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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,
p1 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) = p1 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 2 and let (1) hold for u E VP-1(Q \ F) with
lal 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 -e, IyI >-e, IzI 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) 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 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 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)
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 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:
1 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
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-jn0,q>p, it follows that 1/q 114119 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, 0 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) withC1 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 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
0). Let i1 denote the resulting interval. Then
r
cJ lu'ydx
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 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 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 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 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 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, t0, 1 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
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-pdo1, {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 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 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,
(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)
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
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:00 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
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 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 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 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 1/3, u(x, y) = 0 for 0 < y < 1/6 and 0 < Q < 1. Clearly C Ilullq,s2 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 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 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) 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) 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 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 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 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(!flPdxO
r(x)
\ c; LP(Rn)) } > 0.
The inequality IIUIIP,Bi 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
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 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
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 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 .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 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. 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'
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 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
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 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°
1 - 1 provided the kernel of T satisfies (2) for
1 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'YI > 1 - 101, S7 (y, x) _ (Y
1
a)!
f i1
H(y, z) _ 171 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 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 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)
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 VP '(R
with norm subject to IIE211
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 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 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 1 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) 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 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
) 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 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
1
the former of the last two sums over k > 1 does not exceed
d-2
{,ES:JS-z11 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)
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) 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) 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) 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 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 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) 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) 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 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 letlp0 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 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 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
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' +
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
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
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 '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 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
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 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 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-
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+l 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) 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
dl+k-8d-l-k
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
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 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)
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
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
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
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 0 and observe that IIEtNIIq,I,R- < C IIUNIIoo,l,1l < C.
Inequality (2) implies N ek+1W(Pk)1-!q
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 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
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)
_ 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)0} by a sequence of open balls
B(") = {x E Rn : Ix - x I
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) 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
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 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 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 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
IVokI1 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{-zI1 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) 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 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) 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
2) be a bounded domain in C°". Suppose c1, cp2 E C°'1(G), cp1 < ,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+)
.
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:z0}, 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)
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) 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) 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) 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) 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) 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) 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)
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
0}.
13(1n-1)
x (-1 1)
Suppose u E WP (52), u180 = f . Let 1 z and V. denote the sections of r and V n 52 by the hyperplane z=const. By Theorem 4.1.3 z)IILp(r=)
- C II U(-, z)II'W"(v )
for a.e. z c (0, 1). Integration with respect to z E (0, 1) gives
f
r
V(z)1-Plf(x)IPdsx < CIIuIItP.vi(iy
(7)
Let I - Ip,r be the seminorm defined by (7.2/4-5). The estimate IfIP,r < d IIVuIILp(vn12)
(8)
384
7. Boundary Values of Functions in Sobolev Spaces for Domains with Peaks
follows from Remark 7.2/4. To establish the inequality dsx IdsP
ff
If
P
{x,{Er:1c-zI>M(z,C)}
Ix-
< cIItIIPwy(S 2
),
(9)
where x = (y, z), e = (i, (), M(z, () = max{cp(z), cp(()}, we note that the left part of (9) does not exceed c
fr
(()n-2d(
11(x) I Pdsx
fJ(E(0,1):1(-Z1>M(.'())
I(-
zln+P-2
The interior integral is not greater than I( - zI-Pd(
w(z)
and this in conjunction with (7) yields (9). Let J E (0, 1) be chosen so that the right part of (1) is contained in U. Put
Q=8S2\{xEUnOn:z Wp (Rn) (see Theorem 5.5.1), (1) follows from the same inequality for Sl = R'. In this case the required inequality is well known (cf. Exercise 1.1 (i) for s = 0). 1 In Lemmas 2 and 3 stated below, I' is the same surface as in Lemma 7.3 (cf. (7.1.1/1)), and c denotes various positive constants depending only on p, cp, w. In addition, we assume that W(z) < z for z E (0, 1].
Lemma 2. Let p = n - 1 > 2 and put M(z, () = max{cp(z), cp(()}, z, ( E (0,1),
CZ) _ (cp(z) log(z/cp(z)))1-p, z E (0, 1),
Q(t) = 1 + t2p-2 (log (l + t))-p, t > 0.
If f is a function in Lp(r), then the norm
If (x) I'
(z)dsx + ff If (x)
-f
I pQ
rxr
( M z, ) I (r
I
dsxdS rn+p-2
is equivalent to the norm (f )p,r defined by (f)P,r = J If (x) I P D(z)dsx
+
ff
{x,{Er:2-1 2z}
f
rn-2
dsx
n+P-2
1
c f2.
M,
for
where M = M(z, (). Now, because M(z, () N cp(() for (x, () E H, the left part of (2) is dominated by c ((f)p r + Il + I2) with
Il =J If(b)IpaP J ( {.Er:jc-zj<M(z,()} I(-Zip r py 2' r I2 -
f(f)IPds£ f
(P
Jrr
(I(-zllw(())p-'dz
(() [log(' + I( - ZIN(())lp
and f(C) = {z E (0, 1) : cp(() < I(- zI < (}. Let J(() denote the interior integral in I,. Since the integrand in J(() is not greater than r2-n and W(z) N cp(() if I( - zI < M(z, (), we have JW 1,
where Wk = c,(2-k). Clearly vkDk is a subdomain of the cylinder C(e) =
(Rn-1 \
x R'.
Since cp'(z) = O(z-1cp(z)), we have
ckl(Vxu)I - IDX((liku) ovk 1)I
for x E Dk. After the change of variable x -+ X relation (3) takes the form IIuIIWD(n)
0" k=1
(4)
Here ek = 2kcpk and the norm II' IWD(C(e),E,,) is defined by (4.2.1/1). It follows
from (4) that co
IIuIIwi(n) > CEWkII(µkf) 0 vk II'TWI, (C(-),c) k=1
(5)
7.4. Inner Peak, the Case p = n - 1
391
(see (4.2.1/2) for the definition of II ' IITw'(c(e),Ek)) An application of Theorem 4.6 gives n,
+
ff
11096kll-PII9kI
Ln(8C(`))
I9k(X)-9k(X')IPQ(IX -X'I) IX
dsX ''In+ sx,
-
P-V
8C(e) x8C(e)
where 9k = (µk f) o vk 1, dsX, dsx, are the area elements on 8C(e) and Q is the same function as in Lemma 7.4.1/2. Returning to the variables x = vk1(X), = v 1(X'), we find vk111TWp(C(e),Ek) - (ILkf)p,r'
(6)
A combination of (5), (6) and Lemma 7.4.1/3 yields (2). Thus IAfNI 0 be
chosen so that 8 < 1/2 and 8 (z) < z for z E (0, 8]. We introduce a set of functions {Ak}k>1 satisfying
Ak E Co (2-1-k, 21-k) , Akµk = Iik, I'k1 5 C2k.
Furthermore, let A E C°O([O,oo)), \(t) = 1 for t < 1/2, A(t) = 0 for t > 1. Put ')k(x) _ Ak(z)A(2k+1IyI), k = 1, 2, ... , x = (y, z) E R". By Theorem 4.6 and in view of (6), there is a function vk E WP (C(e)) such _1 that Vk (µk f) ° vk and IIvkIIW,(C(e),ek) 1
< C E (/kf )p,r < c (f )p,r . k>1
So u is the desired extension of f. The proof of Theorem is complete.
I
Corollary. Relation (1) remains true if the term If Ilip(an\v) is omitted. This assertion is provable in the same way as Corollary 7.3.
7.5. Application to the Dirichlet Problem for Second Order Elliptic Equations Here we give an obvious application of the above results to partial differential equations. Let 52 C Rn be a bounded domain with the vertex of a peak on 852. Let aid E Lm (52) for i, j = 1, 2, ... , n and suppose that n
E aii(x)U.i >_
i,j=1
for all 1; E Rn,
7.5. Application to the Dirichlet Problem ...
393
where c = const > 0. Consider the Dirichlet problem
a
n
Cain (x)
axi
i
au
= 0,
ax'
x E 52,
(1)
1
(2)
Ulan = f.
A function u E W1(S2) is called the solution of the problem (1)-(2) if f is the trace of u on 911 and
,
n
au av E aii (x) axe ax, dx = 0
for all
v c C(1).
+J=1
Theorems 7.2, 7.3 and 7.4.2 for p = 2 imply explicit necessary and sufficient conditions for the unique solvability of this problem in the energy space W2 (Q). If one of the following conditions is fulfilled: (i) S2 has an outer peak and 11f112
I f (x) - f
/ (') + {x,CEUnan:IC-zI<M(z,S)}
I2
Ix - £In
dsxds < oo,
(3)
£
where u is an arbitrary connected open subset of t31 at a positive distance from 0 containing 8SZ \ U;
(ii) n > 3, Q has an inner peak and
L
I f (x) - f
f(X)2 dsx W(Z) +
oa
xa
Ix - In
(x)12
dsxdst < oo;
(4)
act x act
(iii) S2 C R3 has an inner peak, the assumptions of Theorem 7.4.2 hold and Ix - Cl
JJ
{x,tcunast:lx-fl>M(z,c)}
+
f
f (X)2
Unact cp(z) log(z/co(z))
{M(z, () log \1 + M z S l) ((
I f (x) - f
dsx + ac1xasl
I2
dsxds£ < oo,
Ix - CI3
then the problem (1)-(2) is uniquely solvable in the energy space.
2
(5)
7. Boundary Values of Functions in Sobolev Spaces for Domains with Peaks
394
Conversely, if 1 is a domain with an outer peak and the problem (1)-(2) is solvable in the energy space, then (3) is valid. Similarly, if Q has an inner peak and the Dirichlet problem is solvable in the energy space, then (4) is valid for n > 3 while (5) is valid for n = 3. In the last case some additional restrictions on the peak stated in Theorem 7.4.2 are imposed.
7.6. Inequalities for Functions Defined on a Surface with Cusp The main goal of the remaining part of Chapter 7 is to describe the space TWp (Q) for a domain Q C R" with inner peak when p > n -1. The principal theorem is stated in the following section while the present one contains several auxiliary assertions. Let cp and w be the same as in Definition 5.5.1. In addition, we assume that Bin-1) and Ow connected. As above, I' designates W(z) < z for z c (0, 1], w C
the surface defined in (7.1.1/1) for n > 3. Furthermore, we consider the circular surface
S={x=(y,z)ERn:zE(0,1), jyj =cp(z)}, n>3,
(1)
and the circular peak (cf. Fig. 31 in Sec. 5.4.1)
G={x=(y,z)ERn:zE(0,1), jyj n - 1 > 2 and suppose v E Wp (G). Then c {v}p,r < {v}p,s + IIovIIL,(c).
(4)
This inequality remains true if F and S replace one another.
Proof. Put B = B(in-l). For almost all z, ( E (0, 1), the function B i) y'-+ u(y) = v(cc(z)y, z) - v(co(()y,
is in Wp (B). Furthermore w(z)p+l-nllov(.,z)IILp(Bp(=))
cIIVUIILP(B) n - 1 > 2. If f (y, z) = 0 for z > 1/2, then
ff
If(x) - f (S) I
dsxds( p(w(z)cP(())2-n
IC - zlp+2-n
{x,(Er:l(-zI>M(z,()}
+If Ip,r ,,, {f}P,r + If Ip,r,
(6)
where x = (y, z), (17, (), dsx, dsf are the area elements on r and I ' Ip,r the seminorm given by (7.2/4-5).
396
7. Boundary Values of Functions in Sobolev Spaces for Domains with Peaks
Proof. Let D = {x : z E (0, 1), y/cp(z) E w}. If If IP,r < oo, then there is an extension u of f from surface r into D such that IIuIIW;(D)
CIflp,r
(see Remark 7.2/2). Put y = W(z)Y, 77 _ W(C)Y', Y, Y' E -y. Since
If (x) - f (() 15 If M()Y', () - f (w(z)Y', z)
+If('p(z)Y',z) -f(w(z)Y,z)I, the integral on the left in (6) is dominated by c If }P r + c I, where
I= fdz f f o
f (x) - f (x') I d7Yd7Y'
ryxry
LZI>(Z)
I(- P+2
(7)
x = (cp(z)Y, z), x' = (W(z)Y', z). The last interior integral with respect to ( is not greater than ccp(x)n-'-P and thus
I 1 is an integer, then there exist an increasing function f E C'((0,1]) and positive constants c1, c2 depending only on cp, l such that W(z) < f (z) < cl W(z) and
P) (z) 1 < c2 z-k(p(z), k =
(1)
for any z E (0, 1]. In particular, if W(z) = o(z) as z -+ 0, then there is a positive constant c3 = c3(l, cp) such that If(k)(z)I 2, is a bounded domain in C°'1. Put 92 = {x = (y, z) E Rn : z E (0,1), y/cp(z) E w}.
(1)
Theorem stated below gives precise conditions on cp and on p, q, 1, n for the space VP (S2) to be continuously imbedded in Lq(I ), 1 < p < q < oo. In what follows c, c1, ... denote positive constants depending only on n, p, 1, w, W. By definition a - b if cl < a/b < c2.
Theorem. Let SZ be given by (1). Assume that 1 < p < q < oo and that l > 1 is an integer. Suppose cp satisfies the additional condition co(2z) ' ep(z), z E (0, 1/2], in case l > 1. When 1 < p < q < oo, the space VP (S2) is continuously imbedded into Lq(SZ) if and only if the expression
CO = s(p
rE 0,1)
(Ir
ldzdz) I 1/q
cp(z)n
(
1
W(z('-1)p/(p-1)
z)(n-1)/(p-1)
is finite. The boundedness of the imbedding operator:
V1(1) -+ Lq(I), 1 < q < oo,
1-1/p
(2)
8. Imbedding and Trace Theorems for Domains with Outer Peaks
418
...
is equivalent to 1/q
Jr
p(z)i-'dz
sup r1-1p(r)1 rE(0,1)
< 00.
In case p > 1 the continuous imbedding operator: VP (0) --+ C(f) fl L. (1) exists if and only if 1
L
z(,-1)pl(p-l)
W(z)(1-n)/(p-1)dz
< oo.
(4)
Finally, the boundedness of the imbedding operator: Vi (1) -4C(52) fl L, is equivalent to sup {r1-1W(r)1-n : r c (0, 1)} < 00.
(S2)
(5)
Proof. Necessity of inequalities (3)-(5) and Co < oo. Suppose VP (1) is continuously imbedded into Lq(I ), q < oc. That is, for any u E VP (S2) IIUIIq,1 < C IIuIIp,i,ui, C = const > 0,
(6)
(as before, Ilpi designates the norm in Vp(52) and II ' IIp,o the norm in Lp(S2)). First we consider the case l = 1. Put u(x) =
f
l g(t)dt, x = (y, z) E Q.
z
Then (6) takes the form q
1
w(z)n-ll f 9(t)dtI dz) Z
1/q
1/p
1
< cc
(fo
cp(z)n-1I9(z)Ipdz)
,
w here g is an arbitrary function for which the integral on the right is finite.
Now the inequality C > cC0 for 1 < p < q < oo and inequalities (3) and (4) for p = 1 and q = oo, respectively, follow by reference to Lemma 5.1.2/1. Let l > 1 and p E (1, oo). Note that quantity (2) will be replaced by the equivalent one if suprE(o,1) is replaced by suprE(0,1/2). Define u on S2 by
u(x) = f(t - z)'19r(t)dt, x = (y, z) E 1
,
(7)
8.2. Continuity of the Imbedding Operator: Vy(n) -+ Lq(f) ...
where
419
(0 for t E (0,r], (l
'
t('-' )l(P
(t)(' -n)/1P-11 for t c (r, 1)
and r c (0, 1/2) is a fixed number. Then u E Vp(SZ) and I(Viu)(x)I for x E Q. According to Lemma 5.1.1, we have
g, (z)
IItIIP,1,n < c I1 V1uIIP,n,
therefore p1
z(1-1)P/(P-1)w(z)(1-n)/(P-1)dz.
Ilullp i , sz < C J
(8)
r
On the other hand,
(fr Ilul1q'n > c
W(z)f-1dz
(f'(t - z)1-19rtdt I q )1/q
9)
Since t - z > t/3 for t > 3r/2 and z < r, the integral over (r, 1) is not less than C
f
t1gr(t)dt. r/2
Because W(2t) - '(t), the last integral is equivalent to the integral over (r, 1) with the same integrand. Hence and from (9) follows the estimate
(Ir Il1Ilq,n > C
1/q
o(z)n-ldz)
f 1 t1-1gr(t)dt
(10)
r
which is also true for q = oo. Combining (6), (8), (10), we obtain
C> c f
r
1 v( z)'-dx
1/q
f
1
1-1/P x(1-1)Pl(P-1) W(z)(r-1)/(P-1) dz I
Therefore C > c Co if q < oo and (4) holds if q = no.
Let 1 > 1 and p = 1. Suppose (6) holds for all u E V1(SZ). Let 1i be a function in C°°([0,oo)) satisfying 1/)1(0,1) = 1, V) 1 (200) = 0. For r E (0,1/2) define u on SZ by u(x) = 0 (z/r), x = (y, z). Then U E V11 (Q) and IIUI11,1,o 5 c II Viu111,i
Qallq,n < c 11VIullp,n
(14)
IaI
which, clearly, imply (6).
Proof of inequality (13). By Lemma 8.1, the derivatives u(s) E L10(0, 1) exist for s = 0, ... ,1. Since ua (z) = 0 in the vicinity of z = 1, the following representation holds (-1)1IaI
ua(z) _ (l - IaI - 1)! f 1(t z
-
z)l-IaI-lu(a'-I0I)(t)dt
(15)
8.2. Continuity of the Imbedding Operator: V(S1) -+ Lq(1l) ...
421
for jc < I and almost all z c (0, 1). Suppose q < oo. From (15) and the estimate 1ua(z)1W(z)1a1
IQa(x)I
(16)
we obtain
c c IIQaIIq,S2
( r1
r1
fo
(z)glal+n-ldz J:
< c1 J
1
u(i-lal) (t) I dt l l
J
t1-1lu('-IQI)(t)Idtl 9
co(z)n-ldz (J 1
o
q
tl-1-lal
f
Z
By Lemma 5.1.2/1, the right part of the last inequality does not exceed q1P
c coq (fe' w(z)n1 u(aI0I) (z) dz)
which is not greater than c Co IIDiu11p 0 in view of Lemma 8.1. Hence (13) follows for q < oo. In case q = oo the proof of (13) is easier. We find from (15) and (16) that
II`v alloo,f2 < c
f
1
z'-1lu('-IaD(z)Idz.
0
By Holder's inequality 1/p cQ(z)n-' Iu(al-14D (z) IPdz I
IIQa II.,n < c Co (foI
and Lemma 8.1 gives (13) with q = oo. Proof of inequality (14). Let Q=
Qa. Ialo is defined by zo = 1, zk+1 + W(zk+1) = Zk, k > 0.
,
8. Imbedding and Trace Theorems for Domains with Outer Peaks ...
422
It is readily seen that
Zk \ 0, zk+lzk 1 -+ 1, w(zk+1)co(zk)-l -* 1. Let S2k = {x : z E (zk+l, zk), I yl < W(z)J, k > 0.
We claim that VP(Qk) is continuously imbedded into Lq(I k) for each k > 0. Indeed, one has W(r + O(w(r))) _ W(r) + o(W(r)) as r -+ 0
because lim,.,0W'(r) = 0. Therefore
)l/q(f r+G(r) Co >
L_(r) '' * )
...
\ 1-1/p
for any small r > 0 with the same integrands as in (2). Hence l-n/p+n/q > 0. Moreover, lp > n if q = oo, which follows from (4) and (5). Thus the inequality ai for some i = 1, ... , n - 1, then
D'rQa(x) = 0. Suppose ry < a (i.e. ryi < ai for each i = 1, ... , n - 1). In this case
I DIQ.(x)I < c w(z)IaI-171Iu(7")(z)I, x = (y, z) E Q. An application of Lemma 8.1 gives IDIQa(x)Ip
0.
Combining the last and (18) implies IIit - QIIq,n,, < C (p(zk)'-n/p+n/gH VzuIIp,12k, k > 0.
(19)
Now estimate (14) follows from (19) and the inequality
M k
ak)1/9
0, q > p.
/f
Thus, the continuity of the imbedding operator: VP (0) - Lq(S2) is established if 1 < p < q < oo and Co < oo. In particular, if q = oo (i.e. (4) holds for p > 1 and (5) holds for p = 1), we have lp > n so that every function in VP (Sl) coincides with a function in C(S2) almost everywhere on Q. In this case Vp (0) is continuously imbedded into C(1) fl Lo.(SZ). The proof is complete. I
The following assertion refines Theorem if either one of the conditions (4) or (5) holds.
Corollary. Let Q be given by (1), where w and cp are the same as in Theorem. If (4) holds for p > 1 or (5) holds for p = 1, then the space VP (S2) is continuously imbedded into C(1l).
Proof. The continuity of a function v E VP (SZ) up to the boundary 3 needs only be verified in the vicinity of the origin. In view of Lemma 5.4.1/1 and Theorem 1.4.2/1, v can be approximated in Vp (1) by functions in C°° (Q), p < oo. By Theorem just proved, v is the limit in C(Q) fl L ,(SZ) of a sequence in COO (Q). Hence v E C(SZ).
8. Imbedding and Trace Theorems for Domains with Outer Peaks
424
...
Note that 0 E EV, (cf. Remark 5.4.2), therefore every function u E VC1. (S2) also belongs to C(S2).
Lemma stated below will be used in Sec. 8.3.1.
Lemma. Let 92 be defined by (11), where cp is an increasing function in C°'1([0,1]) satisfying cp(0) = limZ,o cp'(z) = 0. Let J E (0,1) and suppose u E VP(SZ), u(y, z) = 0 for z > J. Let, furthermore, additional conditions (12) are fulfilled for l > 1. If 1 < p < q < oo, the following estimate holds IIUIIq,S2 < C Ci(S)IIVIuIIp,1
(20)
C(b) = sup {M(r, b) : r E (0, b)} < oo
(21)
provided
with
U\ r
M(r, b) =
1-1/p
1/q / / b ,z(,-1)P/(P-1)
cp(z)"-Idz
I
J
Jr
dz
(22)
P(z)(n-1)l(p-1)
(in case p = 1 the last factor is replaced by r1-1cp(r)1-")
Proof. Repeating the argument of above Theorem, we observe that representation (15) can be written for z < b with integration over (z, b). Since 0 for z > b, further reasoning leads to the estimate IIQaIIq,n < cC(b)IIVIuIIP,n
(23)
instead of (13). To bound the left side in (14), one should define the sequence {zk}k>o by zo = b, zk+1 + (Q(zk+l) = Zk, k > 0.
We have
C(b) > M(5 - W(46) > w(b)i-"/P+n/q
(24)
for sufficiently small b. Hence l - n/p + n/q > 0. Moreover, lp > n if q = oo, which follows from (21)-(22). Therefore, (18) and (19) hold true. Because cp(zk) < p(b), (19) gives 11U
- QIIq,c 1.
The following assertion can be established by the argument analogous to that in Theorem 8.2.2.
Theorem. Under the hypotheses of Theorem 8.2.2, assume also that 1 < p
1. Then there is a sequence {rk}k>1 of positive numbers such that rk -f 0 and 1/q
rk 11,(rk)1-n
rk
(f 0
V(z)n-ldz)
> c, k = 1, 2, ...
(4)
8.3. Compactness Theorem
427
Consider a function z/, E C°° ([0, oo)), fi(t) = 1 fort < 1, ?P (t) = 0 for t > 2. Let V)k(z) = ti(z/rk) and put 'Y)k(z), x = (y, z) E SZ.
Uk(x) = rk
The sequence {uk} is bounded in V1(1) due to cp(rk) - cp(2rk). Furthermore, Uk(x) -4 0 for any fixed x E 0, but I IukIIq,i > c in view of (4). Thus, there is no subsequence of {uk} convergent in Lq(Q), and the space Vl (SZ) cannot be compactly imbedded into Lq(SZ). The necessity of (3) for the compactness of the imbedding V11 (Q) c C(SZ) n Lm (SZ) is verified in the same way.
In case 1 = 1 the argument should be modified to avoid the use of W(2z) c'(z). If (2) fails for l = 1, there are sequences {rk}, {6k} of positive numbers such that lim rk = lim ek = 0 and 1/q
rk 6k))1-n
(co(rk +
cp(z)n-ldz)
(fo
> c.
Let ' 'k be continuous piecewise linear on (0,1), Ok (z) = 1 for z < rk, ?k (z) = 0 for z > rk + 6k, V)k is linear on [rk, rk + 6k]. Then one can put
uk(x) = (W(rk +Ek))1-ntlk(z), X E 1 ,
to obtain the same result as above.
Let 1 < p < q < oo and let the space VP '(Q) be compactly imbedded into Lq(SZ). If (1) fails, there is a sequence {rk}k>1i rk -4 +0, such that M(rk,1) > c where M(r, 8) is given by (8.2.2/22). For each k > 1 define ek E (2rk, 2) by the equation ek/2
fh
h(z)dz =
1
JPA./2
h(z)dz,
with h(z) = z('-1)P cp(z)(1-")P'/P, p' = p/(p - 1). We observe that ek -+ 0. If this were not the case, the integrals 1
h(z)dz
f
ek;/s
would be uniformly bounded for some subsequence {Ok;} so that
M(rk;,1) =
J
/rrki 0
1/9
cp(z)n-'dz)
1/p'
fk1;
\PJ (2
h(z)dz/ I
/2
-+ 0
428
8. Imbedding and Trace Theorems for Domains with Outer Peaks ...
which contradicts the inequality M(rk, 1) > c. Now the following conditions hold:
Pk > 2rk, Qk -* 0 and M(rk, Ok) > c for k > 1. Let SZk = {x = (y, z) E SZ : z < 0k} and put
for xEI \SZk,
0
rek
Uk(X) =
(t - z)'-lgk(t)dt for x E SZk,
J
where 0
for
gk (t) =
t E (0,rk],
t(,-1)/(P-1),p(t)(1-n)/(n-1)
1
for t E (rk, 1).
If Vk = uk/IIUkIIp,z,c,
then there is a subsequence of {vk} (which we relabel as {vk}) convergent in Lq(SZ). Since vk(x) -+ 0 for x E S2, vk - 0 in Lq(SZ). It follows from Lemma 5.1.1 and the definition of uk that 1/p
69 A;
h (z)dz)
IIUkIIP,1,Q - IIoIUIIP,Q (Ik
with the same function h as in (5). Furthermore,
(f rk IIukIIq,I > c
(J
ek
cp(z)n-1dz
0
(1k 11q
rk
> c1 (f O
p(z)n-1dz
- z)l-1gk(t)dt)
(t
1/q
q
)
ek
Irk
h(t)dt.
The last inequality is obvious for l = 1 and is obtained in the same way as (8.2.2/10) was derived from (8.2.2/9) for l > 1. Therefore IIVkliq,Q 2 c M(rk, 0k)
which contradicts the convergence I I vk I l q,n -1 0.
8.3. Compactness Theorem
429
We have established the necessity of conditions (1)-(3) for the compactness
of the imbedding operators mentioned in the statement of the theorem. We now prove the sufficiency of (1)-(3) and (8.2.2/4) for the compactness of the corresponding imbeddings.
According to Lemma 5.4.1/1 and Lemma 8.2.1, we may assume that Q has the form (8.2.2/11) with cp subject to additional properties (8.2.2/12) for
1>1. Let q be a function in C°° ([1, oo)) such that
0 0, one may expect that the norm of the imbedding operator: VP(S2(E)) # Lq(Q(E)), lp
/
yi /
0
Yn-1
y1
/
/
0
< n, q = np/(n - lp),
E
Yn-1
Yn-1
yi
/
Fig. 41. Two examples of perturbation of a peak
grows as a -> +0 (because the imbedding Vp (Q) C Lq(I) fails). Here we illustrate this effect using the model power peak S2 given by (8.2.2/11) with cp(z) = Z A, A > 1. Two examples of perturbation of SZ are examined. The first
8. Imbedding and Trace Theorems for Domains with Outer Peaks ...
432
example is the truncated peak Q(E) of the form (8.3.1/7). The second example
is given by 1 U BE (see Fig. 41). We obtain sharp two-sided e-dependent estimates for the norms of the Sobolev imbedding operators in both cases. The results turn out to be generally different. In what follows E E (0, 1/2). Positive constants c, c1, ... appearing below depend only on n, p, q,1, A. By definition a - b if c < a/b < c1. 8.4.1. Truncated Peaks Let Q(E) be given by (8.3.1/7) with w(z) = zA, A > 1. The following assertion presents sharp estimates for the norm of the imbedding operator: VP (1l(E)) -
Lq(0)). Proposition. Let 1 < p < q < oo and 1 > 1 an integer. If 1 - n/p + n/q > 0 and the last inequality is proper for q = oo and p E (1, oo), then the norm of the imbedding operator: Vp(c (E)) -+ Lq(Q(E)) is equivalent to Eµ
CE =
if µ 0 a small parameter. Proof. It is sufficient to assume e E (0, 1/4). Since Q(E) E C°'1, the continuous imbedding VP(1 )) C Lq(Q(E)) holds by Sobolev's theorem. We observe that
CE
rE(e,1/2) sup
r
(
1/q
W(z)n-1dz) e
1 z(
1
1)Pl(P-1)dz
Y
1
1/P
P(z)(n-1)/(P-1)
To obtain a lower bound for the norm of the imbedding operator, suppose
that IUIlq,O(e) < C Il uIl p,l,Si(e),
q < oo,
(2)
for all u E VP(S2(E)). Let r E (E,1/2) be fixed. We insert the trial function (8.2.2/7) into (2). The same argument as in Theorem 8.2.2 (see the proof of (8.2.2/8-10)) leads to the inequality (fr
C>c
\
o(z)n 1dz)
l1/q
1
Ur
1-1/P
Z(,-1)P/(P-1)
P(z)(n-1)/(P-1)
dz)
,
p>
1,
8.4. Imbedding Theorems for Perturbed Peaks
433
whence C > cCE. In case P = 1 we put
u(x) = ')(z/r), x = (y, z) E Q(E), where
0EC-([O,oo)), Vi(t)=1fort 2. Then (2) yields
(fr
(z)n-1dz
11/q
/
< (i
r E (e, 1/2),
and again C > cCE.
We turn to an upper bound for the norm of the imbedding operator: Vp(1(E))
.+ Lq(f2'E)). Let u c VPI(S2(E)) with u(y,z) = 0 for z > 1/2. The
following inequality should be checked IIuIIq,S2(E) < c Ce IViuI ,w).
(3)
The proof of (3) is quite similar to that of estimate (8.2.2/6) in Theorem 8.2.2. Here we point out only the distinction in the argument. Representation (8.2.2/15) is valid for z E (e, 1). Accordingly, reference to estimate (8.1/3) should be made for f (z) = za and z E (e, 1) in the proof of (8.2.2/13) to obtain IQaUIq,nco < c Ce Vzullp,c ),
lc
I < 1.
(4)
The sequence {zk} in (8.2.2/18) is constructed to be finite. Namely, zo = 1, zk+1 +W(zk+1) = zk for k > 0, Zk > e.
If ZN+1 < e < ZN, then we set ZN = e. Thus, one may assume that 0 < k < N - 1 in (8.2.2/18-19) to validate the inequality Ilu -
QaHIq,ci
0. Since IIUIIP,B1
IaI«
IID°PoHIp,B6
Ial IlokuI p,B6 + (61-n/p + K(n, p, b)) II VtuIlp,Bi k
Inequality (2) (with e = 1) follows because 61-n/P < c K(n, p, 6).
0
Remark. Estimate (1) is sharp in the following sense: if positive C1 (e, b) and C2 (e, 6) satisfy II'UIIP,B< C
and
C2 (E, 6) > c etK(n, p, 6/e).
This can be verified with the aid of the trial functions u(x) = xi 1 and xl-171(IxI/(3b)) if p < n,
a(x) =
77( - log(IxI/b)/log(b/e)) if p = n, 77(3IxI/(2e)) if p > n, x1
is a nondecreasing function in Cm ([0, oo)), rl(t) = 0 for t < 1/3, 1 ,q(t) = 1 fort > 2/3.
where 77
The assertion stated below is a direct consequence of the above lemma.
8.4. Imbedding Theorems for Perturbed Peaks
Corollary. Let (a, b) C R1 and let G. =
437 Ben-1)
x (a, b). If b E (0, a/2), then
for any u E VP (GE ) 1-1
c(n,p, l)II uIIP,G. < (66-1) (1-n)/P /
E EkIIVkuIIP,ca k=O
+et K (n - 11P, 8E-1) I I V z u I I P,G. ,
where
is defined by (3).
Proof of Proposition. To obtain a lower bound for the norm of the imbedding operator: Vp (cl()) -+ Lq(Il )), suppose that IUIIq,f(E)
c (Ev/P-n/q + e')
1
(9)
Nf > cCE.
(10)
Let g E Cm(Rl), g(z) = 1 if z < 1, g(z) = 0 if z > 2, and let u(x) = g(z/E) for x = (y, z) E 0(E). By inserting u into (8), we arrive at C1En/q < IIUIIq,Be ce"if /t 0 : u*(t) > s}. To check (4), we first note that mess (M9) = sup{t > 0: u*(t) > s}
(5)
by the monotonicity of u'. Hence, (2) yields
mesl(M.,) < µ(M,). For the reverse inequality, let e > 0 and t = mess (Me) + e. Then (5) implies u` (t) < s and therefore µ(M3) p
It was shown in Sec. 2.13 that if 1 = 1, then integral inequality (2.13/2) is equivalent to the isoperimetric inequality (2.13/3) for an arbitrary domain IZ C R". Here we give a generalization of this fact. We recall that the capacity cap (F; LP' (0)) is defined by (2.13/1).
Theorem. Let 1 < p < q < oo and let p be a nonnegative Bore] measure on a domain Q.
(i) Suppose there is a constant D > 0 such that ,a(F)PI" < D cap (F; LP' (Q))
(1)
for all relatively closed subsets F C SZ. Then the inequality IIUIILq(n,µ) 0 independent of u. Then (1) is true with D < CP. Proof. (i) By using Lemma 8.5.1/1 and Lemma 8.5.1/3, we obtain P9
P/q C
in I u(x) I gdul
= (1000
p(Nt)d(t4)l
5 fM ,(Nt)plgd(tP), where Nt = {x E IZ
:
Iu(x)I > t}. It follows from (1) that the last integral
does not exceed
D
J0
cap (Nt; Lp(1))d(tP)
which is not greater than c(p) DIIuftLp(fl) by Theorem 2.13.
8. Imbedding and Trace Theorems for Domains with Outer Peaks ...
444
(ii) Let F be a subset of S2 closed in S2 and let u E Coo (9) be any function satisfying u IF > 1. Inequality (2) yields Ft(F)'11 < C 11u11Ln(0)
Therefore (1) holds with D < CP.
1
Remark 1. The validity of inequality (2) implies µ(S2) < co.
Remark 2. Let inequality (2) hold for all u E Ct(S2) f1 LP(S2), where q E (0, oo), p E [1, oo). Then, by Theorem 1.4.1/1, the map u - ulsuPPA
(3)
can be uniquely extended to a linear continuous operator: LP' (Q) -4 Lq(S2,µ) This operator is called the trace operator. By Remark 1.4.1/2, the restriction of this trace operator to the space C(12) nLp(S2) acts as in (3). In particular, if dp = adx, where o, is a Lebesgue-measurable function, positive almost everywhere on 0, then LP (12) C Lq(S2, µ), and the trace operator is the continuous one-to-one identity map: Lp(12) -> Lq(12, p), i.e. it is the imbedding operator of the space LP(S2) into Lq(S2, µ). I
Combining Lemma 1.12.1 with above Theorem, we arrive at the following assertion.
Corollary. Let S2 C R" be a domain with finite volume and let p c [1,00), q E [p, oo). The Poincare type inequality
tint
ll u - tll q Q < C lloullp,c ,
u E Lp(S2),
C = const,
is equivalent to the inequality [mes"(F)]PI9 < D cap (F; Lp(S2)),
D = const,
where F is any relatively closed subset of Q.
8.5.3. The Case q < p
Let S2 be a domain in R", p E [1, oo) and p a Borel measure on Q. We introduce an isoperimetric function (0, µ(S2)] E) t H vµ,p(t) by vµ,p(t) = inf { cap (F; LP (S2)) : F C S2 , F closed in S2, µ(F) > t} .
8.5. Capacitary Criteria for the Continuity of the 'Dace Operator ...
445
Clearly vµ P is nondecreasing and bounded. It is an easy matter to check that condition (8.5.2/1) can be rewritten in the form tP/4
sup
< 00
tE(O,1+(Q)) vµ,P(t)
and that the last inequality is necessary for the continuity of the trace operator: LP(Q) -4 Lq(Q, µ) whenever q E (0, oo) and p E [1, oo). The following theorem gives necessary and sufficient conditions for the continuity of the trace operator
just mentioned in case q < p. For simplicity, we below abbreviate vµ P(t) _ v(t).
Theorem. Let 0 be a domain in R' and µ a Bore] measure on Q. Suppose that p E [1, oo), 0 < q < p.
(i) If
(tP/) p q q dt
v(t)t 9
D
fIA(Q)
< oo,
(1)
then IIUIILg(ll,µ) 0 such that (2) holds for all u E Coo (Q) n LP' (Q), then (1) is valid with C > c(p, q)D1/9-1/P
In the proof of the theorem we need a simple technical observation.
Lemma. Let {v1, ... , vN} be a finite collection in the space C(Q) n L' (SZ), p E [1, oo). Then the function S2 D x H v(x) = max{v1(x),...,11N(x)}
belongs to the same space and N
IIVviIILp(0)
IIVvIILp(sz) ` i=1
Proof. An induction argument reduces consideration to the case N = 2. Here v = (vi + V2 + Ivl - v2I)/2,
8. Imbedding and Trace Theorems for Domains with Outer Peaks
446
...
and hence v is absolutely continuous on almost all straight lines in S2 parallel to coordinate axes (cf. Theorem 1.2.4). Furthermore
Vv = 2 (Vvl + Ov2 + sign(vl - v2)(Ovl - Ov2)) almost everywhere in 0. Therefore IVv(x)I < max{IVvl(x)I, IVv2(x)I}
for almost all x E Q. The last inequality gives IVv(x)I' < IVv1(x)IP+ IVv2(x)IP
thus concluding the proof.
1
Proof of Theorem. (i) Note that (1) implies µ(S2) < oo and that v is a positive function. Let u E C°°(1) nLP(S2) and let Nt be as in Lemma 8.5.1/1. By this lemma and by the monotonicity of p(Nt), one obtains
f
00
I uI qdµ = >2 o i=-00
f
2i+1
µ(Nt)d(t9)
2
00
< E pj (2q(j+1) -
297)
j=-o
where pj = µ(N2; ). We claim that the estimate m
pj(2q(j+l) - 297) < cDl-9/PIIUIILp(0)
(3)
j=r is true for any integers r, m, r < m. Once (3) has been proved, (2) follows by letting m -+ oo and r -* -oo in (3). Clearly, the sum on the left in (3) is not greater than m
,,m2q(+n+l) + r
(lLj-1
(4)
-,uj)2j9.
j=1+r
Let Sr,m denote the sum over 1 + r < j < m. Holder's inequality yields m
-1
m
r
Sr,'. < I > 2P7v(tj-1) l j=1+r
J
(1j-1 - lLj)p°v(µj-1)9 } j-1+r
v (5)
8.5. Capacitary Criteria for the Continuity of the pace Operator ...
447
We have µp/(P-q)
-,L7)PI(P-q)
1. Hence, by the monotonicity of v, the sum in curly braces is dominated by m
E
lµi
1
v(t) q n d(t n
)
9=1+r µi which does not exceed p D/(p - q). Next, since v(i3) < cap (N2., ; LP '(Q)), the sum in square brackets in (5) is not greater than 00
2P E 2Pi cap (N2,; LP(H)).
j_-c In view of Theorem 2.13, the last sum is majorized by c IInIIL, (-). Thus m
(ttj-1 - 1 )2q3
/jj for x E Fj, r < j < s, and µ(Fj) > 2j, the inequality
Since
p({x E S2 : Ifr,s(x)I > -r}) < 2i
implies T > /ij. Hence for t E (0, 23), r < j < s,
fr 8(t)
where f,'.,s is the nonincreasing rearrangement of fr,, (cf. (8.5.1/1)). Then
f
µ(p)
(fr,s(t))gdt>
2 J=r f2j-1
( fT8(t))gdt> Qp2i0 j-r
and it follows from Lemma 8.5.1/2 that s
1.
(7)
j=r
Next, we note that if (2) is valid for all u E Cm (S2) n L1(S2), then (2) is valid for all u E C(S2) n LP' (Q). This is an easy consequence of Remark 1.4.1/2. In particular Ilfr,sIIL4(Q,µ) C C Ilfr,sIIL,(s2)
8.6. Compactness of the Trace Operator: Lp(1) -> Lq(S2, µ)
449
Now (6), (7) and the last inequality give l1/q
y
C>C
C
S
J-r
(Es=, RPL(27)) 1/P
1/q-1/P
2PJI(P-q) (v(2i))q/(P-q)
)
By letting r -+ -oo and by the monotonicity of v, we obtain s
C>C
9 23) (E , (i;)) D
>cI
1/q-1/p
(t)P_Q)
0
91/q-1/p
This completes the proof of the theorem.
1
Statement (i) of Theorem implies the following assertion.
Corollary. Let the assumptions of statement (i) of the theorem be fulfilled. If A is a Bore] subset of SZ, then µ(A) /I
A I u qdµ 0 and II(K(Eil - K(Ei+')) * fiIIL,(µ) ? co/2, i = 1,2,... . Given m = 1 , 2, ... , define
gm(x) = m-1/P lmamfi(x) Then IIsmIIP _ co min {M1/q-1/p' m1-1/P}.
However, the expression on the right is unbounded as m -4 oo, and this contradicts the continuity of operator (1). Lemma 2 is proved. 8.6.2. The Equivalence of the Continuity and Compactness of the Trace Operator: LP(Q) -+ Lq(S2, p), q < p, p > 1 Two lemmas established in the preceding subsection enable us to validate the
compactness of the continuous trace operator: HP -* Lq(µ), q < p, where HP = H, (R") is the Bessel potential space.
Let 1 < p < oo and l > 0. The Bessel kernel G1 is that function on R" whose Fourier transform is G1(6) = (1 +
Ifl2)-1/2
where the Fourier transform is u(0 = f e-`x£u(x)dx
and xl; denotes the inner product in R". The Bessel potential of a function f is defined as the convolution Gi * f, and the space HP consists of all Bessel potentials with f c L. The norm in HP is IIG, * fJIH, = Ill
lip.
8.6. Compactness of the Trace Operator: L,(1) -i Lq((1, µ)
453
It can be shown that Gi is positive on R' and satisfies the following conditions.
If jxj < 1, then
c Ixit-" for 0 < l < n, Gj(x)
n.
If I x I > 1, then Gj(x) < c jxj(1-n-1)/2 exp(-IxD-
We refer the reader to the books by Stein [194] (Chapter V, §3), and by Adams and Hedberg [3] (Sec. 1.2) (see also the work by Maz'ya and Havin [138]) for a treatment of R.iesz and Bessel potentials. Here we only mention that HP(Rn) = WP(Rn) with equivalence of norms when 1 < p < oo and l = 1, 2, .... A proof of this fact can be found in the paper by Calderdn [40] and in the book by Stein [194] (Theorem V.3). Clearly, the kernel Gi can be expressed as the sum of two nonnegative kernels, one of which has a compact support and belongs to L1 and the other belongs to L, for any r > 0. Therefore, the following assertion is a direct consequence of Lemmas 8.6.1/1-2.
Lemma 1. Let µ be a finite Bore] measure on R" with compact support. Assume that 1 < p < oo, 0 < q < p and l > 0. Then the continuity of the trace operator: ' -+ Lq(11) (i.e., the continuity of the operator Lp D f H Gi * f E L9(µ)) implies its compactness. The next assertion concerns Sobolev spaces.
1
Lemma 2. Let Q be a domain in Rn and p a finite Bore] measure with compact support in Q. Suppose that 1 < p < oo, 0 < q < p and that 1 is a positive integer. If the trace operator: LP' (Q) -+ Lq(Q, µ) is continuous, then it is also compact. The space L,(1) can be replaced by WP '(Q) and VP '(Q).
Proof. It follows from the assumptions that the trace operator: WP(Rn) -+ Lq(p) is continuous. According to Lemma 1, this trace operator is compact. Let cp E Co (1k) be a function satisfying cp(x) = 1 for x E supp jt. The linear map
L,(1) D u H cpu E Wp(R") (where cpu = 0 outside f) is continuous, and hence the operator L,(cl) D u H cpu E Lq(Il, µ)
is compact by Lemma 1.
1
8. Imbedding and Trace Theorems for Domains with Outer Peaks
454
...
In the following theorem we establish the compactness of the continuous trace operator: L,'(1) # Lq(SZ, p), where q < p and the support of p is generally not compact.
Theorem. Let SZ be a domain in R" and p a finite Bore] measure on Q. If 1 < p < oo and 0 < q < p, then the compactness of the trace operator: LP '(Q) - Lq(SZ, p) is equivalent to its continuity and to condition (8.5.3/1).
Proof. In view of Theorem 8.5.3, it is sufficient to infer the compactness of the trace operator: LP1 (0) + Lq(S2, p) from (8.5.3/1). Let {Ik}k>1 be an increasing sequence of subdomains of SZ such that Qk CC SZ,
U
1Qk = Q-
Then p(SZ \ Q k) -4 0 as k -+ oo. Using Corollary 8.5.3, we obtain the following estimate for any it E C°°(c) n L1(SZ): C II1IILy(S2,µ) < 6k IIuIIL,(O) + IIUIILq(clk,µ),
k = 1, 2 ...
(1)
Here c = c (p, q) > 0 and ql(p-q)dtI ) 1Iq-11p
//A(O\fzk) Ek =
C J0
(-u)
//
-4 0 as k -oo.
By Lemma 2, every subset of Cm(1), bounded in LP' (Q), is relatively compact
in Lq(I k, p) for all k > 1. Using inequality (1) and a diagonal method, we can select from any sequence in Cm(SZ), bounded in LP(SZ), a subsequence convergent in Lq(SZ, p). The proof is concluded.
1
Remark. The last theorem cannot be directly extended to the case p = 1 > q > 0. Consider the following example.
Let SZ be the cube {x E Rn : x; E (0,1), i = 1, ... , n} and let S denote its (n - 1)-dimensional section by the hyperplane x = 1/2. By p we mean the restriction of to S. According to Theorem 5.1.1, the space of the traces uls of the functions it E L11(Q) coincides with L1(S). Moreover, the norms IIfHIL1(s)
and
inf{IIuIILi(o) : ul5. = f}
are equivalent. Thus, the trace operator
L'(1) D u H uls E Lq(S)
8.6. Compactness of the IYace Operator: Ly(1l) - LQ(1, µ)
455
is continuous for any q E (0, 1). However, this operator is not compact because its compactness would imply the compactness of the imbedding L1 (S) C Lq(S), which is not the case. That the last imbedding is noncompact can be checked in the following way.
For each k = 1, 2.... and each i = 0, ... , 2k - 1 let Ai,k = [2-ki, 2-k (i + 1)). Define 9k on (0, 1) by
90) = (-1)i + 1 for t E Ai,k, i = 0, ... , 2k - 1. Then mess ({t E (0,1) Igk(t) - gm(t)I = 2}) = 1/2 if k # m. Let fk(x) _ gk(x1) for x E S. Clearly { fk} is a bounded sequence in L1(S). At the same time 21-1/q q > 0, k :A m. Ilk - fmII L5(S) > :
Thus, there is no subsequence of { fk} convergent in Lq(S).
8.6.8. Compactness Theorem in the Case q > p
Let SZ be a domain in Rn and µ a Borel measure on Q. Given p c [1,00), q > 0 and a relatively closed subset F C 11, put
'Yp,q(F,
SZ,
µ) =
(µ(F))P/q cap (F; L1(P)) 0
if cap (F; L1(SZ)) > 0,
otherwise.
The following assertion is the principal result of the present subsection.
Theorem. Let it be a Bore] measure on a domain 1 C R. If 1 < p < q < oo, then the conditions (8.5.2/1) and lim sup{yp,q(F,1,,u) : F C SZ, µ(F) < 6} = 0
6-+0
(1)
are necessary and sufficient for any subset of COO(Q) n LP(SZ), bounded in L1(SZ), to be relatively compact in Lq(Q, p).
Proof. Necessity. The necessity of condition (8.5.2./1) is already known (see Theorem 8.5.2). We now check (1). Note that the elements u E COO (SZ)nLp(SZ)
have equicontinuous norms in L. (Q, µ). That is, p/q
U IuIldp /
e(6)IIuIILp(sl),
(2)
456
8. Imbedding and Trace Theorems for Domains with Outer Peaks
...
where e(6) -4 0 as 6 --+ 0 and F is any Borel subset of S2 with µ(F) < 6. Taking F to be closed in 52, we insert into (2) an arbitrary function u E C°°(S2)nLP(Sl)
with uIF > 1 and pass to the infimum on the right part over such functions. This results in -yp q(F, Q, p) < e(b).
Sufficiency. First we remark that condition (8.5.2/1) implies i (SZ) < oo. Let u E C°°(I) n LP(Q) and let 6 E (0, p(1)/2). Put
t(6) = sup It > 0: p(Nt) > 61, where Nt = {x E SZ : Iu(x)I > t}. Then
µ(Nt) < S for t > t(b).
t(d) < oo, p(Nt(5)) > 6, Clearly
IuIILq(D,p) < fN julldy + t(a)4µ(Q), )
and hence the inequality C IIuIILQ(12,µ) < IIvbIILq(O,p) +
t(a)(t(1))11q
(3)
holds where v6(x) = max{lu(x)I - t(d), 0}. We now estimate the first term on the right of (3). By Lemma 8.5.1/1
(fvdii)
P/q
=
(f
t}d(t9)\
tt{x : v6(x) >
I
P/9 .
(4)
Since {v6 > t} C Nt(6)+t for t > 0 and according to Lemma 8.5.1/3, the right part of (4) does not exceed (lt(Nt(6)+t))P19
d(tp)
f"o We have µ(Nt(a)+t) < 6 for t > 0, and by (1), the last integral is not greater than f (b) f cap (Nt(6)+t; LP(1l))d(tp), 0 00
(5)
where f (8) denotes the supremum in (1). Theorem 2.13 (combined with the inclusion Nt(a)+t C Nt) says that expression (5) is dominated by c f (a)IIuIILD(O). Now Iv6IILq(12,M) 5 Cf(a)1/PI U ILi(Sz)
(6)
8.6. Compactness of the Trace Operator: LP' (n) -* Lq(I, it)
457
To bound the last term in (3), we note that f I uI dµ > t(6)p(Nt(6)) > St(S).
This in conjunction with (3) and (6) gives C IIUIIL,(cl,,) 0 and g(6) --> 0 as 6 -+ 0. Inequality (7) implies that a sequence in C°°(12), bounded in LP(12) and convergent in L1(12,µ), is convergent in Lq(12, µ). It remains to observe that, by the assumptions, the trace operator: L1(12) --> Lq(0, µ) is continuous (cf. Theorem 8.5.2) and hence the trace operator: Lp(12) - L1(12,µ) is compact in view of Theorem 8.6.2. 1
Remark. We observe that the simultaneous validity of (1) and (8.5.2/1) is equivalent to
t9lq/v(t) -4 0 as t -* +0,
(8)
where v is the isoperimetric function introduced at the beginning of Sec. 8.5.3.
It is an easy consequence of the definition of v that (8) implies (1) and (8.5.2/1). Conversely, suppose that (1) and (8.5.2/1) hold. To prove (8), we let e be any positive number and choose 6 > 0 such that FL(F')nle
cap (F; LP' (0))
<E
/2 (9)
whenever F is a relatively closed subset of 12 with 0 < µ(F) < 6. We claim that then t7l9/v(t) < E for all t subject to
0 6, (10) implies
t< rEl
\2D/
qlP
µ(F)
and hence
2D tPl q < ei (F)P/q < e D cap (F; LP ' (0)),
the last because of (8.5.2/1). Now (11) is again true. This completes the proof of (8). 1 A combination of the last theorem with Theorem 8.6.2 and Theorem 8.5.2 leads to the following assertion (cf. Theorem 1.10/2).
Corollary. If the trace operator: Lp(1l) -4 Lq(1l, p) is continuous for p E (1, oo) and q E (0, oo), then the trace operator: LP '(Q) -+ Lr (1l, µ) is compact for all r E (0, q).
Comments to Chapter 8 The results of Sections 8.1-8.4 (except for those in Sec. 8.4.2) were obtained in the paper by Maz'ya and Poborchi [152]. Concerning similar results for power peaks see Maz'ya [124], [125], Globenko [75], R. A. Adams [Adm] (Sec. 5.35, 5.36), Fukushima and Tomisaki [68], Tomisaki [200].
8.5. A nonincreasing rearrangement of a function of one real variable was considered in the book by Hardy, Littlewood and Polya [88] (Sec. 10.12). The reader may consult the book by Kawohl [108] and the survey by Talenti [198] for a deeper treatment of rearrangements and their applications. Theorem 8.5.2 is due to Maz'ya [132], [136] (Sec. 4.4.3, 4.6). Statement (i) of Theorem 8.5.3 was established by Maz'ya [132] (see also [136, 4.4.4]). Statement (ii) of this theorem was proved by Maz'ya and Netrusov [141]. A necessary and sufficient condition for the continuity of the imbedding Lp(1l) C
Lq(12), q < p, different from (8.5.3/1) for t = mes,,, was given by Maz'ya [132], [136, 4.4.3].
Comments to Chapter 8
459
In a recent paper Davies [48] has shown, in particular, that there is a biLipschitzian map of the von Koch snowflake domain (see e.g. Falconer [58], p. 37, for the description of this domain) onto the unit disk with the metric ds2 = (1 - lxl)-27(dxi +dx2), y = 1 - log3/log4.
This seems to open an interesting approach to the study of Sobolev spaces on domains with fractal boundaries.
8.6. In case 1 < q < p and supp K compact, Lemma 8.6.1/2 can be obtained from a theorem on integral operators due to Ando [10]. The proof of the Ando theorem is also found in the book by Krasnosel'ski et al [112] (Sec. 5.3, Theorem 5.5). The general version of Lemma 8.6.1/2 and its proof presented here were communicated to the authors by Netrusov in July 1994. Lemma 8.6.2/1 can be strengthened. Let 1 < p < oo, 1 > 0. For any compact set F C R", the capacity of F in HP is introduced by
cap(F;H,)=inf{IIuIIHp:uECO (R"), UIF> 1}. Given a Borel measure p on R", the function (0, p(Rn)) 3 t H /3(t) is defined by
/3(t) = inf { cap (F; Hj,) : F compact, µ(F) > t}.
Let p E (1, oo), 0 < q < p, 1 > 0. Then the best constant in the trace inequality IIullL,(µ) < C IIUIIH1
,
i E Co
,
(1)
is comparable to
µ(R^)(t)(P_)) 1j9-11p Cf
(2)
This result is due to Maz'ya and Netrusov [141]. It is not difficult to show that the conclusion of Lemma 8.6.1/2 remains true without the assumptions µ(R") < oo and supp p is compact provided supp K is compact. With the aid of this assertion one can obtain that the compactness of the trace operator: HP _ L4(µ), q < p, is equivalent to its continuity, i.e., to the finiteness of quantity (2). A non-capacitary description of measures satisfying trace inequality (1) for q < p was given by Verbitsky [206].
8. Imbedding and Trace Theorems for Domains with Outer Peaks ...
460
We note here that if 1 < p < q < oo and lp < n, a necessary and sufficient condition for (1) to be valid is that sup
(rq(1P-n)/Pp(Br(x))
: x E R', r E (0, 1] } < oo,
see Adams [1], [2], Adams and Hedberg [3], Sec. 7.2. In case q = p E (1, oo), l > 0, the validity of (1) is equivalent to any of the following conditions: Gi * (GI * p)P' < C Gi * p
fF
a.e. on Rn;
(GI * p)p dx < C cap (F; HP) for all compact F C Rn,
where Gi is the Bessel kernel, p' = p/(p - 1). See Maz'ya and Verbitsky [155]. In case p = mesa Theorem 8.6.3 was proved by Maz'ya [132], [136, 4.8.2].
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Index absolute continuity on lines, 14 Adams, D. R., 81, 86, 129, 130, 140,
capacity, 129, 139, 156, 161, 201, 443 Carlsson, A., 139, 201
453, 460 Adams, R. A., 56, 63, 76, 430, 458 Agmon, S., 76 Ahlfors, L. V., 79, 84 Amick, C. J., 137 Ancona, A., 139 Ando, T., 459 anisotropic Sobolev spaces, 81, 82 Anzelotti, G., 362 Aronszajn, N., 259 Aubin, T., 86
characteristic function of a set, 1 Chua, S. K., 85, 88 Cianchi, A., 140 classical isoperimetric inequality, 57 coarea formula, 58 convolution, 5 convolution operator, 450 Coulhon, T., 88 Courant, R., 83, 138 Davies, E. B., 458 De Giorgi, E., 122 Deny, J., 79, 80, 83, 87 domain
Babich, V. M., 44, 83, 259 Banach algebra, 64, 114 Belova, N. 0., 81 Berger, G., 138 Besov, 0. V., 81, 82, 88 Bessel kernel, 452 Bessel potential, 452 Bessel potential space, 452 Beurling, A., 79, 259 bi-Lipschitzian map, 46 Birman, M. S., 73 Boas, H. B., 83 Bobkov, S. G., 87 Bojarski, B., 88 boundary trace operator, 75 Brudnyi, Yu., 362 Buckley, S., 88 Burago, Yu. D., 58, 122, 362 Burenkov, V. I., 81, 86, 206, 326 Bylund, P., 362
(E, 6), 85
having the cone property, 22 having the segment property, 18 John, A-John, 83 Lipschitz, 23 Lipschitz graph, 23 of class C, 18 of class C', C°"1 C'"A, 20 of class EVp, 45 special Lipschitz, 45
starshaped with respect, to a set, 20
with the interior segment property, 80 Douglas, J., 259
Edmunds, D. E., 140 equivalent norms in Sobolev spaces, 40, 145
Evans, G. C., 78 Evans, W. D., 83, 326 extension operator, 45
Calderon, A. P., 81, 83, 453 Calkin, J. W., 79 capacitary isoperimetric inequality,
Fain B. L., 326
129, 443 477
478
Index
Falconer, K. J., 458 Federer, H., 16, 46, 58, 86 Fichera, G., 78, 79 Fleming, W. H., 86 fractal set, 362 Fraenkel, L. E., 77, 79, 86, 87, 138 Franchi, B., 88 Friedrichs inequality, 120, 266 Friedrichs type inequality, 139 Friedrichs, K., 78, 121 Fubini, G., 78 Fukushima, M., 458 Gagliardo, E., 56, 63, 79, 86, 87, 207, 210, 260 Garofalo, N., 85, 88 generalized derivative, 9
generalized Poincare inequality, 35, 37, 145
Giaquinta, M., 362 Gilbarg, D., 74, 77 Globenko, I. G., 458 Glushko, V. R., 79 Gol'dshtein, V. M., 79, 84, 86, 124, 129, 326
Gorbunov, A. L., 206 Gurka, P., 140 Gutierrez, C. E., 88 Holder's inequality, 4 Hadamard, J., 259 Hadwiger, H., 58 Hajlasz, P., 87, 88, 139, 362 Hardy type inequality, 139, 251, 268 Hardy's inequality, 6, 73 Hardy, G. H., 442, 458 Harris, D. J., 83 Hausdorff measure, 16 Havin, V. P., 453 Hedberg, L. I., 79-81, 129, 130, 140, 453, 460 Heinonen, J., 88, 129
Herron, D. A., 85 Hestenes, M. R., 83 Hilbert, D., 83, 138 Houdre, C., 87 Hurri, R., 83 Hurri-Syrjanen, R., 83, 88
Il'in, V. P., 81, 82, 86 imbedding operator, 48, 444 inner peak, 298, 299 integral representation, 32, 34, 81,
82
intrinsic metric, 325 isoperimetric function, 410, 444 Jerison, D., 88 John, F., 83 Jones' extension theorem, 85, 298 Jones, P. W., 84, 85 Jonsson, A., 362 Kaimanovich, V. A., 87 Kalyabin, G. A., 206 Kawohl, B., 458 Kilpelainen, T., 129 Kolsrud, T., 137 Kondrashov, V. I., 87 Koskela, P., 83, 85, 87, 88, 139 Krasnosel'ski, M. A., 87, 459 Kufner, A., 77, 140
Ladyzhenskaya, 0. A., viii Landkof, N. S., 168 Latfullin, T. G., 84 Leibnitz formula, 11 Levi, B., 78, 79 Lewis, J. L., 80, 139, 140 Lichtenstein, L., 83 Lions, J.-L., 79, 80, 83, 87, 88 Littlewood, J. E., 442, 458 Ljusternik, L. A., 58 locally finite covering, 25
Index
479
Magenes, B., 88 Martio, 0., 83, 129, 362
Maz'ya, V. G., vii, viii, 56, 58, 60,
63,76,77,79,80,82,83,8688, 129, 130, 138-140, 201, 205, 206, 260, 268, 325, 326, 362, 408, 430, 453, 458-460 Meyers, N. G., 79 Mikkonen, P., 140 Minkowski's inequality, 4, 5 mollification, 7 Morrey, C. B., 79, 80, 86 Morse, A. P., 58 Muckenhoupt An condition, 85 Muckenhoupt, B., 325 multiplicative inequality, 64
Nazarov, S. A., vii Necas, J., 88 Netrusov, Yu. V., 81, 138, 140, 201, 362, 458, 459 Nhieu, D. M., 85, 88 Nikodym's domain, 107 Nikodym, 0., 78, 79, 108, 138 Nikol'ski, S. M., 44, 81, 83 Nirenberg, L., 87 nonincreasing rearrangement, 441 Nystrom, K., 140
operator of the Neumann problem, 69, 114, 120, 430 Opic, B., 140 outer peak, 266
partition of unity, 25 perturbed peak, 431 Plamenevsky, B. A., vii Poborchi, S. V., 205, 206, 260, 325, 362, 408, 458
Pohoiaev, S. I., 86 Poincare inequality, 74, 82 Poincare type inequality, 68, 69, 87, 444
Polya, G., 442, 458 Popova, E. A., 86 power cusp, 308, 311, 315, 318, 430 projector, 40 Prym, F., 259 pseudonorm, 55
quasi-isometric map, 23 quasicircle, 84 quasidisk, 124 Rademacher, H., 46 radius of mollification, 7 reflection of finite order, 43 Rellich, F., 78, 87 removable singularities, 16 Reshetnyak, Yu. G., 79, 81, 86 Riesz, M., 60 Rosin A. L., 326 Rudin, W., 69
Sarvas, J., 83 Schmidt, E., 58 Semmes, S., 88 Serrin, J., 79 Shaposhnikova, T. 0., 86, 87 Shvartsman, P., 362 Sitnikov, V. N., 326 Slobodetski, L. N., 259 small domain, 148 Smirnov, V. I., 79, 205 Smith, K. T., 81 Smith, W., 80, 83, 138 Sobolev's theorem, 55 Sobolev, S. L., 78, 79, 81, 83, 86, 205, 260
Sobolev-Gagliardo inequality, 50 Solomyak, M. Z., 73 spectral synthesis, 80 Stanoyevitch, A., 76, 80, 83, 138 Staples, S. G., 83 Stegenga, D. A., 80, 83, 138
480
Index
Stein, E. M., 45, 84, 85, 299, 300, 353, 453
Stepanov, V. D., 326 Straube, E. J., 83 strong capacitary inequality, 129 Talenti, G., 86, 87, 325, 458 Tomaselli, G., 325 Tomisaki, M., 458 Tonelli, L., 78, 79 topological imbedding, 48 trace inequality, 410, 459 trace operator, 444 Trudinger, N. S., 74, 77, 86 truncated peak, 318, 432 Turesson, B. 0., 140 uniformly Holder function, 3 uniformly Lipschitz function, 3 Ural'tseva, N. N., viii Uspenski, S. V., 260
Vaisala, J., 79
Vasil'chik, M. Yu., 362, 408 Verbitsky, I. E., 459, 460 vertex of a zero angle, 346 Vodop'yanov, S. K., 79, 84, 325 Wallin, H., 362 Wannebo, A., 139 weight function, 271, 279, 292, 301, 307, 311, 425 weighted L9 space, 425 weighted Sobolev space, 85, 87, 271 Wheeden, R. L., 88 Whitney, H., 297, 325 Wiener capacity, 144, 168 Wolff, T. H., 80 Yakovlev, G. N., 362 Young's inequality, 5 Yudovich, V. I., 86
Zalgaller, V. A., 58, 122 Ziemer, W. P., 16, 58, 73, 74, 83 Zobin, N., 325
List of Symbols Symbols listed in order of appearance
Chapter 1 Br(X), B, B(n) ...................
cap (F; L,(1)) .................. 129 1
Chapter 3
Sn ................................ 1
'Y(e,p) .......................... 154
XE ................................ 1
cap (F; Vp (D)) .................. 156
diam (E) .......................... 1
V y (D), p E (1, oo), 1
.\E(AER1, ECRn) ............. dist (E, F)
........................
(D)
....... 157
1 7o(E,p) ......................... 157 1
supp f ............................ 2
cap (F; LIP (D)) .................. 161 cape ........................... 168
E C C Q .......................... 2
Cap (F;VP) ..................... 201
Zn, Z+ ............................ 2
Chapter 4
(n)
P, I
V1u, IViul ........................ 2
' IITW (n) ............ 209 TLp(e), II ' IITL;(n) .............. 209 ['Jp,s, p E (1, o0) ................ 209 Wp-1/P(S), II Iiw,-1/,(s) ........ 209 TWp (e), 11
.............. 2 .................... 2 CI(S2), C°°(S2) ..................... 3 C(e), C'(e), C°°(e) CO, A, Co (e)
Ci',\(e), C1,,\(n) ................... 3 IIc,(n), II ' IIC,,,(.a) ............... 3
N i's ........................... 210
L9(e), Lp,,.(e) ................... 4 II IIL,(n), ' Ilp,n .................. 4
Chapter 5
K * u ............................. 4
Chapter 6
Mhu .............................. 7
TWp (Rn, 8e)
I.
II
LP' (Q), WW(e), Vp (e) ............. II
II wp(n), II
12
II
' IIW;(n,E),
Vp,o(G),
II
II
'
IITW,(n,f)
.........
219
II vp,o(c) .............. 271
................... 358
Chapter 7
IIv;(n) ............... 12
I' ............................... 365
H,,(E) ........................... 16
LP(e) ............................ 39
M(z,() ......................... 366
IIL,(n) ......................... 40
I' Ip,r ........................... 375
II
EVp
Chapter 8
............................. 45
IILq(1,µ) .............. 54
Mt, Nt .......................... 441
LVP (e) .......................... 82
uµ, u' .......................... 441
Chapter 2
vµ,p, v .......................... 445
Wp,r(e,8e) ..................... 121
Gi, HP .......................... 452
L9 (e, /1),
II '
481