1 for k < N - 1 and ON = QN.

where crk = (cok/zk)I, Clearly Qk 1 <

(e/co(E))I and suppEu C Uk 1Gk.

Since every point in Q`E) belongs to at most three elements of the covering {Ok}, (3) implies I

IlojEullp,Rn < C (e/W(E))Ip

Ilze-IV suHI p,n(e)

s=O

The proof of Lemma 5.1.2/3 shows that Hardy's inequalities (5.1.2/3-5) remain valid if G is replaced by the truncated peak Q W. So the last sum is not greater than c IIVzuIIP,Q(n

To conclude the proof of Theorem 1, we establish the estimate IIEII > c(E/cp(E))I for any extension operator E. Let f E C°° [1, oo), f 1(1,2) = 1, f 1(3,m) = 0,

and let uE(x) = f (z/E), x E WE). Then uE E Vp(Q(E)) and f2e

IIEUE IIP,I,R^

? CJ

E

II (EUE) (', z)

IIP,I,R--1

dz.

322

5. Extension of Functions to the Exterior of a Domain

...

By Sobolev's theorem, the space VP '(R'-') is imbedded into Lq(R"-1) for q = (n - 1)p/(n - 1 - lp). Thus, the last integral is not less than

Cf

2E

IIte(., Z)Ilq

z.)dz,

(4)

E

where

nZ(E)

is the section of Q(E) by the hyperplane z = const. Because the integral in (4) is comparable to eW(e)"-1-1P and 1-t n-1 , p,l,f2(e) ^ e p(P(e) P

we have IIEUEI1p,1,Rn/IIueIIp,t,n(E) ? c(e/cp(e))t

thus completing the proof of Theorem 1.

5.7.2. Inner Truncated Peaks, n = 2 Let cp_, cp+ be the same functions as in Definition 5.5.2 and let G(E)

(x, y) E R2 : x E (e,1), 0- (x) < y < W+ (x) Q(E) = R2 \ G(E)

be model domains depending on a small parameter e > 0. The problem analogous to that in the preceding subsection is studied here. Namely, we are concerned with the "speed of degeneration" of a bounded extension operator E : VP '(Q(-)) -* VP '(R

when e -+ 0. The following assertion is established by the argument analogous to that in Theorem 5.5.2. We give the proof for completeness.

Theorem. Let W = cp+ - (p_ satisfy condition (5.2/1) and suppose that the function (0, 1) E) x H w(x)/x is nondecreasing in the vicinity of x = 0. Then inf I I E I I ^' (e&(e)) t

1/p

1 < p < oo, l = 1, 2, ... ,

the infimum being taken over all bounded extension operators acting as in (1).

5.7. Small Perturbations of Peaks in the Vicinity of the Vertex

Proof. We construct a linear extension operator acting as in (1) with 116115 c (Ch*))

1-1/p (2)

One may assume without loss of generality that W+ (x) < x for x E (0, 1]. Put U(C)

Let u E V,

= {(x, y) E R2 : x E (e,1), V+(x) < y < x}.

and let u = 0 outside U+E). The domain

D(') = R2 \ { (x, y) : x E [e,1], y < W+ (x) } satisfies the hypotheses of Theorem 1.6.2, and there is a linear extension operator E+ : Vp(D+E)) - Vp(R2)

whose norm is uniformly bounded in E. By V) we mean the function introduced

in Lemma 5.5.2/1. If

IIE_{(x,y): xE(e,1), yER'}, u+=uID(E), then 0 E

and the function V'E+u+ coincides with u on each of the

sets

{(x, y) : x E (e,1) : y > W+(x)}.

{(x, y) : x E (e,1), y < cP-(x)}, Thus, the formula

u on S2(E),

{ V)E+u+ outside S2(e)

defines a linear extension operator u H Elu E VP '(R'). We now check the estimate IIVkE1ullp,G(e) < C

(3)

where k = 0, ... ,1. If k = 0, (3) follows because IV)I < c. Let 1 < k < 1. By Lemma 5.5.2/1 k

h0'_kvs(E+u+) llp,G(E)

lIOk(E1u) llp,G(1) < C

9-0

(4)

5. Extension of Functions to the Exterior of a Domain ...

324

If v is a derivative of E+u+ of order s, s < k, then representation (5.5.2/6) and estimate (5.5.2/7) are obtained on G(E) by the same argument as in Theorem 5.5.2 (i). Inequality (5.5.2/7) implies that

II v(x,')II

c (x)1/Pxl

L,(w-(=),w+(x))

81-8 V

-1/P

8t1-8

Z

P

(x, t)

for almost all x E (e, 1). Hence 1/p

(ff O(x)

E

1-1/P

k)pIv(x,y)IPdxdy)


(5)

G()

Clearly, (5) is also true for s = k. A combination of (5) and (4) yields (3). Thus, u H Flu is the required extension operator in case u = 0 outside U(E)

Turning to the general case u E Vp(c )), we introduce the domain DE = R2 \ { (x, y) : x E [e, 1], y E [1P- (x), x] }

and note that there is a linear extension operator E : VP (DE) -+ 1'(R2) with norm uniformly bounded in e (see Theorem 1.6.2). Put u = u1b.. Then u - Eu E VP '(Q(6)) and u - Eu vanishes on WE) \ U. According to what has been said above, the mapping VP (52(E)) E) U H Eu = Eu + E1 (u - Eu)

is a linear extension operator with norm satisfying (2). To conclude the proof of Theorem, one should verify the inequality opposite to (2) for any extension operator acting as in (1). We will assume e to be so small that I c '+(x) I < e for x E (e, 2e). Let f, g be functions in C°° (R1) subject to

supp f C (1, 3), supp 9 C (-2, 2), f(3/22) = 9I (_1 1) = 1, and let f (x/E)g(y/e) if x E (e,1), y > W+ (X), uE(x, y)

0

at the remaining points of

(

J L(E).

Comments to Chapter 5

325

Then uE E C°°(1Z(E)) and IIueIIp,1 o(E) < cE2/P-1. Put vE = EuE. We have for almost all x c (3E/2, 2e) 1

1 = vE (x, (p+(x))

<

= (l - 1)! (x)

W(x)1-1/P

p- (x)

'91

a y(-

V+ (x) L_

!-1 aiv

(`p+ (x) - y)

O-j (x, y)dy

(x) 1/p

p

(x, y) dy

Hence IIVIVEII p,G(e) >_ CE 1/p

((p(2e)) 1/p-'

and IIEUEIIp,I,R2IIUEIIP111o(E)

> c (E/cp(2E))'-1/P

This concludes the proof.

Comments to Chapter 5 Most of the results presented in Chapter 5 were announced in the note by Maz'ya and Poborchi [144]. Detailed proof was given in [145] and [146]. In connection with Remark 5.4.2, we mention a result due to Whitney. Let SZ C R" be a bounded domain. Whitney [215] gave a characterization of the restrictions ul, of functions u c L'm(R^). As a consequence, he found [216] a simple sufficient condition on S2 to have the property Loo'

(Q) _ Jul, : u c L'(R")}.

(1)

This condition is do (x, y) < const Ix - yI for all x, y E 0, where do(x, y) is the intrinsic metric in SZ, i.e. the infimum of the lengths of arcs in ) joining x to y. Clearly, domains with outer peaks satisfy this condition. Recently Zobin [222] has shown that if 0 is a planar bounded finitely con-

nected domain, then the equivalence do (x, y) - Ix - yI is necessary for (1) to be true for any fixed 1 > 1. However, there are domains in R", n > 2, for which (1) holds without this equivalence [222]. Vodop'yanov [208] has given sufficient conditions for extendability of functions in anisotropic spaces W,I.(Sl) in terms of anisotropic intrinsic metric in E2. Lemmas 5.1.2/1-2 in case p = q are due to Talenti [196], Tomaselli [199] and Muckenhoupt [161]. The generalizations for p# q (including q < p) were

5. Extension of Functions to the Exterior of a Domain ...

326

obtained by Rosin and Maz'ya (see [136], Sec. 1.3.1, 1.3.2). More general two-weighted inequalities were obtained by Stepanov [195]. Lemma 5.1.2/3 for p = 2 and l = 1 was proved by Maz'ya [135]. We remark that extension theorems in Sobolev spaces with worsening of the class (if a space-preserving extension is impossible) was studied by various authors. Burenkov [35] constructed a linear bounded extension operator: in case 52 E C°,a, A E (0, 1) (here denotes the integral Vp(0) part of a number). Gol'dshtein and Sitnikov [79] considered a planar domain 11 with an outer or inner peak such that a part of the boundary in a neighborhood of the vertex of the peak is in C°-a, A E (0, 1). A bounded extension operator: Wp (S2) -* WQ (R2) was presented for some q < p. Fain [57] described a class of domains 0 C R° with nonsmooth boundaries (including in particular domains in Co,-\, A E (0, 1)) and constructed a bounded extension operator from VP '(Q) either to VQ (R") with q < p or to VP Q(R") where a is a weight function such that olasz = 0. If SZ has an isolated power cusp, exponents q in [57] agree with those in Examples 5.6.1-5.6.4. The following result was communicated to the authors by Burenkov in June 1996.

Let A be the set of all functions A which are positive, nondecreasing on (0, oo) and limt_,+o = 0. Let A E A and 1 < p < oo. By Hy (') (R") we mean the space of all Lebesgue measurable functions f on R" with finite norm IIfIIHP()(Rn) = IIfIILp(Rn) +°#

Rn

{(A(IhI))-IIIf(. +h)

- f(')IIL,(R-)}

Suppose Q C R' is a bounded open set and 1 < p, q < oo. Then the following statements are equivalent: (i) the imbedding WP (0) C L,(Sl) is compact;

(ii) there is a A E A such that the map WP '(Q) E) u H Eou E HQ (') (R"),

where Eou = u on Q and Eou = 0 on R' \ 52, is a bounded operator; (iii) there are A E A and a bounded extension operator: Wp (S2) -4 H9 (') (R").

In connection with this result see also the paper by Burenkov and Evans [37]. Note that (ii) and (iii) are always valid if q < p (cf. Theorem 1.10/2).

CHAPTER6

BOUNDARY VALUES OF SOBOLEV FUNCTIONS

ON NON-LIPSCHITZ DOMAINS BOUNDED BY LIPSCHITZ SURFACES

Introduction In this chapter, as in Chapter 4, we study the traces ulasz of functions u c WP (Il). Here the domains S2 are not generally Lipschitz, they may have cusps at the boundary as depicted in Fig. 35, 36.

We now describe typical results of the present chapter using the domain in Fig. 35 as an example. Let cpl, W2 be in C'([0,1]) and assume that the function cp = cp2 - cpl is positive on [0, 1) and that o(1) = cp'(1) = 0. Domain Q is given by SZ={x=(y,z)ER":yEB('-1)

,

c°i(M)
2,

The lower and upper parts of 81 are denoted by S1 and S2, i.e., Si _ n {(y, Pi(IyI)) : y E B( -1)j, i = 1, 2.

AZ

327

6. Boundary Values of Sobolev Functions on Non-Lipschitz Domains ...

328

Let p c (1, oo) and let f be defined on aSZ with f ISi E LP,io, (Si), i = 1, 2. It is shown in Sec. 6.2 that f is in TWp (SZ) (cf. 4.1.1) if and only if the norm on the right part of the following relation is finite, and, moreover,

IVIITWW(S1)

l i=1 fS;

+f

f(x)IPw(IyI)dsx If(y,w2(IYI))-f(y,wl(IyI))IPw(Iyj)1-Pdy

l

ff

If x) -

dsxdsC

C

l1/P

x-CI
where x = (y, z), = (77, (), dsx, dse are the area elements on a12, a is a positive constant (which can be written explicitly) depending only on n and the Lipschitz constant for cp, and M(y, 77) = max{w(IyI), w(I71I)}.

An analogous result holds for p = 1:

f

II fIITwi(sz)

+f

l

If

f(y,w2(IYI))-f(y,wl(IyI))Idy

if

2

+E i=1

(x)Iw(IyI)dsx

dsxds£

If(x)-fWIM(y,'77)n-1' CC

{x,fESi.Ix-CI
The space TWp (R" \ SZ) for p E (1, oo) is described by the relation

IIfIITWD(R'\ri) - I IfIIWP-11P(Si) i=1,2

+

(f

If(y,W2(IYf(Y,wl(IyP (1 Bl

and the space TW' (Rn \ S2) coincides with L1(aQ).

dy

- IyI)P -l

)1/P

6.1. Ball Coverings of an Open Set Associated with a Lipschitz Function

329

6.1. Ball Coverings of an Open Set Associated with a Lipschitz

Function This section contains auxiliary results which will be used in the following sections to prove trace theorems. We begin with a covering lemma. Lemma 1. Let cp be a nonnegative function on R", cp 0 0, and let cp satisfy the condition

Jcp(x)-W(y)I 0. Then there are two positive constants cl = cl (n), c2 = c2 (n) such that for any a E (0, c1L-1] there exists a covering of the set

G= {xER":cp(x)>0} by a sequence of open balls

B(") = {x E Rn : Ix - x I <

v = 1, 2, ... ,

(1)

with the following properties: the concentric balls of doubled diameters lie in G with their closures and form a covering whose multiplicity does not exceed 2-n-5n-1/2. c2. In particular, one may choose c1 =

Proof. For each i E Z and each k = (k1i... , k") E Zn put

Qi,k = {x E Rn : 0 < x9 - 2-% < 2-t, s = 1, ... , n). We introduce the set of indices

A = {a = (i, k) E Zn+1 : cp(x) > 2-iL for all x E Qa } . A cube Q y, ry E A, is called maximal if the inclusions a E A, Q.y C Qa

imply Q y = Qa. Note that the interiors of two different maximal cubes do not intersect. Let F C A be the subset corresponding to the maximal cubes. Since every point of G lies in some cube Qa, a E A, and every cube Qa is contained in a maximal cube, one obtains

G= UQ,. -yEr

The remaining proof will be performed in two steps.

6. Boundary Values of Sobolev Functions on Non-Lipschitz Domains ...

330

Step 1. Let {Qy}7Er be the collection of the closed cubes Qy concentric with Q7 and such that

diam (Qy) = (1 + d,,) diam (Q7), d = (2/)-1. We now check that the number of the cubes Qy containing the same point is bounded by a constant depending only on n. First we estimate the value cp(x) above and below for x E Q.

Suppose y = (i, k) E F and X E Q. Then 1x - yj < 2-:dnn1/2 for some y E Q7. Hence

W(x) ? W(y) - Lax - yl > 2-i-'L.

(2)

In particular, Qy c G. Next, let Q, be a cube with a = (i -1, j) E Zn+' and let Q7 c Q,,. We have a V A because of the maximality of Q7, and there is a point z E Qa for which W(z) < 2'-'L. Therefore v(x) < w(z) + Liz - zj < 2-'L(2 + n1/2(2 + dn)).

(3)

A consequence of (2) and (3) is the following rather rude but more convenient for our purposes two-sided estimate

2-t-'L < W(x) < 2n+2-'L, x E Q.

(4)

One can observe from (4) that if y = (i, k), a = (m, 1), a, y c F and Q,*, n Qy 54 0, then

ji - ml < n+3. This inequality follows by applying (4) to each cube Qom, Qy and to their common point.

We now fix an a = (m, 1) E F and we bound the number of the cubes Qy, y = (i, k) E r, intersecting Q0. If m < i and Qy n Q,, # 0, then aQ7naQ0 # 0, and the number of such cubes Q7 does not exceed (2`--+2)" Let m > i. Then the cubes Qy and Qa have a common point only if aQ7

6.1. Ball Coverings of an Open Set Associated with a Lipschitz Function

331

intersects the boundary of that (unique) cube of the form Qi,r, r E Zn, which contains Q,,. The number of such cubes Q.y is not greater than 3n. Since Ii - ml < n + 3, the total number of the cubes Q,*, having a common point with Qa is dominated by n+3 c(n) = (n + 4)3n + >(28 + 2)n.

(5)

8=1

Clearly, the multiplicity of the covering {Q,*Y} yEr does not exceed c(n).

Step 2. Here we construct the required coverings of G. It will be shown that such coverings exist when 0 < aL < 2-I 5n-1/2

(6)

Let N denote the greatest integer less than 2n1/2(aL)-1. We also put

ei = 2-`-'aL, i = 0, ±1,... . If y = (i, k) E I', then the points xj,7 = 2-°k + ein-1/2j

(7)

? = (ii,...,j,) E Zn, 1 < jg < N, s = 1,...,n,

(8)

with

lie in the cube Q., and form an ei-net in Q.y. Since min{cp(x) : x E Q y} > 2-iL,

it follows that acp(xj y) > 2ei, and the family of various balls B(j,7) = {x E Rn : Ix - xj,. I < ac'(xj,..)}

(9)

covers G, where y E r and j is subject to (8). Let 5(j,7) be the open ball concentric with B(1') with doubled diameter. When y = (i, k) E r is fixed, the distance between any different centers of n-1/lei. On the other hand, (4) implies these balls is at least diarn (CB(j,7)) < 2n+4-`aL = 2n+5ei.

(10)

332

6. Boundary Values of Sobolev Functions on Non-Lipschitz Domains ...

Therefore, the number of the balls 1313,'0, containing the same point, is dominated by a constant co = co(n) provided -y E IF is fixed. Furthermore, it follows from (6) and (10) that diam (B1i'7)) < 2-idn. Hence BU'7) C Q7 C G, and the multiplicity of the covering {811,7)} does not exceed co(n)c(n) where c(n) is given by (5). We have shown that the balls defined by (7)-(9) form the required covering of G. The proof of Lemma 1 is concluded. I

Remark 1. The family {811,7)} just constructed has the following property. If two balls in this family have nonempty intersection, then the ratio of their diameters is bounded above and below by positive constants depending only on n. Indeed, it has been proved in Lemma 1 that if -y = (i, k) E F, then 8Ei < diam (B1'"71) < 2n+5Ei, ei = 2-`-'aL.

Let 813,7) n B1e,a) # 0 for a = (m, r) E F. In this case Q,*, fl Q,*,, # 0, and

therefore 1i - ml < n + 3, as shown in Lemma 1. Thus, the ratio of the diameters of these balls is in

[2-2n-522n+5]

I

Remark 2. If a E (c1L-1] is fixed, then every compact subset of G intersects only a finite number of balls in the collection {B1i,7)}. Indeed, let F C G be compact. Then Cpl F is bounded and separated away from zero. It follows from (4) that F cannot intersect cubes Q,* y E F, with sufficiently large or sufficiently small edges. Next, for fixed i E Z, the set

F clearly intersects only a finite number of cubes Qi,k, k E Z'. Hence F intersects a finite number of maximal cubes. Furthermore, the number of the cubes {Q,*,}7Er intersecting a fixed maximal cube is bounded by a constant depending only on n. It remains to note that each cube Q7 contains a finite number of the balls {B1i,7)}.

Remark 3. Let G C Rn be an open set, G # Rn. If we put w(x) = dist (x, Rn \ G), X E Rn, then cp is a uniformly Lipschitz function (with Lipschitz constant 1), and G is exactly the same set on which W is positive. In this case the covering of G constructed in Lemma 1 consists of balls whose diameters are comparable to the distances between the balls and the boundary of G. I We now prove the existence of appropriate partitions of unity subordinate to the coverings mentioned in Lemma 1.

6.1. Ball Coverings of an Open Set Associated with a Lipschitz Function

333

Lemma 2. Let the assumptions of Lemma 1 hold and let {B(v)l,,;,, be the covering of G by a sequence of balls of the form (1) constructed in Lemma 1. There is a smooth partition of unity {o"}v>1 for G subordinate to the covering by the sequence of the concentric balls with diameters z diam (B(")) and a constant c depending only on n such that

IVovIdiam(Bl"i)
(11)

Proof. We assume that {B(")}v>1 is the reenumerated sequence {B('7)} of the balls given by (7)-(9). Let rv be the radius of B("). Define 77V (X) = 77 (TV '(X - xv)) ,

2 E Rn, v = 1, 2, ...,

where

77ECo (B312), 0<71<1,

gjBl=1.

Then 71 E Co (B3r.12(xv)) and for x E G the sum v(x) _ E">17!"(x) is bounded above and below by positive constants depending only on n. We v = 1, 2, ... , form the required now show that the functions ov = 71vQ-1,

partition of unity. One should check estimate (11). Clearly IOQ"I 1

The general term of the last sum is nonzero only if supp 71v fl supp 77, # 0 and thus only if rvr9 1 is bounded above and below by positive constants depending

only on n (see Remark 1). Furthermore, the number of nonzero summands involved in the sum is uniformly bounded at any fixed point of G. It remains to note that I V77.1 < c(n) TV 1. The proof of Lemma 2 is complete. I The following simple assertion will facilitate the proof of the main result in the next section.

Lemma 3. Let the assumptions of Lemma 1 hold. Suppose t c G and x E Br(t) with r = a W(t), a > 0. If b = a(l + 2aL)-1 and Ix - yJ < bmax{cp(x),,p(y)},

then y E B2r.(t).

Proof. Since ly - tj < r+bmax{W(x),cp(y)},

6. Boundary Values of Sobolev Functions on Non-Lipschitz Domains ...

334

it is sufficient to check that the last term does not exceed r. We have W(x) < W(t) + Ljx - tj < W(t) + Lr, whence

w(x) < (1 + aL)cp(t)

(12)

and bcp(x) < r. Let us show that the last estimate remains true if x is replaced by y. Assume that W(y) > cp(x). Then V(y) < W(x) + Lax - yI < w(x) + bLco(y).

Consequently W(y) < (1 - bL)-1cp(x).

In view of (12), we find w(y) < (1 - bL)-1(1 + aL)cp(t) = b-1r. The result follows.

I

Remark 4. Let {B(")} be a collection of balls of the form (1), where a = c1L-1 and c1 is the constant from Lemma 1. Put c = c1(1 + 2c1)-1. Then the inclusion x E B(") and the inequality Ix - yJ <

cL-1

max {co(x), c'(y)}

imply that y is in the ball concentric with B(") with doubled diameter.

6.2. Domains Between Two Lipschitz Graphs 6.2.1. Description of Domains and Approximation Lemma We begin with the definition of domains we deal with in this section. Let G C Ri-1 be a domain. Let cp1, cp2 be uniformly Lipschitz functions on G such that cp1 < W2 on G and cp1 = cp2 on G. Consider a domain

1 = {x = (y, z) E R"

: y E G, Z E (cp1(y), cp2(y))}, n > 2,

(1)

(cf. Fig. 35). In particular, G may coincide with Ri-1. In this case Il is a layer between two Lipschitz graphs.

6.2. Domains Between Two Lipschitz Graphs

335

The lower and upper parts of KSZ are denoted by S1 and S2, respectively, i.e.,

S2={(y,co(y)): yEG}, i=1,2.

(2)

Clearly, every function u E WP (SZ), p E [I, oo), has the traces ul sl, ul s2. Moreover, ulsc can be described by Theorem 4.1.1 in a neighborhood of any point in Si, i = 1, 2. We need two lemmas to prove the principal result. In what follows c designates various positive constants depending only on n, p, W1, W2. If a, b > 0,

then a - b means that c-1 < ab-1 < c. Lemma 1. Let G # Rn-1. If u E WP (1), p E [1, oo), and ul Sl = 0, then dx

I Iu(x)IP (dist(ye aG))P


Proof. Let cp = cp2 - cp1. Since W 10G = 0 and cc is uniformly Lipschitz, we have cp(y) < cdist(y,aG), y E G. Clearly

P

dx

<

co(y)P

dy G cc(y)P

',iwa

(v)

(y)

P

Z

au (y t)dt

dz W1(y)

at

By Holder's inequality, the expression on the right does not exceed IIVullip(n) which concludes the proof. I

Let p E [1, oo). We introduce the subspace WP (0, Si, S2) of WP (1) consisting of those u E Wp (1) that satisfy uls, = 0 and uls, = 0.

Lemma 2. The space Wp (SZ, S1i S2) coincides with WP (1), i.e. with the closure of Co (1) in Wp (SZ). Proof. Only the inclusion Wp (SZ, S1i S2) C WP (SZ) needs to be checked.

For u E WP(SZ), let H(u) denote the orthogonal projection of suppu into G. We note that the functions u E WP (SZ, Si, S2) with II(u) contained in a compact subset of G are in Wn (SZ). Thus, it is sufficient to approximate an arbitrary u E WP (SZ, S1, S2) by such functions. Given u, put UN (x) _

6. Boundary Values of Sobolev Functions on Non-Lipschitz Domains ...

336

UN(y)u(x), x E SZ, where N = 1, 2, ..., vN(y) = o(IyI/N) and o is a function in C°° ([0, oo)) subject to

0
UI[01] =1,

0'I[2oo)=0.

Then UN -+ u in WP 1(Q) as N -* oo. Since II(UN) is bounded, the proof is

complete in case G = Rn-1. It remains to show that for G # R"-1 every u E WP (SZ, S1, S2) with bounded support can be approximated in WP (SZ) by

functions in 1I', (S2). Let A be defined on [0, co) by AI[0 1) =

0, Al(2 m) = 1, A(t) = t - 1, t E [1, 2].

For small e > 0, we put u,(x) = .1(e-le(y))u(x), x E SZ, o(y) = dist(y, 8G). Then uE E WP (SZ, S1, S2), ue has bounded support and ue(x) = 0 if o(y) < E. Hence uE E WP (Q). Clearly uE -+ u in LP (SZ) as e -4 0. Furthermore, ` C

IIo(U- u)II L() C I(1 -

(ele))ouII) +

dx, fo(u,c) N(y)P

where Q(u, e) = {x E SZ : y E II(u), o(y) < 2e}. The first term on the right tends to zero as e -4 0 by the dominated convergence theorem. The second

term tends to zero by Lemma 1 and because mes,,(1(u, e)) -+ 0. The proof of the lemma is concluded.

I

The following assertion is a direct consequence of Lemma 2.

Corollary. Let TP (S), S = S1 U S2, be the space of the functions f on S such that f = ul S for some u E WP (SZ); the norm in Tp(S) being given by inf {IIuIIwp(n) : ul s

=

f }.

(3)

LetTWP(SZ) = WP (SZ)/WP(Sl). Ifu E WP(S2) and u = u+WP(S2) E TWP(S2), then the mapping is H uIs is an isomorphism between TWP (SZ) and TP(S). I

Remark. Corollary just stated enables us to identify the spaces TWP(Q) and TP(S). Below the elements of TWP(SZ) will be regarded merely as functions f on S with IIf IITwp (o) given by (3).

6.2. Domains Between Two Lipschitz Graphs

337

6.2.2. Trace Theorems for Domains Between Two Lipschitz Graphs We now turn to the principal results of Sec. 6.2. Let St be defined by (6.2.1/1).

Put ca = cP2 - cal. Then cp is uniformly Lipschitz on G, ca = 0 on 8G and ca > 0 on G. Extending ca by zero to the exterior of G, we obtain a uniformly Lipschitz function on Ri-1 with {y E Rn-1 : W(y) > 0} = G. Below we assume that such extension has been already made. Theorem 4.1.1 says that in a neighborhood of any point in Si, i = 1, 2, the traces ulst of the functions u E WP (Sl) belong to WW- 1/p for p > 1 and to L1

for p = 1. This is a complete local characterization of the traces. However, this information is not sufficient to describe the space TWp (SZ). For p > 1, this description is presented in the following theorem. Theorem 1. Let S2 be a domain given by (6.2.1/1) and let S = S1 U S2 with Si given by (6.2.1/2). Suppose that the function ca = cat - cal is bounded. If p E (1, oo), then IIfIITW,

^ {f}P+ E i=1,2

(IfI+ff(x)IPYds)

1/p ,

(1)

,

where {f} p and If Ii,p are the seminorms defined by {f}P = f lf(y,(P2(y)) G

Iflp,p -

x = (y, z),

-

U

f(y,wl(y))IPCP(y)1-Pdy,

If (x) - f(f)IP

dsydsC

Ix - S

In+P-2

i = 1,2,

(71, (), dsx, ds£ are the area elements on S, M(y, y) = max{ca(y), ca(y)}, y, 77 E G,

and a is a positive constant depending only on n and the Lipschitz constant for W. In particular, one may put a = cl(1 + 2c1)-1L-1, where cl = cl(n - 1) is the constant from Lemma 6.1/1 and L is subject to sup y,nEG,yj4n

Ic(y) - cP(i) I I y - 771-1 < L.

Proof. If f is a function defined on S, we put for brevity fi(y) = f (y, cai(y)), y E G, i = 1, 2.

6. Boundary Values of Sobolev Functions on Non-Lipschitz Domains ...

338

Let U E WP (Q), ul Is = f. One can observe that IlullW,(Q) - IIVUIILD(f1) + E IlfiWl/PIIL,,(G).

(2)

i=1,2

Indeed, if i = 1, 2 and z c (W1 (Y), c'2(y)), Holder's inequality yields

au

MY, z) - fi (y) I P < W(y)p-1

8t

P

(Y' t)

dt

(3)

for a.e. y E G. Integration with respect to z E (WI (Y), c'2(y)) and then over y E G gives IIIUIIL,(o)

- Ilf cc'11PIIL,(G)

< C IIVUIILP(sl)

Hence (2) is valid. Therefore, (1) is a consequence of the relation inf{ IIouIILD(o) : ul s

= f j - {f }P + E If Ii,P

(4)

i=1,2

The proof of (4) will be made in several steps.

Step 1. First we establish the required estimates for the trace f = ul Is of a function u E Lp(1). The inequality {f}P <_

IloullLn(c)

(5)

follows from (3) with z = W3-i(y)Since dsx - dy and ds£ - d77 on Si, the estimate

IA & < C IIoullLP(D),

i = 1, 2,

(6)

is a consequence of the inequality

f

Ifi(y) - fi(j)IP

{y,nEG:Iy-,iI
Iy -

7dyIn+p-2

< C IIVulliP(Q).

Let i = 1. The change of variable

x= (y,z)'-*X = (s,t), s=y, t=z-c,1(y)

(7)

6.2. Domains Between Two Lipschitz Graphs

339

transforms Sl onto {X E Rn : s c G, 0 < t < cp(s)} with IVxul - I Dxul and I D(s, t)/D(y, z) I = 1. Hence, to prove (7) for i = 1, one can assume without loss of generality that cp1 = 0. We shall validate (7) with a = cl (1 + 2c1)-1L-1, cl = 2-n-4(n

- 1)-1/2(8)

By Lemma 6.1/1, there is a sequence of (n-1)-balls {Brk (yk)}k>1 with centers Yk E G and radii rk = c1L-1cp(yk) such that these balls form a covering of G with the open concentric balls {B(k)}k>1 of doubled diameters also lying in G. In addition, the multiplicity of the covering {B(k)}k>1 depends only on n. Furthermore, it follows from Remark 6.1/4 that if y E B,.k(yk), ri E G and Iy

- 771 < aM(y, 77),

then 77 E B(k). Note also that for y c B(k) cp(y) - cp(yk) I < 2rkL,

whence

1 - 2-n-3 < co(Y)/cp(Yk) < 1 + 2-n-3, y E B(k), k> 1.

(9)

We now prove (7) for i = 1. The left part of (7) does not exceed dy J Iy-nl1 f1r,, (yk)

drl

Ifi(y) -

ly - 7,In+P-z'

which is dominated by the sum

If

k>1 B(')xB(k)

Ifl(y) - f1(l)I P

djn -r?ln+P-2.

(10)

Iy

Let

Qk= {x=(y,z):yEB(k), 0 1. Then each cell 1 k is in SZ because of (9) (recall that we temporarily assume cp1 = 0). By Theorem 4.1.2, the general term of the sum in (10) is not greater than c IIDuIIi,(szk). Therefore, expression (10) does not exceed c IIDuIIiD(O).

6. Boundary Values of Sobolev Functions on Non-Lipschitz Domains ...

340

Now (7) with i = 1 is established. The case i = 2 is treated in the same way. Step 1 is complete.

Inequalities (5), (6) show that the right part of (4) is dominated by a constant times the left part of (4). To prove the reverse inequality, we let f be a function defined on S such that fl, f2 E Lp,l,,(G) and the right part of (4) is finite. Our goal is to construct a function u E Lp(11) with uIs = f satisfying

IIVuIILp(cl) s Mp+ > Ifli,pi=1,2

Step 2. Here we define two functions U1, u2 E LP (Q) subject to ui I Si =

flsi, i=1,2,andto II Vui IILp(Sl)

C I f l i,p, i = 1, 2.

(12)

To construct such separate extensions of f, we need a locally finite covering of G by a collection of open (n-1)-balls {B(k)}k>1 with the following properties:

1) B(k) C G for all k = 1, 2,..., and the multiplicity of the covering {B(k)} depends only on n. 2) The open balls concentric with B(k) with half diameters form a covering of G.

3) If y E B(k), then cp(y) - diam (B(k)). 4) If y, 77 E B(k), then

Iy - riI + E IVi(y) - wi(rl)I < aM(y,77)

(13)

i=1,2

with a given by (8). Let us check the existence of this collection {B(k)}. Let L1, L2, L be Lipschitz constants for W1, cp2i ', respectively. If we put

b=min{2c1L-1, a(2+aL+2(L1+L2))-1}, where a and cl are defined by (8), then, by Lemma 6.1/1 and Remark 6.1/2, there is a sequence {yk}k>1 C G such that the balls

B(k) = {y E Ri-1 : Iy - ykI < bco(yk)}, k > 1,

(14)

6.2. Domains Between Two Lipschitz Graphs

341

form a locally finite covering of G and satisfy conditions 1), 2). Furthermore, if y E B(k), then IW(y) - c'(Yk)I <- bLcp(yk),

whence

1 - bL < cp(y)/cp(yk) < 1 + bL,

(15)

and condition 3) holds. Next, it follows from the definition of b that (1 + L1 + L2) diam (B (k)) < all - bL)cp(yk).

If y, y E B(k), the right part of the last inequaltiy does not exceed aM(y, y) because of (15). At the same time, the left part is greater than Iy - iI + E IWi(y) - Wi(77)1. i=1,2

Thus, condition 4) also holds for the balls given by (14). Below we abbreviate cp(yk) = Wk. Let {Ok}k>1 be a partition of unity for G subordinate to the covering by the open balls concentric with (14) with radii 4bcok. In view of Lemma 6.1/2, one may assume that

IVokI
1,

k=1,2,....

Furthermore, let {Ak}k>1 be a set of functions such that for all k > 1 Ak E Co- (B (k)), 0 < Ak < 1, Aktk = Qk,

dist (supP Ak,G \ B(k)) 2 CWk, IVAk1 < CWk

1

We now turn to constructon of the desired extensions u1, u2 of f. Let

S2,k = {x = (y, z) : y E Blkl, z = cpi(y)}, i = 1, 2, k > 1, and let f i,k be the mean value of f on Si,k. Define

Si,k 3 X H fi,k(X) = \k(y) (f (x) - fi,k) . The mapping x = (y, z) r+ 4ix = (y, z - cpi(y))

6. Boundary Values of Sobolev Functions on Non-Lipschitz Domains ...

342

transforms Si,k onto rk = {(y, z) : y E B(k), z = 0} with Ilfi,kIILp(si,k) - Ilfi,k

"Pi

and

1IILp(rk)

[fi,kJp,Si,k - [fi,k o (bi

1]P,rk,

are the seminorms given by (4.1.1/3). By Remark 4.1.1, there is an extension Ui,k of fi,k o iDs 1 from rk to R" such that where

k,

wk'IIUi,kIILp(R-) + IIVUi,kIILp(R')

< C k 1+"P Ilfi,k o

1

i

IILp(rk) + [fi,k o i ]P,rk)

.

Put C(k) = B(k) x R1 and ui,k = Ui,k o 4bi IC(k) Then Ui,k is an extension of fi,k from Si,k to the cylinder C(k) and Wk 1IIUi,kIILp(C(0) + IIVui,klILp(C0k))

< C ((pk 1+1IPIIfi,klI Lp(S;,k) + [fi,k]P,Si,k)

(16)

(the finiteness of the right part in (16) will be clear from the following argument). We define for x = (y, z) E 92

vi (x) = E fi,kak(Y),

(17)

k>1

wi(x) = E O'k(Y)Ui,k(x),

(18)

k>1

the general term of the last sum being zero outside C(k). Then each function

ui=vi+ wi

,

i=1,2,

satisfies the condition ui I Si = f I si .

Proof of estimate (12). Since Ek>1 Vok = 0 in G, it follows that

Vvi(x) = E k>1

LJ i,k - f (Y, Wi(y))] Vo'k(y), X E S2.

6.2. Domains Between Two Lipschitz Graphs

343

Hence, because the number of nonzero summands in the preceding sum is uniformly bounded for any fixed y E G, one has

Iovi(x)I' < C> Ifi,k - f(y, Wi(y))IPlVok(y)I' k>1 and

IloviIIL,(O) < CL(Pk-Pllf

-Ji,kIILD(Si,k)-

k>1

By Lemma 4.1.1, the general term of the sum on the right does not exceed c [f ]p S k . To bound this last seminorm, we use condition (13), which implies that I < aM(y,77) if x,C E Si,k. Now

[f]p,S;,k < cJ Si,k

dsy f CES::J{-zI
The family {Si,k}k>1 forms a covering of Si whose multiplicity depends only on n and therefore

E[f]p,s,

k

< c if IP,P) i = 1, 2.

(19)

i =1, 2.

(20)

k>1

Thus, IIVviIILp(o) <- C

Ifli,P,

Turning to the function given by (18), we obtain from the estimate IVokl C cpk 1 and (16) that

IlowillL,(n) < CE (Wk-Pllfi,k lLn(S:.k) + [fi k]p,Si,k) .

(21)

k>1

The first term on the right is estimated by Lemma 4.1.1: (22)

Wk Pllfi,kllLP(S:.k) < C[f]p,si,k

To bound [fi,k]p Si,k, one can observe that

C [fi,k?,S,,k < [f]9i k + I'

(23)

where

I=

ff

Si,k X Si'k

if (o - fi,klPlak(y) - Ak(ri)IP

ds,dsC SIn+P-2.

Ix -

344

6. Boundary Values of Sobolev Functions on Non-Lipschitz Domains ...

Since

-fi,

I)'k(y) - Ak(r1)I <_ CWk'Iy - 711, y, r, E B(k),

and because dsx - dy for x E Si,k,

f()-f,kldsfy-+II
I
< Cck-PIIf - fi,k1ILP(si,k) < C [f]P,S; k'

(24)

Again Lemma 4.1.1 has been employed in the last inequality. Equations (21)(24) yield

Ilowill p(0) <

C>[f]P,si k.

k>1

This in conjunction with (19), (20) implies (12). Step 2 is concluded. Step 3. Here we construct a global extension u off satisfying (11) and thus complete the proof of the theorem. It will be shown that one can choose the required extension in the form u(x) = ul(x) + (u2(x) - u1(x))(x - Wl(y))&(y), x = (y,z) E SZ,

where ui = vi +wi with vi,wi defined in (17), (18). Clearly, u I S = f because uilsi=fisi,i=1,2.

Turning to inequality (11), first we note that II °u1l LP(0) <-

i=1,2

dx

11VuiIILn(n) + fu lu2(x)

IP

W(y)P.

2


U2(X)

TIP

(25)

i=1

to prove the claim. It follows from definitions (17), (18) (cf. also conditions 1), 3) for the balls given by (14) at the beginning of Step 2) that the integral on the left in (25) is dominated by C E I f 2,k - I1,kI k>1

P.k-P

+C

r W_'IIUi,kI1LP(R°). i=1,2 kk>11

(26)

6.2. Domains Between Two Lipschitz Graphs

345

As it has been shown above (see (16), (22)-(24) and (19)), the last double sum is not greater than CE2i=1 iflpP, To bound the first sum in (26), we observe that its general term is comparable to

k-P LAO If2,k - f1,kIPdy and thus can be majorized by

cf M if (Yo my)) - f (y> C

Pl(y))IPG(y)1-Pdy

= Wk -P Lh) I f (y, Wi(y)) - f i k Pdy.

i=1,z

Since dy - dsx for x E Si,k, the last sum over i = 1, 2 does not exceed Ak-PIIJ -fi,kI1Lp(S:.k)

C

i=1,2

which is not greater than c E j= 1[f]P'S,", by Lemma 4.1.1. Taking (19) into P account, we can dominate the first sum in (26) by the right part of (25). Now (25) is established thus concluding the proof of the theorem. 1 The proof of Theorem 1 also contains the following assertion.

Corollary. There is a linear bounded extension operator: TWP (Q) - WP (1), p E (1, co) Remark 1. We may assume that a E (0, 1] in the definition of I f I i,P. This can be made by choosing the Lipschitz constant L for cp to be sufficiently large (cf. (8)).

Remark 2. Inequality (7) shows that relation (1) remains valid if

I f Ip ,

is

replaced by the left part of (7). The next theorem describes the traces of the functions in W1 (1).

1

Theorem 2. If 0 is the same as in Theorem 1, the following relation holds IIPITWi(S2)

f If

f I f (x)

t=1,2 {x,fES,:Ix-I
- A01

dsxds M(y,rl)"-1

346

6. Boundary Values of Sobolev Functions for Non-Lipschitz Domains ...

with notation introduced in Theorem 1.

Proof. The estimates for the trace of a function in Wi (Q) are obtained exactly in the same way as in Step 1 of the proof of Theorem 1. The further argument is analogous to that in Step 2 and Step 3 above and even simpler since (16) for p = 1 turns into Wk 1IIUi,kIIL,(R^) + IIDui,kIIL1(R^) < C Ilfi,k11Ll(S,,k).

It should be noted that the extension operator: TWi (S2) -* W1(S2) constructed in the same way as in Step 3 is bounded but nonlinear.

6.3. The Space TWp (O) for a Planar Domain with Zero Angle We begin with the definition of the vertex of a zero angle. Suppose 52 is a bounded domain in R2. Let 0 E 8S2 and assume that 812 \ {O} is a curve which can be locally represented as the graph of a uniformly Lipschitz function

in some Cartesian coordinates. At 0 we locate the origin of the Cartesian coordinates x = (xl, x2). Let Pl, IP2 E C°'1([0,1]), 01 (0) = W2(0) = 0,

and let cP = X72 - coi be positive on (0, 1] and satisfy lim p'(t) = 0 as t -+ +0.

Definition. The point 0 is called the vertex of a zero angle on the boundary of S2 if there is a neighborhood U of 0 for which U n Q = {x E R2 : x1 E (0,1), V1(x1) < x2 < W2(X1)

We will also assume for the simplicity of presentation that W(t) < t for t c (0, 1] and that (cf. Fig. 37)

Un8S2=u 1{x:x1 E [0,1):x2=Wi(xl)}.

Fig. 37

6.3. The Space TWp (S2) for a Planar Domain with Zero Angle

347

Note that the vertex of an outer peak in the sense of Definition 5.1.1 for n = 2 is the vertex of a zero angle. Let 0 be the vertex of a zero angle on the boundary of a planar domain 12. If r C i9Q is a curve distant from 0, then, by Theorem 4.1.1, every function u E Wp (S2), 1 < p < oo, has the trace u I r which belongs to Wp-1/P(r) if p > 1 and to L1(r) if p = 1. Thus, the trace uIan is defined almost everywhere. The space TWP (Sl) of these traces is equipped with the norm given by (4.1.1/1). In this section c denotes various positive constants depending only on p and

ft The relation a - b means that c-1 <

ab-1 < c.

Theorem 1. Let 0 be the vertex of a zero angle on the boundary of a planar domain Q. If p c (1, oo), the following relation holds II f II TWp (Q) -

(i f 1 t=1,2

1/p

I fi (t) I Plp(t)dt)

0

+{f}P+ i IfiIP+IIfIIw -=/P(r),

(1)

where fi(t) = f (t, col(t)), i = 1, 2, W _ W2 - W1,

{f}P

I9IP = (

-(f

1/P

1

If2(t) -

ff

fl(t)IPco(t)1-Pdt)

I9(t) - 9(T) IP It

{t,rE(0,1):It-rI<M(t,r)}

dtdr

l/P

- TIP)

M(t, T) = maxi cp(t), W (T)},

r = an \ {x = (x1, x2) E U n aQ : x1 < 1/2} and U the neighborhood from the definition of the vertex of a zero angle.

Proof. Let U E Wp (I1), ides = f . The estimate

{ f}P + E i=1,2

I' I fi(t)IPw(t)dt < c IIuII ;,p(o) 0

is obtained in the same way as (6.2.2/3-5) in Theorem 6.2.2/1.

(2)

6. Boundary Values of Sobolev Functions for Non-Lipschitz Domains

348

...

The inequality

Ilfllw;-l'P(r)
(3)

is a consequence of Theorem 4.1.1. We now turn to the estimate I f+Ip

c Ilullwi(n), i = 1, 2.

(4)

Let 6 E (0, 1) satisfy the following condition: the constant L = L(8) in the inequality

sup {Ico(t)-(p(T)IIt -TI-1:t,TE [0,8], t: T15 L can be chosen so that a = 1 in (6.2.2/8) for n = 2. This is possible because limt..o cp'(t) = 0. We may also assume B6 c U. One can observe that if

t,TE (0, 1), It - TI < M(t,r)

and if either one of the points t or T is in [8/2, 1), then the other point is in some interval (E(8), 1) with E(8) > 0. Indeed, let e.g. r > 8/2. Then t > T - M(t, T). If W(T) > W(t), we have

t > T - W(r) > min{s - W(s) : s E [8/2, 1]} > 0.

If W(r) < ep(t), then t + co(t) > T and hence t > 8/4 since cp(t) < t. Thus, the number e(8) really exists, and we may assume that e(J) E (0, 8/2). Now I fi I p 5c [f ]p Bn\Be(6)

+

If

I fi(t) - fi(l)l

ItdtdT p

(5)

{t,TE (0,6/2): It-r <M(t,r) }

where [ ]p

is the seminorm given by (4.1.1/3) for n = 2. The first term

on the right in (5) does not exceed c IIulIpyj l(n) by Theorem 4.1.1. To bound

the integral in (5), we introduce a cut-off function A E Cm([0, oo)), 0 < A < 1, AI[o 2/3) = 1, Al [5/6 m) = 0.

Next, let 06 = {x E Sl fl U : x1 < 6} and define the function SZ6 E) x H v(x) = A(xl/8)u(x).

(6)

6.3. The Space TWP (ft) for a Planar Domain with Zero Angle

349

Clearly, v(x) = u(x) for xl E (0, 6/2) and v(x) = 0 in the vicinity of xl = b. Let D6 denote the union of Q6 with its mirror image with respect to the line x1 = J. If v is extended to be zero on D6 \ Q6i then v E WP (D6) and, furthermore, IIviiW'(D6) <_ c(6)IIuiJwp(n)

By applying Theorem 6.2.2/1 (see also Remark 6.2.2/2) to domain Da, we dominate the integral in (5) by ciiviiWP(D,). Thus, inequality (4) is estab-

lished. Estimates (2)-(4) show that the right part of (1) is not greater than CII.fIITWP(SO

The remaining argument is devoted to the proof of the opposite inequality.

To this end, we let f be a function on acl and (f) the norm defined by the right part of (1). If (f) < oo, we should construct a function u E WP(S2) subject to (7) ulafl = f and IIuIIw, (o) <_ c(f) Let a be given by (6) and put

h(x) _

.\(xl), x = (x1i x2) E a1 n U,

x E asp\U U.

o

Then supp (1 - v)) f c r and it is readily seen that (cf. (4.1.1/4)) II(1- Of iI wi 1/P(r) < C IIf iiwD-1/P(r).

By Theorem 4.1.1, there is a function u E WP (S2) satisfying ulast = (1 -1G)f

and

IIuII W (fl) < c IIf

Ilwp-1/P(r).

We now define a function w c WP (S2 n U) such that wlasznv II'"Ii1v (sznu) < c { f }P + c

(Ifilp +

= f and

IILP(o,l))

.

(8)

i=1,2

Once w has been constructed, the function u subject to (7) can be given as follows. Let i E Cm ([0, oo)),) z = A and µ = 0 in a neighborhood of [l, oo).

Put ty(x)=p(xl)w(x)for xES2nUand w=0in S2\U. Then wEWp(S2),

wla,, =Vif and

IIWIIwp(sl) 5 CIIwiIW;(onU)

350

6. Boundary Values of Sobolev Functions for Non-Lipschitz Domains

...

Now (7) is fulfilled for the function u = ii + 17v. To construct w, we consider an auxiliary domain

D = {x : x1 E (0,2), cp1(xl) < x2 < cp2(xi)}, where

M1 + t) = cpi (1 - t), t E [0,1], i = 1, 2. A function h is defined on 8D by hI aDnu

= bf,

hl aD\U = 0.

By Corollary 6.2.2, there exists a function w E Wp (D) such that wl8D = h and 2

C II W II w11(D) <

f 1h2(t) - hi(t)IpcP(t)1-Pdt + E IIhiw11PIILp(o,2) i=1 2

o

dtdT

+ E ff Ihi(t)-hi(T)Ip It - TIP

(9)

i=1,2 H

with cc = W2 - co on [0, 2],

hi (t) = h(t, Wi(t)), t E (0, 2), i = 1, 2,

H={(t,T):t,rE(0,2), It-rI
M(t,T) =max{cp(t),cp(T)}, t, -r E (0, 2).

The constant a depends only on the Lipschitz constant L for W. In particular, one may put a = 2-6(1 + 2-5)-1L-1 E (0, 1] (cf. (6.2.2/8) and Remark 6.2.2/1). Clearly w = ?P f on 8 fl U. We claim that the right part of (9) is not greater than the right part of (8), and hence wlslnU is the required function

satisfying (8). Only the last term in (9) needs to be estimated. Note that if (t, T) E H, then w(t) - cp(T) and also It - 7-1 5 aL max{t, T} < 2aL < 1/6.

6.3. The Space TWP (Q) for a Planar Domain with Zero Angle

351

Here we have used that W(t) < Lt. Since hi(t) = 0 for t > 5/6, the integrand in the last integral in (9) is nonzero only if t, r E (0, 1). Therefore, the integral over H in (9) does not exceed

ff

c

A(r)Plfi(t) - fi(r)IP

Itdtdr

{(t,r)EH:t,rE(0,1)}

+C f

IA(t) - A(r)IP

Ifi(t)IPdtf 1

It - rP

dr

with A given by (6). The former term in this sum is not greater than c IfiIP, whereas the latter is dominated by c The proof of Theorem 1 IIfiw1"PIIL(o,l)

is concluded.

1

Remark 1. Relation (1) remains valid if r is any connected curve in 852 such that t is distant from {O} and dist (851 \ r, 90 \ U) > 0. This is provable by using an appropriate cut-off function instead of that in (6). The constants in (1) generally depend on P. I

The following assertion is established by an argument similar to that in Theorem 1 with reference to Theorem 6.2.2/2 in place of Theorem 6.2.2/1. Theorem 2. Let SZ be the same as in Theorem 1. Then IIf IITW; (sl)

f I fi(t) IW(t)dt + Il f II L, (r)

f 1 I f2(t) - f1(t) Idt + o

+ i=1,z

i=1,2

if

0

dtdr

Ifi(t) - fi(r)I M(t, r)

{t,rE(0,1):It-rI<M(t,r)}

with notation introduced in Theorem 1.

1

Remark 2. Let ) be the same as in Theorems 1, 2. Theorem 2 says that the space L1(81) is continuously imbedded into TWi (l). We now show that the inclusion L1(,9 ) c TWi (52) is proper.

Indeed, suppose L1(85Z) = TWi (52). Then IIfIIL1(

) <- C IIUIIWi (c1)

(10)

for any u c Wi (52) with f = ules1. Let e > 0 be sufficiently small and let v E C°'1([0,1]) be defined by v = 1 on [0, e], v = 0 on [e + w(,-), 1], v is linear

352

6. Boundary Values of Sobolev Functions on Non-Lipschitz Domains ...

on [E, e + cp(e)]. Put u(x) = v(xi) for x c Sl fl U and u = 0 on SZ \ U. By inserting u into (10), one obtains E+'P(E)

CE <

1

cp(t)dt + W(E)

re+ (e)

J

cp(t)dt.

However, this inequality contradicts the conditon W(t) = o(t) as t -> +0.

1

Remark 3. Let SZ be a planar domain with the vertex of a zero angle on the boundary. Then TWi (S2) does not coincide with a weighted space L1(852, u), where o, is a nonnegative bounded measurable weight function on 9I which is separated away from zero on any part of 8S2 distant from 0. Indeed, assume that the continuous imbedding L1 (8s2, o,) C TWi (S2) holds. Let f E L1(BSZ,o,). We define g on 8S2 by

g(t, 02(t)) = 0 for t E (0,1)

and g = f on the remaining part of M. Then, by Theorem 2, IIfOIIL1(8n) 2 IIg0'IILI(8n) 2 CII9IITWi(Q) 2 CJ 1If1(t)Idt 0

(we use the same notation as in Theorem 2). The estimate IIf°IIL1(80) 2 C IIf2IIL1(0,1)

is established in a similar way. Thus L1(80,a) C L1(8Q) C TWi (0). Hence the equality TWi (S2) = L1(8S2, o) implies TWi (SZ) = L1(8S2) which contradicts Remark 2.

6.4. Traces of Functions in W, ',(fl) for Domains Complementary to Those Between Lipschitz Graphs Here we describe the boundary traces of the functions in W" A' 1 < p < oo, for domains complementary to those considered in Sec. 6.2. Let G C Ri-1 (n > 2) be a bounded domain in C°". Suppose c1, cp2 E C°'1(G), cp1 <
(1)

6.4. 'Daces of Functions in Wp (1) for Domains Complementary to ...

353

By Theorem 4.1.1, every function u E Wp (1), 1 < p < oo, has the trace ul s, where S = S1 U S2 and Si is given by (6.2.1/2). It can be shown (cf. Exercise 1.6 for s = n - 2) that Lemma 6.2.1/2 and hence Corollary 6.2.1 with Remark 6.2.1 hold for the domain SZ just described. Below the sign - and the letter c have the same sense as in Sec. 6.2.

Theorem 1. Let SZ be given by (1) with a bounded domain G E C°"1. If p c (1, oo), then

IIlIITW,(f) - E Ilfllw'-1/p(Si) i=1,2

+(f I f (y, P2(y)) - f (y, P1(y)) Ipdist (y, 8G)p

1/p

(2)

1)

where Si = {(y, cpi(y)) : y E G}, i = 1, 2.

Proof. First we note that the function cpl can be extended to be uniformly Lipschitz and bounded on R1-1 (see Theorem 1.6.2 for p = oo; cf. also Stein [194], Chap. VI, Theorem 3). Then, with the aid of the map S2 (y, z) N (y,z - W, (y)), the proof of (2) is reduced to the case cpl = 0, cp2 > 0 on G, cp2 = 0 on 8G. In this case we abbreviate cp2 = WLet u E Wp (Sl), ul esz = f. The estimate 1,2,

IIf IIw'-./,,(S,) < c IIuIIw1(S2), i

follows from Theorem 4.1.1. To prove the inequality dy

JG

dist (y,

8G)p-1

< c IlullwD

(n),

(3)

we observe that it will suffice to estimate the integral over a narrow boundary

strip in G. There is a covering of 8G by a finite number of neighborhoods such that in each neighborhood the part of 8G is the graph of a uniformly Lipschitz function in some Cartesian coordinate system. Let us fix any such neighborhood U. Using a bi-Lipschitzian map of U onto an (n - 1)-cube (i.e., a map whose coordinate functions are uniformly Lipschitz and the same is true for its inverse), we reduce our consideration to the case U fl G = {y E R"-1 : yn_1 E (0,1), lyil < 1, i = 1, ... , n - 2}

(4)

U \ 0 = {y E Rn-1 : Yn-1 E (-1, 0), Iyil < 1, i = I,-, n - 2} .

(5)

and

354

6. Boundary Values of Sobolev Functions on Non-Lipschitz Domains ...

Then (3) follows from the inequality

J

(6)

Let L be a Lipschitz constant for cp. With any point y = (y', yn-1) E U fl G, y' = ( Y 1 ,- .. , yn_2), we associate the rectangle P1P2P3P4 with vertices

P1 = (y', yn-1, Lyn-1), P2 = (y', -yn-1, Lyn-1),

P3 = (y', -yn-1, -Lyn-1), P4 = (y

P3

, Yn-1,

-Lyn-1)

P4

Fig. 38

(see Fig. 38). Clearly C If (y, ,(y)) - f (y, 0)1' _< If (y, iv(y))

- u(Pi)I'

3

+

I u(PP) - u(Pi+1) I' + If (y, 0) - u(P4) I'

(7)

i=1

for y E U fl G. We now bound the integral over U fl G (with weight y,'-P) Of each term on the right of (7). Since W(y) < Lyn-1 for y c U fl G, one has If (y, W(Y)) - u(Pi)IP =

Lya-1 Vu I w(y)

8z

P

J(y, z)dz

which is not greater than Lyn-1

(Lyn-1)P-1 W(y)

P

dz 8z (y' z)

6.4. Traces of Functions in WP (S)) for Domains Complementary to ...

355

by Holder's inequality. Therefore JUnG If (y, (P(y)) - u(Pi) I Pyn-Pdy < C IIuIIW1(o).

The other terms on the right of (7) are treated similarly. Hence (6) follows.

In order to conclude the proof of relation (2), one should construct an extension u E WP (1) of a function f defined on 8SZ such that c llullW (l) is dominated by the right part of (2). Recall that it is sufficient to assume cp1 = 0 (and then we write cp instead of W2).

Consider a covering of the edge {(y,0) : y E 8G} by a finite collection {Uk}k 1 of neighborhoods Uk C R' with the following property. If Uk is the projection of Uk onto the plane z = 0, then the family {Uk}k 1 is a covering of 8G, and each set Uk n G can be represented by the inequality yn-2) in some Cartesian coordinate system where I.'k is a Yn-1 > ,bk(yl,

,

uniformly Lipschitz function. For some b > 0, the union Uk 1Uk contains the strip S(6) = {(y, z) E 9Q : y c c, dist (y, 8G) < b} . (8) We now turn to construction of the required extension f i-+ u for a function f with finite right part of (2). It will suffice to examine the case supp f c S(b). The general case then follows by using Theorem 4.1.1 and a smooth cut-off function.

Fix a k, 1 < k < N, and put U = Uk. One can assume without loss of generality that (4) and (5) hold. Let W be extended by zero to Rn-1 \ G and let

17_={(y,0):yEU}, r+={(y,cp(y)):yEU}, S_={(y,0):yEUnG}, S+={(y,W(y)):yEUnG} Given f, define f_ on r_ by f- IS_ = f IS_ and f- (y, 0) = f (y1,

, yn-2, -Y.-I, 0) if y E U \ G.

It is readily seen that Ilf-llww-1111(r_) < c If Il wn-1"D(s_)-

Next, define

f (x) for x E r+ n S+,

f_(x)for xEr+\s

6. Boundary Values of Sobolev Functions on Non-Lipschitz Domains

356

...

We claim that then f+ E WP-11P(F+) and the estimate IIf+I1P

_1/P(r+)

<< c II f II wi1/P(9 +)+ C II f II W11-1/P(S-)

+c J

(12)

py1

UnG

yn-1

is valid. Indeed, in view of (11), inequality (12) follows if we dominate the integral

f

If+(x) nr- ds{ f r

Ix

dsx - CIn+P-2

(13)

by the right part of (12) (here dsx, ds£ are the area elements on r+). Expression (13) does not exceed

cf diif

If-(77,0)-f(y,co(y))IPly

d n+p 2

Iy

which is not greater than c

f {y,17EUnG}

+cf

UnG

I f (y, 0) - A q, 0) IP I_

dyd - yln+P-2

If (y, 0) - f (y, P(y)) I Pdy JUG In - yln+P-21

where _ (r11, ... , 7n-2, -77n-1) The integral over {y,'/1 E UnG} is majorized by c IIf IIyy1-), and the last integral over 77 E Uf1G is dominated by cyn=P Therefore, (10) holds.

We now introduce a smooth partition of unity {µk}k 1 for the strip (8) subordinate to the covering {Uk}k 1. Furthermore, let \k E C o (Uk), \kµk = µk, 0 < Ak < 1 , k = 1, ... , N.

The previous argument shows that for each k = 1,. .. , N there exist local extensions fk) of f from the surfaces given by (10) (with U = Uk where Uk is the projection of Uk) to the surfaces F = r(k) given by (9) such that Ilakf+ )Ilwp-1/P(r(h)) _< c E IIfIIWD-1/P(Si) +

1=1,2

6.4. Traces of Functions /in WP (1) for Domains Complementary to ...

+c

JG

If (y, c'(y)) - f (y, 0) IP dist (ydeG)P-> .

357

(14)

By Theorem 4.1.1, for each k = 1, ... , N there are extensions WP 'IP(r±k)) D \kftk) H u+k) E Wp (R.")

subject to Ilu(k)IIW'(Rn)

< cIIAkffk)IIWn-"

(rfl).

Note that the constants in (14) and in the last inequality generally depend on the collection {Uk } (but do not depend on f). For k = I,-, N define

0 outside Uk, Uk (x) =

jIlk (x)U- (x) if x = (y, z) E Uk f Q, z < 0, Ilk (x)u+k) (x) if x = (y, z) E Uk fl SZ, z > W(y)

Then uk E Wp(]) and IIUkIIpWP(Q) is majorized by the right part of (14). Now

1 Uk is the required extension of f from 9) into Q. The proof of I Theorem 1 is concluded.

u=

The description of the space TWi (SZ) does not differ from that for SZ E C°'1

Theorem 2. Let Q be the same as in Theorem 1. Then TWi (SZ) = L1(85Z).

Proof. It is again sufficient to assume cpl = 0. We observe that the estimate IIf IILl(8Sl) < C II f II TWi (Sl)

(15)

follows from Theorem 4.1.1. In order to prove the reverse inequality, one should extend f by zero to the set {(y, 0) : y V G}. Then, by Theorem 4.1.1, there are functions ul E Wi (SZ1), U2 E Wi (SZ2), where

Q1={x=(y,z)ESZ:z<0}, 522={xE ):z>0}, v'1Ian, = f Ian,, u2Ion, = f 18Q2 and

IIUiIIWi (Qi) < C 11 f IILi(aci,), i = 1, 2.

358

6. Boundary Values of Sobolev Functions on Non-Lipschitz Domains

...

Define u on SZ by u(x) = ul(x) for z < 0 and u(x) = u2(x) for z > 0. Then u E W1 (12). Furthermore, uIan= f and IIullW (n) <_ CIIfIILI(en).

Hence the inequality opposite to (15) follows.

I

Remark 1. Suppose cpl = W2 in (1). It is natural that in this degenerate case functions in WP (52) generally have different traces on the "upper shore" and the "lower shore" of 852. Given u E Wp(1), put

u_ = ul {(Y,z).yEG,zv(y)}' where cP = cot = cp2. Let S = {(y,W(y)) : y E G} and f+, f- E Lp(S). One can modify the proofs of Theorems 1-2 to establish the relations inf {IIulIw;(n) : u+I s

= f+, u-I s =

f-} ,,, IIf-IIWD-11,,(S) + II.f+Ilw' lie(s) 1/p

+

(IIf+(y, P(y)) - f- (y, P(y)) Ip disc (y,BG)p-1) inf {IIuIIwi (n) : u+Is = f+,

u-Is = f-} -

'P>

IIf-IIL1(S) + IIf+IILI(S)

Remark 2. Let 12 be the same as in Theorem 1. It is of interest to note that the norms in the trace spaces TWp (1) and TWp (R" \ 1) are generally noncomparable for p > 1. The intersection TWp (12) fl TWp (R" \ 52) can be easily interpreted as the space TWp (R", 812) of the traces of functions in Wp (R") on 812 with norm IIfIITW1(R",an) = inf {IILIIwp(R") : u1an = f}.

Combining Theorems 1 and 2 with Theorems 6.2.2/1-2, we arrive at the following normalizations of TWp (R", 812): IIf IITW1(R",an) - IIf IIL1(Sn) and

1/p

IlfIITWp(Rn,an)

{ f I f(y, W2(y)) - f(Y,

+ : IIfIIwp-1ip(S;), i=1,2

Wz(Y))Ipw(Y)1-'dyl

6.5. A Planar Domain with the Vertex of an Inner Peak on the Boundary

359

where p E (1, oo) and the same notation as in Theorem 1 has been used.

6.5. A Planar Domain with the Vertex of an Inner Peak on the Boundary In this section we deal with domains SZ described in Definition 5.5.2. Let 0 be the vertex of an inner peak. By Theorem 4.1.1, every function u E WP Al p E (1, co), has the trace u I r E Wp-l/P(r) if I' C 8S2 is a curve distant from 0. Hence ulan is defined almost everywhere. The space TWp (52) of all such traces is normalized by (4.1.1/1). Below positive constants c depend only on p, I and the relation a - b means that a/b is bounded above and below by such constants.

Theorem. Let 0 C R2 be a bounded domain with inner peak. If p E (1, oo), then IIlIITWW(si)-

(Ill II(a

dsMdsN

+

l1/p

ff If(M)-f(N)IPP(M,N)P/

(1)

OS2 x aO

where dsM, dsN are the length elements on Q and e(M, N) the distance between the points M, N along 852.

Proof. One can assume without loss of generality that Un8S2 = {O}US_US+, where U is the neighborhood from Definition 5.5.2 and

Sk,={(x,cpf(x)): xE (0,1)}.

(2)

If f is a function on 852, then f+, f_ are defined on (0,1) by f± (x) = f (x, W± W), x E (0, 1).

We observe that the norm on the right of (1) is equivalent to the norm IIIf AI = 11f IIWD-lia(r) + (f),

where IF = 852 \ {(x, y) E U n 852: x < 1/2} and 1

WP - IIfliW, 'ID(S-)+IIf IIWD1

d

'"(S+)+ If+(x)-f-(x)IP-1

6. Boundary Values of Sobolev Functions on Non-Lipschitz Domains ...

360

The equivalence of the norms follows from the relations

,o(M, N) ' x + t if M = (x, W+(x)) E S+, N = (t, W- (t)) E S-, xl-P

J

dt

1

(x + t)P

0

X E (0, 1),

and the inequality dxdt \/I f- (t) 1P (x + t)p /

C 1f If+(x)

1/p

- (ff Q

Q

(ff If (x)-f-(t)IP

dxdt Ix

dxdt I f+(x) - .f- (x) I1 (x +

1/p t)P

)

1/P

- tIP

Q = (0,1) x (0,1).

Q

Thus, relation (1) is equivalent to IIfIITww(Q) - IBf l-

(3)

Let e > 0 be chosen so that B2E C U, B2E fl I' = G. Using a smooth cut-off function and Theorem 4.1.1, one can readily show that (3) is a consequence of the relation IfIITWP(ci) ^- (f)

(4)

for f satisfying supp f C B. To check (4), we introduce an auxiliary domain

D=R2\{(x,y):x E [0, 2], W-(x)
Given f on 8S1 with supp f c BE, define g on 8D by gleDnan -

f8Dnasl, gI aD\8S2 = 0.

Let ' E Co (B2E), 0 = 1 on B. If U E Wp (S2), ule, = f, and v is introduced by

vI DnBa,

= 'Pu'

VID\B2

0,

6.5. A Planar Domain with the Vertex of an Inner Peak on the Boundary

361

then v E WP (D) and, furthermore,

vlaD=9, IIvIIW,(D) Conversly, if v E WP (D), V

OD

CIIUIIWI(o)

= g, and if u is defined on I by

ulnnBsc

-

i,bv,

0,

UIa\BZ,

then u is in WP (1l) and

Ulan = f,

IIHIIWW(n) <- CIIVIIW1(D).

Thus, the inclusions f E TWP (S2), g E TWP (D) simultaneously hold and f II TWy (S2) - 119I1TWD (D)

Now (4) follows from Theorem 6.4/1 which concludes the proof.

I

Remark 1. If SZ is the same as in Theorem, then TWi (1) = L1(a1). This equality is a simple consequence of Theorem 4.1.1.

Remark 2. Obvious modifications in the proof of the theorem enable us to consider a domain with a cut, i.e. the case cp+ = cp_ (cf. Remark 6.4/1). We formulate the corresponding result using the following example.

Example. A disk with deleted radius. Let Il be the unit disk with deleted radius {(x, 0) : 0 < x < 1} and let (r, 0) be polar coorinates. One can show that a function f defined on 8Q (and having generally different values for 0 = 0 and 0 = 27r) is in TWP (1), p E (1, oo), if and only if r27, f21r If (1'0) - f (1, o`)IP

r27r

If(1,B)IPdO+J

J0 +

IB - aIP

0

f' d, 0

7,,21f(r,0)-All0)IPdO+ f 0

(1 - r)P + OP

0

dada

ldr r2, If(r,2ir)-f(1,e)IPdO 3n,2 (1 - r)P + (21r - O)P

+f1(If(r,0)IP+If(r,27r)IP)dr+ f 1If(r,2ir)-f(r,0)IPrd-l <00. 0

0

Remark 3. Combining above Theorem with Theorems 6.3/1-2 and Remark 1, we arrive at the following normalizations of the space TWp (R2, 8S2) for a domain 0 with inner peak (cf. Remark 6.4/2): If II TWi (R2,8S2) - If IIL1(8Sl)

6. Boundary Values of Sobolev Functions on Non-Lipschitz Domains ...

362

and

IIfIITWP(W,aQ)^

fo

1

If(x,

+(x))-f(x,w-(x))I1w(x)1-1dx

1/p

+ Ilf llw; 11n(s_) +

P

II f II Wp-1/n(S+)

+ II f iiWP-1/n(r) }

,PE

(1, CO)

Here cp = cp+ - W_, S+, S_ are given by (2) and r is any open connected subset of 8SZ, distant from 0 and containing 9 \ (S+ U S_ U {0}).

Comments to Chapter 6 The contents of sections 6.1, 6.2 and 6.4 are taken from the paper by Maz'ya, Netrusov and Poborchi [142]. Theorem 6.3/1 under some additional conditions on the form of the cusp was established by Yakovlev [218] using a different

approach. Theorem 6.3/2 was proved in the authors' paper [148]. The description of the space TWp (a), p > 1, for some domains with cusps was also obtained by Vasil'chik [205]. In connection with Theorem 6.4/2, we mention the works by Burago and Maz'ya [31] (see also Maz'ya [136, Chap.6]), and by Anzelotti and Giaquinta

[11], where the traces of the functions in BV (1) and Wi (S2) were studied under weak restrictions on domains. Theorem 6.5 is due to Yakovlev [217]. Since late 70s the traces of differentiable functions on subsets of R" with nonintegral Hausdorff dimension (such subsets are also called fractals) have been intensively studied. See Jonsson and Wallin [106], Jonsson [105], Bylund [39] and Wallin [212], [213] for an account on this subject. Netrusov [164], [166], [167] considered the problem of extension of func-

tions in Ly-spaces on subsets of R' to spaces of smooth functions on R'.

He showed, in particular, that for every set F C R' with locally finite (n-l)-dimensional Hausdorff measure there is a bounded (nonlinear) extension W11(R"), l < n. operator: L1(F) We point out a recent paper by Hajlasz and Martio [86] where the traces of functions in Wp (Rn) on a wide class of subsets of Rn have been described in terms of Sobolev spaces on metric spaces. Finally, we mention the work by Brudnyi and Shvartsman [28], where, in particular, the trace space (C1 (R") fl L , (R")) I F has been characterized for

closed F C R".

CHAPTER 7

BOUNDARY VALUES OF FUNCTIONS

IN SOBOLEV SPACES FOR DOMAINS WITH PEAKS

Introduction In this chapter the trace space TWp (52), p E [1, oo), (cf. 4.1.1) is studied for

a domain with the vertex of a peak on the boundary. It turns out that the boundary values of functions in Wp (52) are characterized for such domains by the finiteness of the norm \ 1/p

(f )p,an = 1

1 If (x) l pq(x)dsx + ff if(x) - f (e) I PQ(x, C)dsxdsg I

,

anxao where q and Q are nonnegative weight functions and ds, ds£ the area elements on 852. We now describe the results of Chapter 7 using the following domain which is a special but typical case of domains we shall deal with. Suppose cp is an increasing function in C0,1([0, 1]) such that cp(0) = limz,o W'(z) = 0. Put

Q = {x E R, xn E (0, 1), xi +... X2 n_1 < (p(xn)2} , n > 3. The case n = 2 was examined in the preceding chapter. Let p E (1, oo) and let f be a function on 852 vanishing outside a small neighborhood of the origin. Then f E TWP (52), p E (1, oo), if and only if (f )p,ac < oo, where 0 < q(x) < const fp'(xn),

Ix -

e12-n-p

if Ixn - Snl < M(xn, Sn), xn,

E (0,1),

0 otherwise and M(z, () = max{cp(z), cp(()}. In addition, the norm (f )p,ac is equivalent to Ilf lITwp (n) A necessary and 363

364

7. Boundary Values of Functions in Sobolev Spaces for Domains with Peaks

sufficient condition for f to belong to TWP (Rn \ 0) is that (f )p,as2 < oo with cp(xn)1-P for 1 < p < n - 1, (c(xn)Ilog(w(xn)/xn)I)1-P

for p = n - 1,

q(x) _ cp(xn)2-n for p > n - 1

and Q(x, ) # 0 only if xn, n E (0, 1). For these pairs x, C E 852, Q is defined as follows. If p < n - 1, then Q (X, S)

=

IX

-

I2-n-P

In the case p > n - 1 and Ix - EI < M(xn, £n) the weight Q is determined by the same formula. Finally, if Ix - I > M(xn, fin), then Q(x, S) = Ix - S1-1M(xn, X

Sn)2(1-P)

Ix-eII

(log(1+ M(xn, Sn))//)

, p=n-1,

and

Q(x, C) = Ix -

CIn-P-2 (co(xn)co(Sn))2-n,

p > n - 1.

The norm (f)p,asz with these weights q,Q is equivalent to IIfIITw'(R.^\sl). In the case p = n-1 some additional restrictions are imposed on ep (not excluding power cusps). A function f defined on 8S2 and vanishing outside a small neighborhood of the origin is in TWi (52) if and only if (f ) 1,aQ < oo with 0 < q(x) < const W'(Xn),

Q(x, F) =

M(xn, fin) 0

1-n

for xn, Sn E (0,1), Ixn - enI < M(xn, W,

o therwise.

Furthermore, the norms IIf IITwi (Rn\si) and (f),,&2 are equivalent. The characterization of the space TWi (Rn\S2) is the same as for SZ E TWi (Rn \ SZ) = L1(8S2).

CO,1:

7.1. Traces of Functions with Gradient in L1

365

In all cases mentioned above there is a bounded extension operator:

TWp(SZ) -+ WP(SZ) or TWp(R"\1) -*Wp(R This operator is linear for p > 1 and nonlinear for p = 1.

7.1. Traces of Functions with Gradient in L1 In this section the space TW1(S2) is studied for an n-dimensional domain, n > 2, with the vertex of an outer or inner peak on the boundary. 7.1.1. Outer Peaks

Let SZ be a bounded domain in R", n > 3, and let 0 be the vertex of an outer peak on 8S1 in the sense of Definition 5.1.1. In addition, we let 8w and 8S2 be connected and iv C B(in-1) For convenience, we will also assume that p(z) < z for z c (0, 1] and that

U n esi = r u {o}, r= {(y, z) E R" : z E (0, 1), y/w(z) E awl,

(1)

where W is the function describing the cusp and U the neighborhood from 1 Definition 5.1.1. We now make a general remark concerning the trace spaces TWP (0), p E

[1, oo), for domains with outer or inner peaks. Since 8SI \ {O} is locally a Lipschitz graph, every function u E WP '(Q) has the trace ulest defined a.e. on 8Sl. It can be shown (cf. Exercise 1.6) that the set {u E Wp (1) : ul asZ = 0} coincides with bVP (S2), i.e. with the closure of Co (1) in Wp (S2). Therefore, the space TWP()) _ {f = ule, : u E W1P (1)} normalized by (4.1.1/1) is isomorphic to the factor-space WP (SZ)/4VP(1).

1

In this section c, co, c1, ... denote positive constants depending only on n and ft the relation a - b means that co < a/b < c1.

Theorem. Let 0 be the vertex of a peak directed into the exterior of a domain SZ C R". The following relation holds

I

I f II Twi (n) - f

(z) I f (x) I ds. + fs If (x) Ids. + (f), nasl

(2)

7. Boundary Values of Functions in Sobolev Spaces for Domains with Peaks

366

where

it

dsxdsg

I1(x) - M) I M(z, ()n-1

{z,CEUn8S2:I(-zl <M(z,()}

x = (y, z), _ (71, (), dsx, dsC are the area elements on 852, U is the neighborhood from Definition 5.1.1, S = 852 \ {x c U (1852: z E [0,1/2]} and M(z, () = max{W (z), W(()}.

Relation (2) remains valid if the first term on the right of (2) is omitted. Proof. Let u E W, '(Q), ul ao = f . Then, by Theorem 4.1.1,

IIfIIL,(s) _CIIUIW;(f)

(3)

Let I' be defined in (1) and -y = 8w. We have

fr

(z) I f (x) Ids. < c

f

v(z)"-2

1

o

(z)dz

f

I f (W(z)Y, z) I dyY.

ry

By means of the change of variable W(z) = t, the right part of the last inequality takes the form C

f

t"-2dt

I.

If (tY, P-1(t))Id'YY

(4)

0

The mapping s = y, t = cp(z) transforms u n i onto the cone K = {(s, t) t c (0, w(1)), s/t E w}. Put v(s, t) = u(s, cp-1(t)). Then IIVIIWi(K) <_ CIItIIw1(Uns1).

Next, we observe that quantity (4) does not exceed cIIvIIL,(aK) which is not greater than c I I V I I W (K) by Theorem 4.1.1. Hence

f'(z)If(x)Idsx < c IIuIIwl A-

(5)

It follows from (1) and the definition of (f) that (f) = 21(f) where

I(f)= v(dsi_1 f

If(x)-f(f)IdsC.

(6)

7.1. Traces of Functions with Gradient in Ll

367

Thus, the inequality (f) < c IIuIIwi (o) is a consequence of the estimate I(f) _< d IIVuIIL1(Unn).

(7)

We now check (7). Let z1 E (0,1) and let 2co(z) < z for z c (0, z1]. If zo = 1, zk+1 = Zk - 2co(zk), k > 1,

then {zk}k>o is a decreasing sequence with 0,

Zk

W(zk+l)W(zk)-1 -+ 1, zk+lzk1 -4 1.

We assume z1 to be so small that lp(zk_1) < 2W(zk) for all k > 2. In this case z - W(z) > zk+1 for z c (zk, zk_1), k > 2. Put

a=min{z-cp(z):zE [z1i1]}, r1={xEF:zE (a,1)} and

I'k = {x E F : z E (zk+l, zk-1)}, k > 2, Uk = {x E r : z c (zk, zk_1)}, k > 1. Then the inclusions x E 0k,

I (f)

03

f k T( )n-1 fr If (x) - f(f)Ids < cE [f]l,rk, dsx

_ k=1

where

E r, (E (z - cp(z), z) imply E Fk. Hence C

k

k=1

is given by (4.1.1/7). Applying Theorem 4.1.2 yields

[f]l,r < c IIouIILI(ok), k > 1, with

521={xEUfl1 :zE (a,1)}, SZk={xEunQ:zE (zk+l,zk_1)}, k>2.

Thus, inequality (7) follows. Estimates (3), (5), (7) show that the norm on the right in (2) is dominated {O}) by c IIf II Tw; (st) To verify the reverse inequality, we let f E L1,10 be such a function that the sum of two last terms in (2) is finite. An extension u of f from %' into Q should be constructed to satisfy

cIuIIwi(o) <_ IIfIIL1(3)+(f)

(8)

7. Boundary Values of Functions in Sobolev Spaces for Domains with Peaks

368

First we consider the case f is = 0. Define to = 1, tk+1 = tk - v cp(tk), k > 0,

(9)

where

v = min{1/6, (2IIcp

IIL-(o,1))-1}.

Putting for brevity Wk = cp(tk), we observe that

t2 > 2/3, tk % 0, tk+ltk 1 -4 1, cok+lck 1

1.

Furthermore, tk

Wk - (Pk+l = f

W'(t)dt < uokII c IIL0(0,1) < c/,

t"+1

and thus Wk < 2Wk+1 for k > 0. Hence

tk-1 - tk+1 = V (wk-1 + Pk)

6v cpk+1 5 Wk+1

(10)

and therefore

(tk+l, tk_1) C (z - cp(z), z + cp(z)) if z E (tk+l, tk_1), k > 1. Let

sk = {x c r : z c (tk+1,tk-1)}, k > 1,

Gk={xEUnQ:zE(tk+1,tk_1)}, k>1, A Z I

I

Fig. 39

(11)

7.1. Traces of Functions with Gradient in L1

369

(see Fig. 39). It follows from Remark 4.1.1 and Corollary 4.1.1 that for each k > 1 there is a function wk E Wi (Gk) subject to wkISk = f - .fkf Wk 1llwkllL,(Gk) + IIVt kIIL,(Gk) < C[f]1,Sk,

(12)

where f k is the mean value of f on Sk and is given by (4.1.1/7). Let {Eck}k>1 be a smooth partition of unity for (0, t1] subordinate to the covering {(tk+1,tk_1)}k>1. Clearly one may assume Iµkl < ccpk1. Put

v(x)=>fkIk(z), x=(y,z)EUn1,

(13)

k>1

w(x) = > µk(z)wk(x), x E U n Q,

(14)

k>1

(E f) (x) = v(x) + w (x), x E U n Q,

E f l n\U

= 0,

(15)

the general term of the sum in (14) being zero outside Gk. Then v(x) = w(x) = 0 for z > t1. Hence and from (13)-(15) follows the equality E f Iasz = f. We claim that also IIEfIIw;(o) <_ c(f). (16)

To prove the claim, we observe that

vlGknGk+l = fk+1 + (fk - fk+1)Ak, k > 1, whence k+1

IIDvIILl(GknGk+i) < CWk_1 A - fk+115 C1

E Ilf - fiUUL1(Si). i=k

By Lemma 4.1.1, the last sum does not exceed Ls +k [his,. Therefore

IIovIIL,(uno) < CE [f]1,Sk

(17)

k>1

In view of (12), an analogous estimate holds for w: IIVwIIL1(Uno) < CE [f]1,Sk k>1

(18)

370

7. Boundary Values of Functions in Sobolev Spaces for Domains with Peaks

By (11), the sum on the right in (18) is not greater than

>ISkI-1 f dsy f k>1

{(Er:j(-zj<M(z,()}

Sk

where ISkI is the area of Sk. Since ISkI _ M(z,()s-1 for x E Sk, E I' with IC - zI < M(z, (), the last sum is dominated by c (f ). Combining this with (15), (17), (18), one obtains IIV(Sf)IILI(n) < C(f)-

Now (16) holds by Lemma 5.1.1, so u = E f is the desired extension of f if

fls = 0. Turning to the general case, we let f be in L1,1.,,(8SI \ {O}) with (f) + I

I f IIL1(s) < 00. By Theorem 4.1.1, there is a function u E W1 (St) subject to

ulon = Xf,

(19)

IIuIIWl (o) <_ C IIf IILI(S),

where X is the characteristic functon of S. Put g = (1 - x) f and define Eg by (13)-(15) (with f replaced by g). Then the function u = u + Eg satisfies ul asz = f, and (16), (19) yield C IIullW (s1) <- If IILr(S) + (g).

Now (8) is a consequence of the estimate (9) -< C(IIfIIL1(S)+ (f))-

(20)

To check the last, we observe that (f) - I (f) with I (f) given by (6). Next, I9(x) - 9(0I <- I f (x) - f (0I + If (x)IIX(x) - X(e)I therefore

I (9) < I (f) + Jr If (x) I W(Z)n-1

f

(z)

IX(x) -

Ids

,

(21)

where I'(z) = { = (77, () E IF : ( E (z - cp(z), z)}. If x E IF \ S, the integral over r(z) equals zero and in the case x c r n S this integral does not exceed

7.1. Traces of Functions with Gradient in Li

371

ccp(z)n-1. Hence the last term in (21) is dominated by c 11f IIL1(rns) Thus, (20) is valid. The possibility to omit the integral over U n X2 in (2) follows from (5) and (8). The proof is concluded. 1

Remark. Relation (2) remains valid if S is any connected open subset of as2 at a positive distance from 0 containing as2 \ U. This can be shown in the same way as above by choosing an appropriate coefficient v in (9). The constants in relation (2) generally depend on S. 1 The following assertion gives an explicitly defined seminorm equivalent to - IITL'(n) (cf. (4.1.1/2)). II

Corollary. Under the assumptions of above Theorem, the relation holds III IITL;(n) - (f) + [f]1,s, where

is the seminorm defined by (4.1.1/7).

Proof. Let 7 denote the mean value of f on S. Corollary 5.1.1, Theorem 1.5.4 and relation (2) with omitted first term on the right yield I f I ITLI'(n) -IIf - f II TW (n) - (f) + IIf - f II L, (S) I

It remains to observe that III - f IIL1(s) ' [f]l,s

Let S2 C R^, n > 2, be a domain with an outer peak and let r be a nonnegative function in L(a52) which is separated away from zero in any part of as1 distant from 0. By L1(,9 , o,) we mean the weighted space with norm IIfIIL1(en,a) = II°fIIL1(8n)-

If v(x) < ccp'(z) in the vicinity of the vertex, then the above theorem implies the validity of the imbeddings L1(8 ) C TWi (S2) C L1(aIl, a). We now show that these imbeddings are sharp and thus it is impossible to characterize the space TWi (S2) in terms of L1(au2, o,).

Let L1(a1 , a) be imbedded in TWi (S2). We divide ry = aw into two disjoint

parts -yl and y2 with equal (n - 2)-dimensional areas. Put (0,

i = 1, 2.

7. Boundary Values of Functions in Sobolev Spaces for Domains with Peaks

372

Given f E L1(91 Q), define g on r9Q by g(x) = f(x) for x E r1 and g(x) = 0 for x E asp \ r1. Since g E L1(aQ, a), the above theorem yields IITfIIL1(r1) = II°91IL1(oQ) >- C1II9IITW, (n) >- c2(9)

rl

> C3

dsx

If (x) I (ds

frz(z) ds(> C4

f

1

If (x) Ids.,

(77, () E F2 : ( E (z - cp(z), z)}. In the same way one

where 172 (Z)

obtains II°f IILI(r,) >- C IIf IIL,(r,)

Thus, the imbedding L1 (a1, a) C TWi (0) implies L1(aQ, u) C L1(ac ). Now let TWi (c) be imbedded in L1(8 , a) and suppose that the weight u is "stronger" than cp' in the following sense: lim ess inf { (cp (t)) -1 l

z-+0

J7o, (cp(t)Y, t)dryy : t E (0, z)1 = 00.

(22)

Let e > 0 be a small number. Define u on S1 by u = 0 outside the set {(y, z) E U n S2, z < e + ap(e)}, and on this set u(x) = 1 for z < e and u is linear for z E [e, e + cp(e)]. Inserting u into the inequality II0UIIL,(8o)
results in E

f cp(z)"-2dz 0

f

z)d yy < c

P(e)n-1

w hich contradicts (22).

In particular, the space TWi (1) is not imbedded in L1(aI) and hence does not coincide with a weighted space L1(ac , a) (cf. Remark 6.3/3).

7.1.2. Inner Peaks

Let S2 be a bounded domain in Rn, n > 3, and let 0 E 91 be the vertex of a peak directed into S2 in the sense of Definition 5.5.1. For convenience, we impose the same additional requirements as for outer peaks: w C B(n-1), au) and 8 are connected and W(z) < z for z E (0, 1]. The space TWi (1) is characterized by the following theorem.

7.1. 'Daces of Functions with Gradient in L1

373

Theorem. Let S2 be a domain with the vertex of an inner peak on the boundary. Then TWi (S2) = L1(an).

Proof. We may assume without loss of generality that the neighborhood U from Definition 5.5.1 has the form U={x =(y, z) E R' : z E (-1,1), IyI < 1} and that (7.1.1/1) holds. Let u E Wi (S2), uleo = f . The norm of f in L1(0 1 \ U) is majorized by c IIuIIwi (n) in view of Theorem 4.1.1. Furthermore, according to Theorem 4.1.3, the estimate IIf(', z)IIL1(r=) < C Hu(', z)II w; (u.)

(1)

holds for almost all z E (0,1), where rz and Uz denote the sections of U f18SZ and UnS2 by the hyperplane z=const. Integration of (1) over z E (0, 1) implies that the norm of f in L1(U fl acl) is dominated by c IIuIIwi (Sa) Thus IllIIL1(8O)
For the proof of the reverse inequaltiy, one should construct an extension u E W1(0) of a function f E L1(852) satisfying Ilullwi (St) < C IIf IIL1(ocl)

(2)

Let B2E C U. It will suffice to define the required extension off when supp f c BE. The general case then follows using a cut-off function and Theorem 4.1.1. We can assume W(2e) < E. Introduce a sequence {zk}k>o by zo = 2e,

Zk+1 = Zk - W(zk), k > 0.

Clearly

Zk \ 0, zk+lzk

''(zk+1)W(zk)-1 - 1.

Let {Uk}k>1 be a smooth partition of unity for (0, zl] subordinate to the covering {(zk_1i zk_1)}k>1i such that IrkI < ceo(zk)-1. We put

rk = {x E U n an: z c (zk+l, zk_1)}, k > 1. By Remark 4.1.1, for each k > 1 there is an extension operator Ek : L1(Fk) -* W1 (Rn) subject to IIV(Ek9)IIL1(Rn) + (p(zk)_1IIEk9IIL1(R°) < C II9IIL1(rk), 9 E L1(rk)

7. Boundary Values of Functions in Sobolev Spaces for Domains with Peaks

374

Let V, E Co (B2E), z/b = 1 on Be. We define u on 0 by u(x) = 0 for x c SZ \ B2E and

u(x) =

(x) 1: ak(z)(F'kfk)(x), x = (y, z) E Q fl B2e, k>1

where fk = fIr,,. Then uIan = f and one can easily check that (2) holds.

This concludes the proof. I The following assertion is a direct consequence of the theorem just proved.

Corollary. Let S2 be the same as in Theorem. Then IIf IITLi(n) '" [f]1,aci,

where the left and the right parts of the relation are given by (4.1.1/2) and (4.1.1/7), respectively.

Proof. Let f be the mean value of f on 852. By Theorem 1.5.4 (clearly 0 satisfies the cone property) and because TWi (S2) = L1(8St), we have Ill IITLi(o) '" Ill - f II TWi (S2) '" If - f IIL1(oSt) ^' {fl1,aci

thus establishing the desired relation.

7.2. The Space TWp (1 ), p >1, for a Domain with Outer Peak Let 11 C Rn, n > 3, be a bounded domain with an outer peak (cf. Definition 5.1.1). We preserve the additional assumptions stated at the beginning of Sec. 7.1.1. In this section positive constants c, c1, ... and the constants in equivalence relations depend only on n, p, Q. The space TWP (S2) is described by the following theorem.

Theorem. Let 0 be the vertex of a peak directed into the exterior of the domain I. If p E (1, co), then 1/p

IIf

II TWw

(Q) -

UnOn

(z)If (x)IPdsx)

+ IIf IIwl-1/p(a) 1/p

+

ff

{x,cEUnaQ:Jc-zJ<M(z,S)}

dsxds{

If (x) - f (e)IP Ix - eIn+P-2

/

(1)

7.2. The Space TWp (1), p > 1, for a Domain with Outer Peak

where

375

Q=81 \{x=(y,z)EUn8S2:0
and the remaining notation is the same as in Theorem 7.1.1. The removal of the first term from the right part of (1) does not affect the validity of (1).

Proof. The argument will be similar to that in Theorem 7.1.1. Let u E Wp (S2), ulesl = f. Since Iulp E W1(S2) and IIIul"Ilwi(c) <_ C IIUII' n ,

the estimate

J

n 8o

(z) I f (x) I pds< c II u lI Wp (a)

(2)

is a consequence of (7.1.1/5). Next, Theorem 4.1.1 implies IIfIIwD-1ID(Q) <_ cIIU11W11

).

(3)

Let F be defined in (7.1.1/1) (recall that by assumptions (7.1.1/1) holds). For brevity, we put 0(f ; x, S) = 11(x) - f (e) I pl x -

I2-n-p

(4)

and introduce the seminorms I f Ip,r, I (f) by

If ff

Iflp,r =

{x,fEr:l(-zl<M(z,()}

I(f)p =

A (f; x,

)dsxdst.

(6)

{x,(Er:o
Since If Ip,r "I(f), the estimate IfIp,r < cIIuilwP(cl) follows from I(f) < cIIVUIlLp(nnu).

(7)

For the proof of (7), we let {zk}k>o, {Fk}k>1, {Qk}k>1 and {Ilk}k>1 be the same as in Theorem 7.1.1 (see the proof of inequality (7.1.1/7)). Then I(f)p < > f dsx k>1

°k

f

rk

376

7. Boundary Values of Functions in Sobolev Spaces for Domains with Peaks

ff A(f;x,. )dsxdsC.

k>1 rkxrk

By Theorem 4.1.2, the general term of the last sum is not greater than c IlVuIILp(Ok). Hence this sum is dominated by c IIDuIIip(slnv) and thus (7) holds.

Estimates (2), (3), (7) imply that the norm on the right in (1) does not To prove the reverse inequality, one should extend a exceed C I I f II TWp (n) function f E Lp,1oc(8SZ \ {O}) with If Ip,r + If II Wp-1/P(o) < 00

into SZ in such a way that the extension (denoted by u) is subject to IlullWi(a)
(8)

First we examine a special case when 8 E (0, 1) is a fixed number indepen-

dent off and supp f C {x E U n 81 : z < 8}. In this case a linear operator f -* E6 f E Wp (S2) will be defined to satisfy the conditions

E6flasz-f, Ea1Is1\U=0 and

Il Eaf ll w, c0 < c If ip,r

(9)

with c = c (n, p, W, w, b). Construction of the operator E6. We again use the sequence {tk}k>o given by (7.1.1/9) where v = min {(1 - 8)/2, 1/6, (2IJ'P IIL_(o,1))-1}.

(10)

Putting Wk = co(tk), one can readily show that t2 > 6, tk+ltk 1

1,

Wk+11Pk 1

Also (7.1.1/10-11) hold. Let {µk}k>1 be a smooth partition of unity for (0, t1] subordinate to the covering {(tk+l,tk-1)}k>1. Next, let Ak E Co (tk+l,tk_1) and Ak{bk = µk for k > 1. Clearly we may assume 0 < Ak,,uk < 1,

lA k I

+ Iµkl

cWk 1

7.2. The Space TWp (1l), p > 1, for a Domain with Outer Peak

377

and

dist (suPP .k, R1 \ (tk+ 1, tk-1)) ? C (Pk, k > 1.

Furthermore, let Sk and Gk be the same as in Theorem 7.1.1 (cf. Fig. 44). For each k > 1 define fk on Sk by

fk(x) = )tk(x)(f(x) - 1k), where f k is the mean value of f on Sk. We claim that Wk1P-'IIfkIILp(Sk)+[fk]P,Sk

with

1,

(11)

given by (4.1.1/3). Indeed, IIfkIILp(Sk) < Ilf -fkIIL,(S,) 5C(pk-11P[f1P,Sk

by Lemma 4.1.1. Note that c [ fk]P sk < If ]p sk + J where

J= ff If

fklPlak(z) - Ak(C)IP IX

d slds£ n+P-2

Sk X Sk

Since I Ak

c Wk 1, we have

j < cW

I f (S) - fkV Pdst Sk

sk 2 dsln-2 I

The interior integral does not exceed C Wk- Combining this and Lemma 4.1.1

implies J < c[f]psk. Thus, (11) is valid. Inequalities (11) and [f ]p sk < I f Ip r (which follows from (7.1.1/11)) give

fk E Wp-11P(Sk). Let fk be extended by zero to 8Gk (see Fig. 44). It is readily seen that then fk E Wp-11P(8Gk) and that the following estimate holds Wk

pllfkll Lp(8Gk) + [fk]p 8Gk

/

< C I wk PIIfkIILa(Sk) + [fk]p,S,)

,

k > 1.

By Remark 4.1.1, for each k > 1 there is a linear extension operator ,c (k)

: Wp-1/P(8Gk) -- Wp (Rn)

(12)

378

7. Boundary Values of Functions in Sobolev Spaces for Domains with Peaks

satisfying

Wk IIIE(k)9IIL,(1) + IIvE(k)9IILa(R°) C(Wk1+1IPII9IIL,,(aGk)

<

+ [9]P,ack), 9 E WP-lIP(8Gk).

This in conjunction with (11) and (12) yields ok 1IIE(k)fkIILD(R^) + IIvE(k)fkIIL,(R^) <_ C VIP, sk'

(13)

We now define E6 f on 0 by (E6 f) (x) = 0 if x E 1 \ U and

(E6f)(x) = 1: fkftk(z) +E{lk(z)(c(k)fk)(x) k>1

(14)

k>1

if x = (y, z) E U n Q. It follows from this definition that (.66f) (x) = 0 for z > t1 because f (x) = 0 for z > t2 > 6. Hence and from (14) also follows the equality E6 f I8

= f,

Turning to the proof of (9), we let v(x) be the first sum in (14) and w(x) the second sum. The estimate IIoVIIL,(Unl) < cF, [f]P,sk

(15)

k>1

is provable exactly in the same way as (7.1.1/17) in Theorem 7.1.1, and the estimate (16) IIVwIIL,(unll) < cy, [f]P,ak k>1

is a consequence of (13). By (7.1.1/11), the last sum does not exceed

f k>1

dsy k

J{eEr:pc-zI<M(z,M A(f; x, )dsC

where 0(f ; x, ) is given by (4). Therefore

EVIPSk

CIfIP,r.

k>1

Inequalities (15)-(17) and Lemma 5.1.1 lead to (9).

(17)

7.2. The Space TWp (f1), p > 1, for a Domain with Outer Peak

379

Construction of the extension f H u in the general case. Suppose that f c LP,i.C(8SZ \ {O}) and that the right part of (8) is finite. Let ?p be a cut-off uniformly Lipschitz function on a such that 0 < t 1,V) = 1 in a neighborhood of 8SZ \ o and

supp' c {x = (y, z) E 8S2 n U, z < 3/4}. Then supp (1 - ?/') c a and it is easily seen that II(1 - Of II Wp-1/P(te) < C II{I II Wp-1/P(0)

(cf. (4.1.1/4)). By Theorem 4.1.1, there is a function u E Wp (S2) satisfying

uI8S2=(1-V)f,

IIuIIWP(S)
(18)

We claim that the desired extension u of f, subject to (8), can be given by u = u + E6(V)f) for d = 3/4. Clearly ul en = f, and (9) with (18) imply that c IIuIIW, (n)

II]

J

IIWn-1/P(Q) +

IV)flp,r.

Now the claim follows from the estimate If Ip,r < c (If Ip,r + If IILP(rny))

To check the last, we observe that I Ip,r -

I (h)f )P < IMP +

Jr If (x) I PJ(x)dsx,

(20)

where dsC

J(x) = I(Z) IVG(x) - v)(011 X - n+p-2 C I

I

and r(z)

(71, () E IF

: C E (z - cp(z), z)}. If x E IF \ o, then O(x)

IfxErnfu,then J(x) < c

Jr(z)

Ix - f I2-nds4,

and J(x) < c which is verified using the substitution W(()-171 = Y, cp(z)-1(z - () = t.

380

7. Boundary Values of Functions in Sobolev Spaces for Domains with Peaks

Hence the last term in (20) is dominated by c II f IIiP(rno) This established (19) and thus finishes the proof of relation (1). That (1) remains valid without the first term on the right follows from inequalities (2) and (8). 1 We now make several remarks concerning the theorem just established.

Remark 1. The proof of Theorem shows that relation (1) holds for any connected open a C % such that a is distant from 0 and o, D 892 \ U. The constants in (1) generally depend on Remark 2. If F is the surface defined in (7.1.1/1), f c LP,10C(F), f (x) = 0 for z > 5 and If Ip,r < oo, then (14) determines an extension operator from r into the circular peak G = {x = (y, z) E Rn : z E (0, 1), IyI < W (z)} (we recall that w C Bii 1)) In addition, IIE6f II w, (c) <_ c If IP,r,

where c = c(n, p, cp, w, 6). This is verified in the same way as (9).

Remark 3. Let IF and f be as in Remark 2. Then the right part of (14) is zero if z > d+ W(6). Indeed, it follows from the definition of {tk} (see (7.1.1/9) and (10)) that t2 > J. Hence 6 E [ti+1, ti) for some i > 2 and thus (E6 f)(y, z) = 0 for z > tz_1. Now the monotonicity of cp and (7.1.1/10) give 5 + p(6) > ti+1 + W(4+1) >_ tt_1.

Remark 4. Let D = {(y, z) E R° : z E (0,1), y/cp(z) E

B(si-1)

\ ail,

where cp and w are as in the theorem (and as in Definition 5.1.1). Let F be the surface defined in (7.1.1/1) and I Ip,r the seminorm given by (4)-(5). If u E WP (D) and ulr = f, then IfIp,r
7.3. Boundary Values of Functions in W1 ((I) for a Domain ...

381

Corollary. Under the hypotheses of Theorem, {the relation IlfIITL1(S2)' IfIP,r+[f]P,a

is valid, where I - IP,r is defined by (4)-(5) and The proof is similar to that of Corollary 7.1.1.

is given by (4.1.1/3).

7.3. Boundary Values of Functions in WP (1k) for a Domain fl c R" with Inner Peak, p E (1, n.-1) In this section we describe the space TWp (0), p E (1, n - 1), for a domain SZ C Rn with the vertex of an inner peak in the sense of Definition 5.5.1. Let W and w be the same as in Definition 5.5.1, w C Bin-1), 8w connected. Let r be the surface introduced in (7.1.1/1). The following Lemma presents an auxiliary extension operator from r into Rn, n > 3.

Lemma. Let p E (1, co) and 0 < b + Bp(d) < 1. Let f be a function in LP(I') such that f (y, z) = 0 for z > b and If IP,r < oo, where I - IP,r is the seminorm defined by (7.2/4-5). Then there is a linear mapping f F4 E f E Wp (R") and a positive constant c = c (n, p, cp, w, b) such that E f I r = f , supp(Ef) C {x = (y, z) E Rn : z E [0, 5 + w(8)], I yI < 8cp(z)}

(1)

and

cIIEfIIwi(R^) <- Iflp,r+ f

w(z)1-Plf(x)IPds,,.

(2)

Proof. Let {tk}k>o be given by (7.1.1/9), (7.2/10). Put Wk = cp(tk),

Sk = {x = (y, z) E r : z E (tk+1, tk-1)}, k > 1. Furthermore, let AI f k, µk, g(k) have the same sense as in (7.2/13-14). We introduce a sequence {7Pk}k>1 C Cl([0, oo)) such that i'k(t) = 1 for t E [0, Wk-1], 00) = 0 fort > 2Wk-1 and 1,0k 'l < c 1 . Put

Qk(x) = Ak(z),bk(IyI), x = (y, z) E R", k > 1. Then Qk(x) = µk(z) if x E I'. Hence F-k>1 ak(x) = 1 for x E r, z < t1. Note also that Wk-1 < 2cpk for all k > 1 because tk-1

(t)dt < V Wk-1IIW'IIL-(0,1)

Wk-1 - cok = it k

Wk-112

7. Boundary Values of Functions in Sobolev Spaces for Domains with Peaks

382

(cf. (7.1.1/9), (7.2/10)). Therefore supp ok C {x E Rn : z E (tk+l, tk-1), IyI < 8q'(z)}.

(3)

We claim that the required mapping can be given by

Ef =1: fkok+1: Ukc(k)fk k>1

(4)

k>1

The argument in Remark 7.2/3 shows that (E f) (x) = 0 for z > b + Bp(d) (in particular, (Ef)(x) = 0 for z > t1). Hence and from (3)-(4) follow inclusion (1) and the equality E f I r = f . Turning to the proof of (2), we denote by v and w the first and the second sum in (4), respectively. Every point in RI belongs to at most two sets in the collection {supp ok}k>1 so that IIVVIILp(R°) < CY:

I

I < C tOk 1, the general term of the last sum does not exceed c

f

I f (x) I

Sk

where dsx is the area element on F. Thus IIVVIILp(R") C C

fr

(x)Pcp(z)1-Pdsx.

If

Also

IIVwIIL,(R^) C C

(, I-C PIIe(k)fkl1Lp(R") + IIOe(k)fklILp(Rn))

k>1

A combination of this inequality with (7.2/13), (7.2/17) gives IIVWIILp(R^) < C If

Ip,r.

Now the last estimate, (5) and the Friedrichs inequality II Ef IILp(R^) < C Ilo(Ef)IIL9(R, )

(5)

7.3. Boundary Values of Functions in WP '(fl) for a Domain ...

383

imply (2). This concludes the proof.

1

Remark. Clearly F in the lemma can be replaced by the surface {(y, z) E

Rn:zE(0,1),jyI=cp(z)}.

I

We now state the principal result of this section. Suppose 52 C Rn (n > 3) is a bounded domain with an inner peak in the sense of Definition 5.5.1. In what follows we will also assume that the additional requirements mentioned at the beginning of Sec. 7.1.2 hold. Below c denotes various positive constants depending only on n, p, 52; a - b implies c-1 < ab-1 < c.

Theorem. Let 0 be the vertex of a peak directed into the domain SZ C Rn.

If 1