By Victor I. Burenkov

ISBN-10: 3815420687

ISBN-13: 9783815420683

Read Online or Download Sobolev Spaces on Domains (Teubner-Texte zur Mathematik; 137) PDF

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Extra resources for Sobolev Spaces on Domains (Teubner-Texte zur Mathematik; 137)

Example text

SOBOLEV SPACES (BASIC PROPERTIES) 31 We shall also need the following variant of Sobolev spaces. Definition 6 Let 12 C R" be an open set, l E N, 1 < p < oo. The function f belongs to the semi-normed Sobolev space w,(f2) if f E L, 0c(12), if it has weak derivatives D: f on fl for all a E l satisfying I a I= I and IID*fIILP(n) < oo. 23) 1a,=( The space wp(fZ) is also a complete space (the proof is similar to the proof of Theorem 3). Thus w,(1l) is a semi-Banach space, because the condition If llwP(n) = 0 is equivalent to the following one: on each connected component of an open set ) f is equivalent to a polynomial of degree less than or equal to l - 1 (in general different polynomials for different components).

C9 jxja-1Q1jlogjxjI"-O, where cg > 0 does not depend on x. Finally, use that for some clo, c,, > 0 I/2 f g(axl)dx = cio f 9(p)p"-idp, B(0,1/2) f 1/2 g(IxD)dx = cii B(0,1/2)nK 0 f g(p)p"-idp. 0 Example 9 Let 1 < p < oo. Under the suppositions of Example 8 jxjµ(logjxj)" E Wo(eB(0, 2)) if, and only if, p < -n/p, v E R or p = -n/p, v < -1/p. ENo,e:v=0orp

48) and II Da(Aaf)II L,(C) 5 C66'-'Q1 11f1014(01), lal ? 49) where c5, c6 > 0 do not depend on f, 5, G and p. (For instance, one can set ca = II(IIL,(n") and ce = max Is1=IaI-1 CHAPTER 2. APPROXIMATION BY COO-FUNCTIONS 64 Idea of the proof. 46) applied to Dwf E w1 I°I(Q). 45). 21)) D°(A6f) = 6IRI-I7I(D°-Iw)6 * Dwf , where ry E too is such that 0 < ry < a and (rye = 1. 0 Let S2 be an open set and let the "strips" Gk be defined as in Lemma 5 if i1 0 R° and as in Lemma 6 if 11 = )R". Moreover, let {ii*}kEZ be partitions of unity constructed in those lemmas.