WEAK CONVERGENCE 7 If U denotes as usual a bounded smooth open subset of R , it follows from (1.4) and Standard extension theorems that (1.5) u\\ L p lV) c\\n wl ., (ü) for each 1 ñ q* and / e Wl,q(U), the constant C depending only on q , ç , and U. 2. Compactness theorems. Our next assertion sharpens the preceding Ob- servation, demonstrating that the embedding W ,q (U) c L (U) is in fact compact if 1 q n , 1 p q*: THEOREM 5 (RELLICH'S COMPACTNESS THEOREM). Assume that the se- quence {fk}™ is bounded in Wl,q(U). Then {fk}°?_ is precompact in L (U) for each 1 ñ q*. Á proof of this assertion is available in Gilbarg-Trudinger [68]. We will later devote considerable effort to understanding just how com- pactness fails for the critical case ñ = q*: see §D below and Chapter 4, §A. THEOREM 6 (COMPACTNESS FOR MEASURES). Assume thesequence {ßk}T_ is bounded in Jt(U). Then {ßk}°° is precompact in W~ ,q {U) for each 1 ú 1 *. oo PROOF. 1. Using Theorem 4 we may extract a subsequence {/^ } . c CO {ì^ } so that ßk — * ì in J?{U), for some measure ì € . 2. Set q =q/(q-\) and denote by Â the closed unit ball in W^q (U). Since 1 q 1*, we have q n\ and so Â is compact in CQ(U). Thus Í(å) — given å 0 , there exist functions {/,} _ c CQ(U) so that min I ö - ö. I ^ _ å liN(e) l C(U) for each ö e Â. 3. Thusif 0 €# , \ öÜì. - öÜì\ 2å sup |ì , |(ß/) + / 0. Üì, - É ö.Üì \Ju J Ju É j j \Ju J Ju for some index 1 / Í (å). Consequently, ' = 0 , lim sup and so ì ßò - ì in W~Uq{U) / öÜì]^- \ öÜì Ju j Ju

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