Lower semicontinuity in Sobolev spaces below the growth exponent of the integrand

Jan Kristensen

Research output: Contribution to journalArticle

18 Citations (Scopus)

Abstract

Let there be given a non-negative, quasiconvex function F satisfying the growth condition lim supA?8 F(A)/|A|p = 0 (*) for some p ? ] 1, 8 [. For an open and bounded set O ? Rm, we show that if q? m-1/m p and q > 1, then the variational integral F(u; O):= ?O F(Du) dx is lower semicontinuous on sequences of W1,p functions converging weakly in W1,q. In the proof, we make use of an extension operator to fix the boundary values. This idea is due to Meyers [26] and Mal? [22], and the main contribution here is contained in Lemma 4.1, where a more efficient extension operator than the one in [22] (and in [14]) is used. The properties of this extension operator are in a certain sense best possible.

Original languageEnglish
Pages (from-to)797-817
Number of pages21
JournalProceedings of the Royal Society of Edinburgh, Section A: Mathematics
Volume127
Issue number4
Publication statusPublished - 1997

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