COMPACTNESS OF ORLICZ-SOBOLEV SPACE IMBEDDINGS FOR UNBOUNDED DOMAINS by Ian Graham Cahill B.Sc, University of Victoria, 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of MATHEMATICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1975 In p resen t ing t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e fo r reference and study. I f u r t h e r agree tha t permiss ion fo r ex tens i ve copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . It i s understood that copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l ga in s h a l l not be a l lowed wi thout my w r i t t e n pe rm iss i on . Department of The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 ABSTRACT In t h i s thesis we are concerned with the compactness of im-beddings, for unbounded domains, of Orlicz-Sobolev spaces. Donaldson and Trudinger [6] have given a gen e r a l i z a t i o n of the Sobolev imbedding theorem and the Rellich-Kondrachov compactness theorem for Sobolev spaces to the case of Orlicz-Sobolev spaces on bounded domains. Clark, Adams, and Fournier [1], [2], [3], and [5] have given extensions of the Sobolev compactness r e s u l t s to c e r t a i n unbounded domains. In this thesis we consider s i m i l a r extensions f o r Orlicz-Sobolev spaces. i i i TABLE OF CONTENTS Chapter Page INTRODUCTION 1 1. Notations 1 2. Orlicz and Orlicz-Sobolev Spaces 4 3. Imbedding Theory 11 1. IMBEDDINGS OF w \ , (G) 19 A 1. Necessary Conditions for the Compactness of WmEA(G) -»• L_ (G) where A-< B 19 2. A Sufficient Condition for Compactness of lA^CG) - L A(G) 28 3. Applications to Certain Bounded Domains 34~ 2. IMBEDDINGS OF ( W™ L A(G) 37 1. Necessary Conditions for Compactness of ^ V G ) * L A ( G ) 3 7 2. Sufficient Analytic Conditions for Compact-ness of W™L A(G) -»• L (G) 4 6 3. Sufficient Geometric Conditions for Compact-ness of W™L A(G) + L A(G) 50 REFERENCES 66 Acknowledgeme nt s I would like to give thanks to Dr., J. Fournier for his guidance and encouragement during the writing of this thesis. I would also like to thank Dr. R. Adams for reading the thesis and Maryse E l l i s for typing i t . 1 INTRODUCTION 1. Notations This s e c t i o n provides a l i s t of the n o t a t i o n s used. U s u a l l y we f o l l o w Adams [4]. U n f a m i l i a r concepts mentioned w i l l be defined l a t e r . The reader may use t h i s s e c t i o n as a reference and begin reading s e c t i o n 2. (a) M u l t i - i n d i c e s We w i l l denote by a the m u l t i - i n d e x ( a ^ , a ) and l e t I ot I = E a. . I f D. = — f o r 1 < j < n, l e t . , j i 3x. — — 3=1 J 3 a a i a n D = D„ *•• D , which i s a d i f f e r e n t i a l operator of 1 n order |a|. (b) Spaces and Norms ( i ) R w i l l be r e a l Euclidean space w i t h the norm I f x = ( x 1 , " * , x ) e R n , then Ixl = ( £ x 2 ) 1 ^ 2 . 1 n 1 1 . - i 3 3=1 ( i i ) L.(G) w i l l be the O r l i c z space defined using the N-f u n c t i o n A and the open set G c R n and w i t h the norm || • || . n . I f ueL (G) then ,|| u | | A j G = inf{k>0| f A ( ^ ) d x < 1} G (Luxemburg norm). ( i i i ) E A(G) w i l l b e the closure in L^(G) of the set of bounded functions with compact support. (iv) w V (G) w i l l be the Orlicz-Sobolev space obtained using the N-function A and open set G cR n. It has norm || • || where i f u £ W™ L (G) , then m, A, G A ' ' |a|<_m (v) W mE A(G) w i l l be the closure i n WmL^(G) of the set of bounded functions w i l l compact support. rn OO m (vi) W L.(G) w i l l be the closure of C (G) in W L (G) (vii) Let u be a positive, continuous, increasing function on the interval (0, °°) , and let y(t)^-0 as t ->• 0; then the space C (G) w i l l consist of functions u e C(G) for which the norm ||u, y g , || ^ 1 ^ 0 ( 0 ) 1 1 , ^ ^ 1 ^ x^y i s f i n i t e . m \ — ( v i i i ) If 0 < X £ 1 we define C ' (G) to be the subspace of Cm(G) consisting of those functions (J) for which, for 0 <^ | ot| <_ m, Dacf) satisfies in G a Holder condition of exponent X; that i s , there exists a constant K such that D%(x) - D%(.y)|£ K | x-y :,yeG. C m' A(G) has the norm || <J>; C m , X(G) || I t>; Cm(G) || + max sup l A ( x ) - D ^ ( y ) | 0<|a|<m x,yeG J x-y| x^y Imbeddings If S and T are Banach spaces, then S -»• T w i l l mean that S continuously imbedded in T. If Sc T, then the only imbedding that we discuss is the natural injection S T. This imbedding is said to be compact i f the unit b a l l i n S maps onto a precompact set in T. Measure If E i s a Lebesgue measurable set in Rn, we w i l l denote its Lebesgue measure by u(E). Domains By a "domain G" we w i l l mean an open set G in Rn. G r For a domain G we define G = ' { x £ G x > r } . r — If G is a domain then 8G w i l l denote the boundary of G. Cone Condition (Adams [4]) A domain G is said to satisfy the cone condition i f there exists a f i n i t e cone C (the intersection of an open b a l l in Rn centered at the origin with a set of the form 4 {Ax|x£B, A>0} where B is an open b a l l not containing the origin) with the property that each x eG is the vertex of a f i n i t e cone C x contained in G and congruent to C. (i) By a strongly Lipshitz domain G we w i l l understand a domain G where each x £ 9 G has a neighbourhood such that there exists a rectangular coordinate system, with origin at x, and a Lipshitz continuous function F such that G n u x = { q , ? n ) e U x U n < F ( V — , > . 2. Orlicz Spaces and Orlicz-Sobolev Spaces In this section we l i s t the definitions and a few of the prop-erties of Orlicz spaces and Orlicz-Sobolev spaces. More details can be found in Krasnosel'skii and Rutickii [ 9 ] , Adams [ 4 ] , or Fournier [ 7 ] . An N-function is a continuous, convex, even function A : R"*" satisfying lim = 0 and lim = » . . . . . (1) t-K) t t-*» t The Orlicz class K. (G) consists of the functions u from G to R"*" such A that A( u(x)|)dx < °°. The Orlicz space L (G) is the linear h u l l JG A of K (G) with the Luxembure norm. It can be shown that E A ( G ) c K A ( G ) c L A ( G ) E (G) is the maximal vector subspace of the class K.(G), (Adams, [4, p. 236]). There are some ways of ordering N-functions. We say A^ i f there exists X Q > x and k > 0 such that for a l l x >^ X Q , A^(x) _< A(kx) A 2(X X) If A -< A and A -< A , then we say A ~ A . If lim . . = 0 0 for 1 1 1 1 1 Z y^co Ai^x> every A > 0 then we say A ^ " ^ A2' N o t ; i - c e that, for bounded domains G, A 1 -< A„ implies that L (G) c L (G) and A . . - ^ A„ implies that 1 Z A ^ A ^ 1 Z L (G) C L (G) . A 2 ^ A X Every N-function A can be represented in the form : M A(x) = a(t) dt J 0 where a(t) is a nondecreasing, right-continuous function. This is proved, from the definition of an N-function, by Krasnosel'skii and Rutickii [6]. Let a(t) = sup { s|a(s) <_ t} . We then define A(x) = a(t) dt 0 A and A are related by y < A(x) + A(y) (Young's inequality) By applying Young's inequality to u(x) , v(x) , N J-L- and — 1—^—^ and U H A , G I I v A ,G 6 integrating over G we get juvdx < 2 ||; u ||A || v ||~ (Holder's inequality) for u e L A(G) and v eL A(G) We say that an N-function A satisfies the condition i f there exists x^ > 0 and k > 0 such that A(2x) _< kA(x) for x >_ x^. We say that i t satisfies the global condition i f there exists k _> 0 such that A(2x) < kA(x) for a l l x > 0. Example 0.1 Let ,. 2" 2 n when 2 _ n _ 1 < x < 2~n a(x) = x when x > 1 Then A(x) = a(t) dt satisfies the condition but not the global A condition . These restrictions w i l l be needed in some of the theorems of this thesis. In general, the condition A( | u (x) | ) dx -»- 0 as n -> 0 0 ,1 n 1 G 7 does not imply that u | | ^ 0 as n->-°°. If A and G have further properties, however, then the im p l i c a t i o n does hold. Lemma 0.1 Suppose that A s a t i s f i e s the condition and that y(G) i s f i n i t e ; or suppose merely that A s a t i s f i e s the global condition. Then the condition ' A(|u (x) |) dx ->- 0 as n -> implies n G u || , 0 as n -»• » n A,U Proof: Suppose that A(|u (x) 0 as n -> 0 0. If A s a t i s f i e s J a n the global condition i t i s clear that | A(2|u n(x)|) dx + 0 as n since t A(2|u(x)|)dx <_ (constant) G A(|u(x)|) dx G If u(G) < 0 0 and A(2x) <_ MA(x) for a l l x >_ x Q proceed as follows: Suppose A(|u(x) |) dx <_ £ Choose x^ so that Let Then Now A ( x 1 ) = £ 1/2 ; x = {y £ G | |u(y)| < x± } G 2 = {y £ G | x± < |u(y)| < x Q } G 3 = {y £ G | |u(y)| > x Q } [ A(2|u(x)|) dx < [ + [ + [ A(2|u(x)|)dx JG ' " J C 1 J G 2 S [ A(2|u(x)|)dx < A(2x ) y(G ) < A(2 X ; L) y(G) A(2|u(x)|) dx < A(2x Q) y(G 2) [ A(2|u(x)|) dx < M| A(|u(x)|) dx < M £ J a J a But, as £ -»- 0, x.^ 0; hence A(2x^) ->• 0. Also e > f 1/2 ' A(|u(x)|)dx >_ A( X ; L) y(G 2) = £ u(G 2) G2 - i / o Thus U(G2) <_ e . Combining these estimates we have | A(2|u(x) [ ) dx <_ A(2x 1 ) u (G) + A(2x Q) e 1^ 2 + Me , G which is independent of u and tends to 0 with e. Thus, i f A satisfies the global A 2 condition, or i f |G | < and A satisfies the A 2 condition, then | A(|u n(x) | ) dx -> 0 as n + G => | A(2|u n(x) |) dx ^ 0 as n-^ By induction oil k, j A(2 k|u n(x) |) dx +0 as n -> In particular, for fixed k and a l l sufficiently large n, | A(2 k|u n(x)|) dx £ 1 For such n, II u II . „ < •—-" n " A,G — „k Since k can be taken as large as we like this completes the proof. Some other relations between the mean and the norm w i l l be useful. Remark 0.1 We note that with no restrictions on A and G the condition ti II 0 as n -»-n 11 A,G implies that | A(|u n(x) | ) dx •*• 0 as n -»• G Indeed, suppose II u H; A j G < e < i . Then | A(|u(x)/e|) dx < 1. G But.A i s convex and A(0) = 0. Therefore A(|u(x)|) < e A ( i ^ l L ) . and A( |u(x) | ) dx <_ e G Also, i f the sequence '{ A(|u (x)|) dx } is bounded away from 0 then J a n so is the sequence {||u ||A ^} 11 Remark 0.2 If {j A(|u n(x)|) dx} is bounded then so is the sequence To see this suppose that A(|u(x) | ) dx _< K ; without loss of generality assume K > !• Then A(|u(x)|/K) < (1/K) j A(|u(x)|) dx < 1 Hence U HA,G 1 K 3. Imbedding Theory The material of this thesis has i t s origins in the imbedding theory of Sobolev spaces. The key result in this theory is a collection of imbedding properties called the Sobolev Imbedding Theorem. The basic case of this theorem says that i f G is a domain satisfying the cone condition then 12 (a) i f p < n, W 1 , P(G) -> L P*(G), where p* = np/(n-p) (b) i f p > n, W 1 , P(G) -»• L o o(G)nc(G) (c) i f p > n and G i s also strongly L i p s h i t z and bounded then W 1 , P(G) -»• C 1 , X ( G ) for 0 < X < 1 - n/p . This r e s u l t i s due la r g e l y to S.L. Sobolev but includes refinements due to C.B. Morrey and E. Gagliardo. See Adams [4] for a d e t a i l e d proof. Compactness of some of these imbeddings i s shown i n the Rellich-Kondrachov compactness theorem. We w i l l now state the basic case. Suppose G i s a bounded domain s a t i s f y i n g the cone condition. Then (a) i f p < n, the imbedding W ' P(G) -* L q(G) i s compact f o r a l l q < p* (b) i f p > n and G i s strongly L i p s h i t z , the imbedding W 1 , P(G) C 1'^(G) i s compact f o r a l l X < 1 - n/p. This theorem i s also proved i n Adams [4]. The above r e s u l t s were extended to Orlicz-Sobolev spaces by Donaldson and Trudinger [6]. For a bounded domain G s a t i s f y i n g the cone condition we have the following r e s u l t s : 13 (a) If oo _ 1 A X ( t ) x t(n+l)/n dt then W L A(G) + LA*(G) where we define the Sobolev conjugate A* by setting A^Ct) f t A - V ) 0 T (n+l)/n dx (In the proof i t is assumed without loss of generality that A (t) 0 t (n+l)/n dt < 0 0 This is justi f i e d since an equivalent N-function satisfying this can be chosen and, since G is bounded, the space w V (G) w i l l not change.) Also, i f B • « A* then the imbedding W^L^G) -* L g(G) is compact. (b) If ^ A-^t) 1 t (n+l)/n dt < 0 0 then W L A(G) -* L^G) fl C(G) 14 (c) If G is strongly Lipshitz, and i f r 0 0 .-1. f A X ( t ) J 1 t ( n + l ) / n dt < 0 0 , then W ^ C G ) -»• C (G) , where .00 ' -n t M o r e o v e r i f V i s a p o s i t i v e , c o n t i n u o u s , i n c r e a s i n g f u n c t i o n o f t > 0 s u c h t h a t l i m ^ r p y = 0 t h e n 1 W L A(G) -»• Cv(G) is compact. Clark, Adams, and Fournier [1], [2], [3], and [5] have generalized the Rellich-Kondrachov compactness theorem to certain unbounded domains. It is the purpose of this thesis to extend some of the compactness results of Donaldson and Trudinger [6] to certain unbounded domains, using many of the techniques of Clark, Adams, and Fournier. We w i l l now give a condition which is sufficient for the compactness of the imbedding S -»• L^(G) where S is a Banach space such that S c L^(CJ) . The condition is also necessary but this w i l l not be proved since i t w i l l not be needed later. 15 Lemma 0.2 (Fournier [7]) Let G be a domain in Rn. Let S be a Banach space such that S -+ L A(G) . For u £L A(G) and r > 0 let u(x) i f |x| > r P u(x) = { I I "J } • r 0 i f I x I < r Let B = { u e S | || u || < 1 } We then have that B is precompact in L^(G) i f the following two conditions hold: (a) (I-P )B is precompact in L (G) for a l l r. TO A (b) || P || 0 as r + °° (|| P || is the norm of P^ as an operator from S to L A(G)) Proof: If (a) and (b) hold then I : S L A(G) is a norm limit of compact operators I - P and is therefore compact. • 16 We w i l l now give a technique for proving compactness of im-beddings which can be used when the N-function A satisfies the global A 2 - condition. Lemma 0.3 Let A be an N-function satisfying the global condition. Suppose that for every £ > 0 there exists r > 0 such that A(|u(x) | )dx < £ || u G for every u £ S and suppose that condition (a) of lemma 0.2 is satisfied. Then ,S L^CG) is compact. Proof: If we pick a sequence {e^} such that £ n -* 0 as n -»• °°, then we can find a sequence such that | A(|u(x)|)dx < £n|| u ||s , G r n that is f A(|P u(x)|) dx G ^ u Hs < £ n Since we are concerned with the norm of P we w i l l consider only u r such that || u || = 1. We then get A(|P u(x) | ) dx < £ G r n n 17 By examining the proof of Lemma 0.1 we see that because A satisfies the global A„ condition there exists a sequence {£"*"} with -*• 0 as n -> 0 0 2 n n such that P u II < e 1 r 11 A,G n n which means (b) of Lemma 0.2 is satisfied. We have assumed (a) to be satisfied; so, by Lemma 0.2, S -*• L A ( G ) i s compact. • We fini s h this section with a result on interpolation. Lemma 0.4 (Adams [4, p. 241]) Let G have f i n i t e volume and suppose that A and B are N-functions such that B "^ "^ A. If the sequence ^ uj} i s bounded in L.(G) and convergent in measure on G then i t is convergent in A L B(G). Theorem 0.1 Let y(G) < °°. Let A, B, and C be N-functions such that A-< B -4"^ C. Let S be a Banach space such that S •* L e(G) and S -> L (G) . Suppose S ->• L (G) and S -»• L.(G) is compact. Then S -»• LT_(G) is compact. 18 Proof: Since y(G) < 0 0 we have S. + L C(G) -» L B(G) so that S •* L B(G) . To see that this is compact notice that i f {u^} i s a bounded sequence in S then i t is bounded i n L (G) Since S + L (G). Since S + ^ A(G) is compact we w i l l have a subsequence {u^ } convergent in measure. i l i (Here we have convergence in norm => convergence in mean => convergence in measure). Therefore by Lemma 0.4 we have that S L (G) is compact. p — i 19 CHAPTER 1 IMBEDDING THEOREMS FOR WmL^(G) In this chapter we w i l l present results due to Fournier [7] on the compactness of the imbeddings between spaces WmLA(G) and various Orlicz spaces over a domain G which is not necessarily bounded. The results of Donaldson and Trudinger [6] are extended to unbounded domains in much the same way as the results for Sobolev spaces are extended by Adams and Fournier [3]. 1. Necessary Conditions for the Compactness of WmEA(G) - L B(G) where A •< B . For our f i r s t condition we need some preliminary definitions. Consider a tesselation of R n by n-cubes of side h. H w i l l always denote a cube in the tesselation under discussion. N(H), called the neighbourhood of H, w i l l be the cube of side 3h concentric with H and having i t s faces parallel to those of H. F(H), called the fringe of H, w i l l be the shell N(H) - H. Definition 1.1 (Adams and Fournier [3]) Suppose G is a domain. Let X > 0. H w i l l be called A-fat i f y(H n G) > X y(F(H) n G) . 20 If H i s not X-fat i t w i l l be c a l l e d X-thin. Theorem 1.1 Suppose that A < B and WmE.(G) -»• L^G) i s compact. • A B Then f o r any t e s s e l a t i o n of R n and any X > 0 there are only f i n i t e l y many X-fat cubes. Proof: Fix a t e s s e l a t i o n of Rn, using cubes of side h. R e c a l l that A -< B means that there are numbers k and x^ SO that A(x) <^ B(kx) for a l l x > XQ. Without loss of generality assume that k _> 1. Choose d >_ x Q so that B(kd) h n > 1 . oo Now consider a X-fat cube H and a function (j) £ CQ supported by N(k") with cp = 1 on H and f D^ cp | <_ M for a l l a _< m. (M depends on h and m but not on H.) Choose c so that B(kc)y(HflG) = B(kd)h n = (constant) > 1 . Then c > d (because y(H flG) < h") . Let u = c$. Then B(k|u(x)|)dx :> B(kc)y(HH G) > 1 JG so that II . . 8 > 1 (see K r a s n o s e l ' s k i i and R u t i c k i i [9, p. 73, Lemma 9.1]) and li» I I . . 0 i I • But i f \ a |. < m, then A(|D au(x)|/M) < A(c)y(N(H) flG) < B(kc)y (H fl G) • (1+ j) G 1 = (constant)(1+ y) . 21 Thus, by the above observations, II D u|| <_ (constant) A ^ LT for a l l d < m . Now suppose that the tesselation contains an i n f i n i t e sequence, {H^}, of distinct A-fat cubes. Passing to a subsequence we can arrange that the sets N(H ) are disioint. Let {u } be the function n J n corresponding to {H }. Then {u } is bounded in WV(G) and n n A u IL > l /k > 0 for a l l n. n "B,G — Since the have disjoint supports the sequence {u^} is not precompact in L_(G). • Corollary: The conclusions of Theorem 2 and Theorem 4 in Adams and Fournier [3] are valid. That i s , i f WmEA(G) - L B(G) compactly with A-< B, then |i(G) < °°. Also, for each r > 0, let S r be the surface {xeG | |x| = r}. Let A r denote the (n-1) - dimensional surface area of S . Then r (a) Given e , 6 > 0 there exists an R so that for :. r > R u(G r) < 6 u{x e G | r-e <_ | x | < r} . 22 (b) If is positive and ultimately nonincreasing as r tends to i n f i n i t y then, for each e > 0, A . /A tends to zero 3 ' ' r+£ r as r tends to i n f i n i t y . Note that y(G )tends to zero more r -kr rapidly than e for any r. Proof: The proof is exactly the same as that in Adam's and Fournier [3] and is based on the fact that there are only f i n i t e l y many X-fat cubes. Remark 1.1: The proof of Theorem 1.1 also shows that i f WmEA(G) -»- Lg(G) for some B with A <•< B, then there is no sequence {K^} of X-fat cubes for which AG) "*" 0 as j 0 0. For, given a X-fat cube H choose c so that B(c) y ( H n G ) = 1 . Let u = c<j> is the function defined in the proof of Theorem 1.2). We then have || u || ^ ^ > 1 because f B(|u(x)| dx > B(c) u(HflG) = 1 . G Now f i x e > 0. For |a| <_ m [ A( | Dau(x) | ,/ £M) dx < A(c/e)u(N(H) n G) J G < (1 + 1/X) • y(H QG) • A(c/e) = (1+1/X) Ai°/^ , because y(HflG) = — B ( c ) , u ^ a ^ B ( c ) Since A « B , A(c/e) B(c) as c -V 00 But as u(Hfl.G) -»• 0 , c ^ - o o Thus, for u(HflG) sufficiently small, A(|Dau(x)|/eM) dx < 1 for a l l Then for a l l Thus a < m Dau/£M||AjG < 1 a < m u . 0 < eM and "m,A,G UHB,G > 1 Since £ is arbitrary this contradicts the continuity c tft^CG) + L B(G) 24 Thus i f W^CG) Lg(G) for some B>"V-A, then either (a) y(G) = 0 0 and y(G^ - G ^) does not tend to zero or (b) y(G) < 0 0 and u(G^) tends to zero very rapidly. We w i l l now present a theorem which provides a necessary condition for compactness in the case that the N-function A does not satisfy the A^condition. Theorem 1.2: Suppose there exists a function u in WmLA(G) such that A( |u(x) I) dx = 00 , JG yet D au is bounded on every bounded subset of G for a l l a with |a| _< m. Then the imbedding WmEA(G) -y L A(G) is not compact. Proof: From the hypothesis i t i s clear that we can form a disjoint sequence of shells { Q > n n n=l with Q*. = G * " GK (a n + 2 < b < a ) n a b n-1 n n n n such that l A(|u(x)|dx > 1 for every ri. Q flG n 00 Choose a C function <f> so that n 1 i f x£ Q n * n ( x ) { 0 i f Ixl < b -1 or | x |' > a + 1 , 1 1 n 1 1 n and so that, for a constant K independent of n, we have D <b < K rn 0 0 — for a l l n and a l l a with |a| <_ m. Let TJ = u • <{>n . Then JL,A,G - K ' I l - u ' lm,A,G Hence ty e w V ( G ) for a l l n. However rn A | A(|^ n(x)|) dx > 1 , so that < M * J I A . G > is bounded below. Since and ty^, have disjoint supports for n ^ n ' , ||* N- V " I A , G > M * n l l A , G > 1 Now we have ity } bounded in WmEA(G) but not precompact in L A ( G ) thus WmE(G) + L A(G) is not compact. • An example of a domain G and an N-function A where the situation of Theorem 1.2 arises has been given by Fournier [7]. Example 1.1: Let 7 _ 2 G = {(x,y) eR | x > 0, 0 < y < e x } . 2 If we let A(x) = e -1, the situation is that of Theorem 1.2 and the imbedding i s not compact. To see this, let u : (x,y) x. Note that ueL^CG), although A( | u(x) | ) dx = °° . JG t i i r°° 2 2 . , u(x)L j r -3x /4 -x , A( J—^ dx = . [e - e ] dx < 1 J G J 0 so that - I'U II A,G 1 2 • Similarly, D°u £ L A(G) for a l l a . bounded subset of G, for a l l a . Clearly, D u is bounded on every a 27 The last necessary condition we give is a simple geometric one. Theorem 1.3: If WmEA(G) -> L A(G) is compact then G has only f i n i t e l y many components, Proof: Suppose G has a sequence of components Define -1 1 u = A ( *~ v) on Q n U(Qn) n and u = 0 elsewhere on G. n This means that A(|u n(x)|)dx =••= y(Q n) A(A 1( 7^ -y ) ) = l G n We now have a sequence, { u } . n n = 1 bounded in W™EA(G) yet not precompact in L A ( G ) . • 28 2. A Sufficient Condition for the Compactness of W^ Y^ CG) L^(G) In this section we give a sufficient condition for the com-pactness of imbeddings WmLA(G) - L A(G) , with the restriction that A must satisfy the condition. First we need a lemma. Lemma 1 . 1 : (Adams [ 4 , p. 2 4 8 , Lemma 8 . 3 1 ] ) Let U E W | ^ ( G ) and let f satisfy a Lipschitz condition on R. Let g(x) = f(|u(x)|). Then 8 £ W l o c ( G ) ' a n d Djg(x) = f'(|u(x)|) • sgn u(x) • D u(x) . Definition 1.2: (Adams and Fournier [ 3 , p. 5 2 6 ] ) . By a flow on G we mean a C^ map <J> : U -> G, where U is an open set in G x R"*" containing Gx{0}, and where <J>(x, 0) = x for a l l x in G. Theorem 1.3: Suppose A satisfies the condition. Let G be an open set in Rn for which: (a) there is a sequence ( H i , . of open subsets of G such that x ^ N N=l for a l l N the imbeddings w l L A ( V LA<V are compact. (b) there is a flow <J> : U -»• G such that i f G„ = G - H then: N N 29 ( i ) G Nx[0,l] cU for each N ( i i ) the function <j) :x (j)(x, t) i s one to one for a l l t <\ ( i i i ) <()(x,t) | <_ M for a l l (x,t) i n U; (c) the functions d (t) = sup | det (J)1 (x) | ^ X £ G N s a t i s f y : ( i ) d N ( l ) -> 0 as N -»• 0 0 and ( i i ) I d (t)dt -»- 0 as N -* 0 0 ; 0 (d) y(G) < » Then the imbedding W^"LA(G) L A ( G ) i s compact. Proof: Since y(G) < 0 0 and A s a t i s f i e s the condition, EA ( G ) = LA ( G ) • Thus W 1L A(G) = W 1E A(G) , and i t i s enough to prove that W 1E A(G) -> L A(G) i s compact. We proceed as i n Adams and Fournier [3], Let ii;eW?"'"'"(G) loc We want to estimate N>(x)| dx . GN For each x i n G„T we have N 30 Now •g- iK<f> (x)) dt o 9 t 11 | |i|)(ct)i(x))| dx < d N ( l ) | |^(<})1(x))| |detcf>|(x)| dx Also 1 d N ( 1 ) k ( y ) | dy < d N ( l ) | |^(y)| dy GN J ° ^ ijj((j)t(x)) dt dx dx V (<|>t(x))| | ^ * t(x) | dt JN < | dt 0 | Vi|> (* t(x)) | Mdx < M d N(t) dt Vij; (4>t(x)> | | det <J>J.(x) | dx < M { d N(t) dt} {J |V^(y)|dy } . Letting 6 N = max(d N(l) , M | d N(t) dt ) we have 31 i | K x ) | d x < 6 " •[ JxpCx) | + [ Vip(x) | > dx <_ 6 N" Y " 1,1,G and 6„ + 0 as N ->• N Now suppose uEC^(G) is a bounded function, and || u II •]_ A G 1/2 Let lp(x) = A(u(x)) (extend A to be even on (- °°, °°)) . Then f dx f A(|u(x)|) J a. N < 6 „ { [ ( A ( | (x)|) + I |Da A(u(x))|) dx } W JG lct|=l Since we have U"A,G K 1 • A(|u(x)|)dx < 1 Fix a with |oc[ =1. Lemma 1.1 applies, with f = A, because u is bounded and A satisfies the Lipschitz condition in every f i n i t e interval ot ot ^ [9, p. 5]; therefore D A(u) = A'(u)«D u . Let A be the function complementary to A. Then by Holder's inequality D aA(u(x))|dx = I | A' (u(x)y| |Dau(x)| dx ,ot < I | A - ( U ) | | X G . | | D U | | A > G 32 since < II A'(u) lfy>G , B A " IIA.G i II»III.A.G i 1 / 2 To estimate || A'(u) |K we use Krasnosel 1skii and Rutickii [9, p. 73, lemma 9.1]. Recall that their norm is at most twice the Luxemburg norm; thus " « A , G i 1 By their lemma 9.1, A(A'(u(x))dx < 1 Therefore c \ and then A'(u) ||^ G < 1 Combining these estimates, we have that, i f u is real valued, U II 1,A,G 1 1 / 2 > A(|u(x)|) dx < (n + 1) 6 N , which tends to zero as N tends to i n f i n i t y . Let 33 u i f x e G N V X ) = {0 i f X E H } N Then | A d u ^ x ) ! ) dx < (n + 1) 6 N • G It follows, by the argument of Lemma 0 . 1 , that, with the above assumptions on u, " " A . G - £ N ' where . eXT -> 0 as N 0 0 N Thus for arbitrary real-valued u i n we have IK"A,G i 2 S « « II 1.A.G • ( 1 > Finally, since c1 (G) fl{set of bounded functions) is dense in W*EA(G) , we have (1) for a l l u in W^E^G) . Now by ( 1 ) , an argument similar to Lemma 0 . 2 , and the assumption W \ ( V * LA<V is compact for a l l N, we have that 1 W E A(G) -»• L A(G) is compact. • Remark 1.2 I t i s clear that i f wh^CG) •+ L A(G) i s compact then W mL A(G) - L A(G) i s compact. Remark 1.3 The hypothesis of theorem 1.3 i s the flow hypothesis i n Adams and Fournier [3], Notice that the examples of domains s a t i s f y -ing the flow hypothesis given by Adams and Fournier apply to the O r l i c z space case. For example W mL A(G) - L A(G) i s compact for the horn of the i r example 2, provided A s a t i s f i e s the &2 condition. Remark 1.4 The domain described i n Example 1 . 1 . s a t i s f i e s the hypothesis of Theorem 1.3. The noncompactness of the imbedding does not contradict the theorem, because the N-function considered i n Example 1.1 does not s a t i s f y the condition. 3. Applications to Bounded Domains The flow condition can also be used to show the compactness of W ^ G ) + L A(G) for some bounded domains not s a t i s f y i n g the cone condition. 35 Example 1.2: (Given in Adams [4, p. 172, Ex. 6.49]) Let G = {(x,y) eR 2 : 0 < x < 2, 0 < y < f(x) } where f e C1(0,1) is positive, nondecreasing, has bounded derivative f , and satisfies lim f(x) = 0 . x+0+ Let U = {(x,y,t) eR 3 : (x,y) EG, -x < t < 2-x } and define the flow <f> : U -* G by Thus cKx.y.t) = { x+t, ^f^-y} j . i | / , f(x+t) det <J)t(x,y) = - j ^ r - . If then G* = {(x,y) eG | x > 1/N } d N ( t ) = S U P 1 ffx+tY N 0<x<l/N *lx+t; satisfies d N(t) < 1 for 0 < t < 1 and 36 lim d N ( t ) = 0 i f t > 0 . N+co Hence also l i m [ d (t) dt = 0 by dominated convergence. Since G* i s bounded and has the cone property, and since the boundedness of -TT- i s assured by that of f , we have, o t by Theorem 1.6, the compactness of W mL A(G) + L A(G) whenever A s a t i s f i e s the An condition. 37 CHAPTER 2 IMBEDDINGS OF W™LA(G) In this chapter we w i l l give some necessary and sufficient conditions for the compactness of the imbeddings W^L^G) L^CG) . The results are analogous to those given for Sobolev spaces by Clark [5] and Adams [1], [2], and [3]. 1. Necessary Conditions for the compactness of W^ L^ CG) -*• We begin with a simple geometric condition on a domain G given by Clark [5]. We say that a domain G is quasibounded i f dist(x, 3G) -»- 0 as - |x| + °° in ' G . Clearly G is not quasibounded i f and only i f i t contains an in f i n i t e sequence of disjoint congruent balls {B j • Note that an n=l unbounded domain satisfying the cone condition is not quasibounded. oo Let <j) £ CQ (G) with support in B^ for one of these balls and with r A(<|>(x))dx > 0 . B, Consider the sequence {(j) } of translates of <j> with <p having support n n in B . {d> } is bounded in W^ L. (G) but bounded away from zero in n n U A L.(G) . Since the supports are disjoint this means {(J) }iis not pre-compact in L A(G) which means 38 woVG) r L A < e > i s not compact. Quasiboundedness i s , therefore, a necessary condition for compactness. We w i l l now give a necessary a n a l y t i c condition on G for the compactness of ^ L A ( G > * V G ) where A s a t i s f i e s the condition. This i s analogous to C™' P defined i n Adams [1]. F i r s t we w i l l need some d e f i n i t i o n s . I f H i s an n-dimensional n 0 0 cube of side h i n R and E a closed subset of H we denote by C (H,E) the class of a l l i n f i n i t e l y d i f f e r e n t i a b l e functions on H which vanish i d e n t i c a l l y i n a neighbourhood of E. We define the f u n c t i o n a l XA,H ( U ) " For each 6 > 0 , A( I h | a ||D°u(x)|) dx H l.<|a|<m I» ( 2 n 1 / 2 u ) C m . (H,E) - i n f A,6 fi n over a l l u i n 00 C (H,E) with A(|u(x)|)dx = 6h n H 39 D e f i n i t i o n 2.1: A domain G w i l l be said to s a t i s f y the C"' condition i f for every e > 0 and 5 > 0 there e x i s t s h^ > 0 such that f o r every h, 0<h< hp there e x i s t s r > 0 such that for every n-cube H of side h meeting G^ we have C™ r. (H,H-G ) > h/e A,o v ' r We w i l l also need a lemma due to Fournier [7] which i s analogous to lemma 1 i n Adams [1]. Lemma 2.1 Suppose that DCH has p o s i t i v e measure and that u e C (H) Then r A(|u(x)|) dx H y(H) {] A(2|u(x) |) dx + C I A(2n 1 / 2h|grad u(x) | ) dx } J D J H Proof: Let x £ H and y £ D . Then where u(x) = u(y) + v = (x-y)/|x-y x-y v • grad u ds and s measures distance along the l i n e from y to x. Let (r,a) denote polar coordinates centred at y so that the boundary of H i s given by r = f (a) , a e Z . By the convexity of A we have AO|u(x)|) = A(|u(y) + f I x-y v • grad u d s ) i i \ Ix~yI < Y A(2|u(y)|) + \ A(2|J v • grad u ds |) Integrating the inequality over H we have A(|u(x)|)dx < 1/2 ( A(2|u(y)|)dx H JH + r r|x-y| A(2| v • grad u ds | ) dx H JO < 1/2 h n A(2|u(y).|) + < to f £ < 0 > -,t(0) rr r dr {A(2 | grad u | ds ) } J 0 J 0 Using Jensen's integral inequality we get A(|u(x)|)dx < hnA(2|u(y)|) r r f ( a ) n - i . da• r " "dr { E Jo A(r2 j grad uj ) ds } < hnA(2|u(y)|) + da f (a ) n-2 , r r dr { ( A(n 1 / 2 h 2|grad u|)ds} J 0 since r < n 1 / 2 h F i r s t note that r < f ( a ) in the last inside integral and then that 1/2 f ( a ) < n h . We then have A(|u(x)|)dx < hnA(2|u(y)|) H + f , 1/2, n-1 rf ( a ) , d a ( n _ i O A(n 1 / 2h|grad u|) Z n-1 JO dr = hnA(2|u(y)|) + . 1/2. .n-1 (n h) n-1 f A(n 1 / 2h|grad uj) H x-y i n-1 dx i f we now integrate over D and use the fact that D | x-y n-1 C1(n){y(D)} 1/n (see Hellwig [ 8 , p. 5 7 ] , we get y(D) A(|u(x)|)dx < h 1 1 A(2|u(y)|)dy H , 1/2. .n-1 + i s _ J y C.(n){y(D)} n-1 D 1/n A(2n 1 / 2h|grad u|) dx H Since . {y(D)} 1 / n < ( h n ) 1 / n = h this becomes 42 y(D) A(|u(x)|)dx < h n A(2|u(y)|)dy H n-1 + h n ( n _ 7 n-1 Now, letting n-1 C^n)) j A(2n 1 / 2h|grad u|)dx 2 C (n) n-1 we have the lemma. • Theorem 2.1 If W^ L.CG) -»• L.(G) is compact then the domain G OA A m satisfies the condition C. A ,m Proof: Suppose that G does not satisfy the condition C~ . Then there exists k = — < 0 0 and 6 > 0 such that for every h. with e J 0 0 < h^ < 1 there exists h < h^ such that there exists a sequence 1.H.J- of mutually disjoint cubes of side h meeting G such that J j=l •^m CA .(H.,H.-G) < kh . A,6 2 3 By the definition of capacity for each cube H^. there exists a function 00 u. £ C (H.,H.-G) such that J 3 3 r A(|u.(x)|)dx = 6h n H. 2 3 and 43 A< I h > l | D a 2 n 1 / 2 u ( x ) | ) l.<Ja|<m H. 6 h n dx < k h Certain l y [ A(2n 1 / 2h|grad u(x)|) dx < k 6 h n + 1 H. Also Now l e t D. be the set J {x|A(2|u (x)|) < .6/2} . By lemma 2.1 we have 6 h- n 1 <f u ( D j ) + c i k 6 h I 1 + 1> jf.n 6C.kh o h + 1 2n+l y ( o . ) which implies that C.kh 2n+l y ( o . ) 44 or y(D..) < C 2 h n + 1 Choose h small enough that C 2h < 1/3 We then have y(D..) < 1/3 y (H ) Choose functions w. £ C_ (H.) such that w.(x) 1 on 3 0 3 3 a subset of having measure no less than (2/3)y(H_.) and such that sup max sup |Daw.(x) | = k* < 0 0 , j |a[<m x£H_ 2 for a constant k* that is independent of j . Then and on v. = u. w. £ C.(H.n G) C C.(G) 3 3 3 0 3 0 A(2|v (x)|) > 6/2 S. fl (H. - D.) , 3 3 3 a set of measure not less than (l/3)h n 45 Then on A(|v (x)|) > C > 0 S. fl (H. - D.) 3 3 3 for some constant C^, so that H. J A( v.(x) ) dx _> h C By remark 0.1 (b). J We have A( | D V (x) j • I D3w_. (x) |) dx < C**(h) for a| , |@| < m and by remark 0.2 i t follows that is a bounded sequence in W L (G). Since the functions {v.} have disjoint supports we have (J A j j k " A,G — 3 n so that the sequence ^ v j ^ is n o t precompact in 46 L A(G) and WQ L^G) •+ L^G) cannot be compact. • 2. Sufficient Analytic Conditions for Compactness of V A ( G ) ^ L A ( G ) We give an analytic condition on G which i s , again, analogous to C m' P in Adams [1]. n Definitions 2.2 If H is an n-dimensional cube of side h in R define the functional max l<Ja|<m JH A(|Dau(x)|) dx m e * and V £ > 0 define the (C ' ) capacity of a closed set E in H by (C™' ) (H,E) inf oo u e C (H,H-E) H A domain G w i l l be said to satisfy the (C™) condition i f there exist a constant C such that for every e > 0 there exists h <_ 1 and r > 6 such that, for every cube H of side h meeting G^, we have (CA'E)*(H,H-Gr) > Ch ITi " Theorem 2.3 If a domain G satisfies the (C^) condition, then the imbedding W0LA<G> * L A ( G ) is compact. Proof: By lemma 0.2 we need only show that || P || 0 as r 0 0 . Given e > 0, choose r and h as i n definition 2.2. Let u e W^L^CG) such that || u || < 1 . 11 "m,A — TO ^ Tesselate G with cubes H.. By the definition of the (C.) condition r l 3 A we then have A ( Ju&LL ) d x G e < Z 1=1 0 0 ( 1 • A( 1?J£1 ) dx H. £ 1 00 . < 1/C Z max A(|Dau(x)I) dx i=l l<|:a|<m 'H. —' -•— 1 00 r < C, max Z A(|Dau(x)|) dx l<|a|:<m i=l -"H. < C n because || u || . < 1 — 2 1 ' m,A — 48 Hence P r U l l A , G ± £ C 3 » for a l l such u, and P < e C, r — : De f i n i t i o n s 2.3 If H i s an n-dimensional cube of side h i n R , define the fu n c t i o n a l - r m t ^ ^ H ( U ) = l<|a|<m A(h' a'|D au(x)|) dx H and, given e > 0 , define the C H by ,m,£ capacity of a closed set E i n C™' £(H,E) i n f u £ C°°(H,H-E) T A . H ( U ) A( u(x) /£) dx H A domain w i l l be said to s a t i s f y the condition i f , for every £ > 0, there e x i s t s h < 1 and r ^> 0 such that, f o r every n-cube of side h meeting G r, C m ' £ (H,H-G ) > h A r 49 Remarks 2.1 (a) It is easy to see that, for a domain G, so that C™ i s a sufficient condition for compactness. (b) Like Adams' C m' P , and C. ~ have been constructed so n ' A A , 0 that the capacity of a closed set E c A is invariant under magnification of H and E by the same factor. (C™ , £) is not invariant. m^ (c) The term h on the right side of the C„ condition can be M replaced by <j)(h) for any function (J) such that <J)(h) 0 as h -»- 0. (d) If there exists a constant D such that for every a > 0, b > 0, and c > 0 A(cb) A(b) then i t can be shown that, for a domain G, Cm <=> am <=> (Cy . A A A In this case is necessary and sufficient for compactness of W 0 L A ( G ) - LA< G ) In particular, when tP A(t) = J L . , the conditions C™, C™ , and (C™) are a l l equivalent to the condition C m' P given by Adams [1 ] . 3. Sufficient Geometric Conditions for the Compactness of WmLA(G) •* L A(G). i Probably the simplest sufficient geometric condition on G is f i n i t e volume. By examining the proof of compactness given by Donaldson and Trudinger [6] one can see that i f ]i (G) < 0 0 , then W Q L A ( G ) -> L A ( G ) is compact. Next we give two geometric conditions due to Clark and Adams which w i l l be shown later to imply another that is sufficient for compactness. Definition 2.4 (Clark [5]) A domain G w i l l be said to satisfy Clark's Condition 1 i f to each R ^> 0 there corresponds positive numbers d(R) and <5(R) satisfying (a) d(R) + 6(R) -> 0 as R 0 0 0>> M | I < M < - f o r a n R 6(R) — (c) for each x e G with |x| > R there exists y such that | x-y | < d(R) and Gfl{z||z-y| < 6(R)} = <|> . 51 Definition 2.5 (Adams [2]) A domain G w i l l be said to satisfy Adams condition 2 i f there exists RQ > 0 such that to each R > R^ there correspond numbers d ( R ) , 6(R) > 0 such that (a) d(R) + 6(R) -> 0 as R ->• °° , (b) d(R)/S(R) < M < 0° for a l l R > Rfl , (c) for each x e G such that |x| >R >_ RQ the b a l l B 3 d ( R ) ( x) is disconnected into two open components C^, and C^ by an n-1 dimensional manifold forming part of the boundary of G in such a way that each of the two open sets c - f l B j ^ \ W i = 1>2 contains a b a l l of radius 6 (R) x d (R) Definition 2.6 A domain G w i l l be said to satisfy the condition F i f there exists a 6 > 0 such that, to every e > 0, there correspond numbers h and R, with 0 < h <_ e and R > 0, such that for every n-cube H of side h meeting G , K U , (H,G) n-l (3) where u (H,G) n-l is the maximum, taken over a l l projections P onto the n-1 dimensional face of H, of the n-1 measure of P(H-G). 52 Theorem 2.4 If a domain G s a t i s f i e s the condition F A' then W S V G ) * V G > i s compact f o r m >_ 1. Proof: Suppose we are given £ > 0. Then consider £^ = e/2n. We then have numbers h and R depending on £^ as described i n d e f i n i t i o n 2.6. Pick a cube H meeting G . Let P be the maximal K pro j e c t i o n r e f e r r e d to i n the d e f i n i t i o n of y n_^(H,G) and l e t E = P(H-G). Without loss of generality we may assume that the (n-1) dimensional face F of H which contains E i s p a r a l l e l to the x^, ' " , x^ coordinate plane. For each point x = ( x 1 , x") i n E where x' = x, and x" = (x , x ) l e t H be the segment of length h i n H which i s contained i n the l i n e through x normal to F. By the d e f i n i t i o n of oo P there exists y £ H „ - G. Suppose u £ C„(G). We then have This implies I! X H A(|D u(x' , x") dx') by Jensen's inequality. Integrating over H^ ,, we get H „ x A(|u(x',x") |/e')dx < A(|D l U(x',x") | )dx' H Now notice that Du(x',x") | < |grad u(x',x") so that we have A( Iu(x',x ") |/£')dx' < A(|grad u(x',x") | ) dx' H „ x Integrating this inequality over E and denoting {x'|x = (x',x") £H for some x" } we obtain A(|u(x)|/e') dx < C A(|grad u(x)|)dx H xE -"H 53 Now rec a l l lemma 2.1 which says that A(|u|/e). < h ^ U(E) { f A(2|u|/e) + C f A(2n1 / 2h (.|gradu |/e)} J„l „ JH H1xE Recall that and _ 1/2,.- ,2nh, , 1-2 , 1/2 , v . 1/2 2n h/e = (——) / n = h/(n e') <_ 1/n 2|u|/e = 2|u|/2ne' < |u|/n 1 / 2e' , so that we have n-1 HA<H'e> i V 7 ( H ^ ) { j A(|u|/n 1 / 2e') + cf A([grad u j / n 1 7 H xE J H 1/2 Now we use inequality (3) and (4) , replacing u by u/n , to get ( A ( M / £ > < u ^(H,G) { C 2 { H A ( ' g r a d U ' / n l / 2 ) + C{ A ( | S ^ 1 / 2 U | ) } < C 1/6 f A(|grad u|/n 1 / 2) (5) Note that grad u| , | <x I •\ i<) £ m a x ID u| • K-|a|<m =•>' A(|grad u|/n 1 / 2) <_ A( max |D°\ I | ) l<Ja|<m. max A(|Dau|) 1<I a I<m => f A(|grad u|/n 1 / 2) < max A(|D a U| J H l<|a[<m _< (constant) Z 1 <|a|<m f A(|Dau|) H < (constant)' max A(|Dau|) ~~ l<|a|<m •'H so that we have, from (5), A(|u(x)|/e ) dx l<[a <m A(|Dau(x)|) dx This means A(|u(x)|/e) dx. > t h 55 m * Thus the domain G satisfies the (C^) condition and W 0 L A ( G ) + L A ( G ) is compact for m > 1. • Remark 2.2 Evidently, for a domain G, each condition we have given i s weaker than the proceeding condition. Finally we give a geometric condition which ensures the compact-ness of W 0 L A ( G ) -* L A ( G ) and which varies with m. When m > 1 we w i l l assume, as in Adams [4], that 1 a _ 1 ( T ) - dT < - . . 0 T(n+l)/n We w i l l need some notation. Given an N-function A we define a sequence of N-functions BQ, B^, *'* as in Adams [4, p. 258]. B Q(t) = A(t) 56 At each stage we assume that f l (B k ) V ) 6 (n+l)/n dx (6) replacing B^, i f necessary, by another N-function equivalent to i t near i n f i n i t y and s a t i s f y i n g (6). Now l e t 0) = CO (n,A) be the smallest non-negative integer such that r B- (T) , < oo 1 T(2n+l)/n d T It can be shown that w(n, A) < n. D e f i n i t i o n 2.6 We w i l l say that a domain G s a t i s f i e s the K™ condition i f m > co(n,A) and there exists a sequence of p o s i t i v e integers { k(r) | r = 1, 2, 3, ••• } with the following properties: (a) S r 1 1 - 1 ( k ( r ) ) ? _ 1 / n < « r=l for some £ such that 0 < E, < 1/n , (b) for every x £ G^ we have d(x,3G) < ^ where d(x,8G) is the distance between x and the boundary of G. Theorem 2.5 If a domain G satisfies the K™ condition then A W 0 L A ( G ) ^ L A ( G ) is compact. In the proof of theorem 2.5 we w i l l use the following lemma. Lemma 2.2 Let e > 0 and 0 < £ < 1/n. Suppose B is an N-function such that •1 T(2n +l)/n d T < 0 0 and A i s any N-function. Let H be a cube of side h such that ^(n/(n+l))£ < £. Then there exist positive constants c^ and C2> independent of h, such that f M ^ ! ) d x JH < c {h n f B(|grad u(x)|)dx + h n + 1 / n _ ? } CO for a l l functions u in C (H) which satisfy the inequality • B(|grad u(x) | ) dx <_ c and vanish in a neighbourhood of some point y £ H. 58 Proof: Suppose H i s a cube of side h such that ^ n ^ n + ^ ^ < £ i L e t tH be a concentric cube of side th. For every x e H we have, from Adams [4, p. 255], that u(x) - h -n r u(z) dz | • — < n 1/2 •1 r _ n h 1 _ n dt | grad u(z)| t — d z 0 J t H £ Let X Now (n/n+l)£. We then have h^ < e or ~ < h"*" n X |grad u(z)| t n ^ 1-n (7) < B(|grad u(z)|) + B(t V n X) by Young's in e q u a l i t y and (7) . Thus |u(x) - h n u(z)dz | • — J H e < n 1 / 2 ^ d t f {B(|grad u(z)|) + B ( t ~ n h 1 " n _ X ) } dz 1 0 J tH <_ n 1 / 2 { B(|grad u(z)|) dz + h 1 1 f 1 ^ ^ ( t ^ 1 " 1 1 ^ ) dt } J0 59 < n 1/2 B( grad u(z) ) dz H ^ . ..-1/2. 1/n-X/n-X + n n B ( T ) .(2n+l)/n dx 1-n-X < n 1 / 2 B(|grad u(z)|) dz + n - l / 2 h l / n - C oo % B(X) 1 X (2h+l)/n dx Now i f x, y e H and x-y < h < 1 then u(x) - u(y) < 2n 1/2 B( grad u(z)| dz H + 2 n - l / 2 h l / n - e .00 'Xi B(X) 1 X (2n+l)/n dx I f we choose y such that u(y) = 0 then t00 |u(x)|/e <_ 2 n 1 / 2 B(|grad u(z) |) dz .CO <\j + 2 n - l / 2 h l / n - g B^X) d T 1 x (2n+l)/n 60 < c { [ B(|grad u(z)|) dz + h 1 / n~^} In Using Jensen's inequality we get A( Jiiool) < 1/2 {A(2c, [ B(|grad u(z)|) dz)+ A(2c„h 1 / m ?)} H < 1/2 A(2c -c ) • — B(|grad u(z)|)dz C l JH + 1/2 h 1 / n ? A(2c 3) < c. { j B(|grad u(z)|) dz + h 1 / n ? } Now integrate over H to get A(|u(x)|/£) dx H 1 c 4 (h r B(|grad u(z)|) dz + h n + 1 / n _ ? } H • 61 Proof of theorem 2.5: Suppose we are given £ > 0. Choose R > 0 such that ( k ( r ) ) " ( n / n + 1 ) ? < £ for every r > R. Consider a unit cube I centered at the origin. Tesselate R n with unit cubes by adding successive layers of cubes T . Notice that n 1 / 2 R r>R r 2 R where r takes on only integer values. Now subdivide each unit cube in T with cubes of side r h(r) k(r) ' Let H = H(r) be one of these cubes of side h = h(r) where r > R. Since d(x,9G) < b^ r^ there exists y £ H - G . Suppose and C O u E C (H, H-G) Then Let U l l m , A , G ± 1 u £ C (H,y) . / 1/2 £' = £/n 62 Applying lemma 2.2 with B = B^, £ replaced by £', and u replaced by u/n , we obtain f A(|u(x)|/n 1 / 2e') dx i =6 t h ° , [ grad u(x) ] > n+l/n-£ u V 1 /1 } d x + h } H n or A(|u(x) | /e) dx < cfi {h [ B (|grad u(x)|/n 1 / 2) dx + h n + 1 / n " ? } . (9) Now suppose that I is a unit cube in T . Sum the inequality (9) over I to get A ( I i ( x ) l ) d x < c ? {h n I B w(|grad u(x)|n 1 / 2)dx + h 1 / n " ^ } Now sum this over to get A(|u(x)|/e) dx + c g r ( h ( r ) ) " } F i n a l l y we have A(|u(x)|/e) dx T R 1 1 r>R A(|u(x)|/e) dx 1 c10 { h n f B n ( l ^ r a d " ( x ) l ) dx Z r ^ C h C r ) ) 1 7 1 1 ^ } . r > R Now suppose that U I' m,A,G < c 11 Then grad u 1/2 B ,G n to' < 1 Hence Jgrad u(x)l < L H W n 1 / 2 Notice also that I r (h(r)) ^ < c r>R Thus we have | A(|u(x)|/e) dx < c 1 3 G 1/2 This implies that || P .|| < c e . nl/2 2 R We then have || P || ->• 0 as r -* 0 0 so that, by lemma 0.2, W 0 L A ( G ) * L A ( G ) is compact. • Remark 2.3 In general we can weaken the condition K™ by replacing (a) in the definition by 65 0 0 i i / c- r00 B (T) , ( h ( r ) ) 1 _ n " W e REFERENCES R.A. ADAMS, Capacity and compact imbedding. J. Math. Mech., 19 (1970), 923-929. R.A. ADAMS, Compact Sobolev imbeddings for unbounded domains, Pacific J. Math., 32 (1970), 1-7. R.A. ADAMS and JOHN FOURNIER, Some imbedding theorems for Sobol Spaces, Canad. J. Math., 23(1971), 517-530. R.A. ADAMS, Sobolev Spaces, Academic Press, New York, 1975. C.W. CLARK, An imbedding theorem for function spaces, Pacific J. Math, 19 (1966), 243-25. T.K. DONALDSON and N.S. TRUDINGER, Orlicz-Sobolev spaces and imbedding theorems, J. Functional Anal. 8 (1971), 52-75. J.J. FOURNIER, Compact imbeddings for Orlicz-Sobolev spaces, Unpublished notes, U.B.C., 1971. G. HELLWIG. Differential Operators of Mathematical Physics, Addison Wesley, Reading, Mass., 1967. M.A. KRASNOSEL'SKII and Ya. B. RUTICKII, Convex Functions and Orlicz Spaces, Noodhoff, Groningen, the Netherlands, 1961.
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Compactness of Orlicz-Sobolev space imbeddings for unbounded domains Cahill, Ian Graham 1975
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Title | Compactness of Orlicz-Sobolev space imbeddings for unbounded domains |
Creator |
Cahill, Ian Graham |
Publisher | University of British Columbia |
Date Issued | 1975 |
Description | In this thesis we are concerned with the compactness of im-beddings, for unbounded domains, of Orlicz-Sobolev spaces. Donaldson and Trudinger [6] have given a generalization of the Sobolev imbedding theorem and the Rellich-Kondrachov compactness theorem for Sobolev spaces to the case of Orlicz-Sobolev spaces on bounded domains. Clark, Adams, and Fournier [1], [2], [3], and [5] have given extensions of the Sobolev compactness results to certain unbounded domains. In this thesis we consider similar extensions for Orlicz-Sobolev spaces. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-01-28 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080117 |
URI | http://hdl.handle.net/2429/19247 |
Degree |
Master of Science - MSc |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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