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

Published online by Cambridge University Press:  14 November 2011

Jan Kristensen
Jan Kristensen
Affiliation:
Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, U.K
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Abstract

Let there be given a non-negative, quasiconvex function F satisfying the growth condition

for some p ∈]1, ∞[. For an open and bounded set Ω⊂ℝm, we show that if

then the variational integral

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 Maly [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.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 127 , Issue 4 , 1997 , pp. 797 - 817
Copyright
Copyright © Royal Society of Edinburgh 1997

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