Calculus of Variations and Geometric Measure Theory
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M. Gori - P. Marcellini

An extension of the Serrin's lower semicontinuity theorem

created on 13 Dec 2001
modified by gori on 27 Dec 2006

[BibTeX]

Published Paper

Inserted: 13 dec 2001
Last Updated: 27 dec 2006

Journal: J. Convex Anal.
Volume: 9
Number: 2
Pages: 475-502
Year: 2002

Abstract:

In this paper we present a new extension of a celebrated Serrin's lower semicontinuity theorem. We consider an integral of the calculus of variation $\int_{\Omega }f\left( x,u,Du\right) dx\,$ and we prove its lower semicontinuity in $W_{loc}^{1,1}\left( \Omega \right) $ with respect to the strong $L_{loc}^{1}$ norm topology, under the usual \QTR{it}{continuity} and \QTR{it}{convexity} property of the integrand $f(x,s,\xi )$, only assuming a mild (more precisely, \QTR{it}{local}) condition on the independent variable $x\in \QTR{Bbb}{R}^{n}$, say \QTR{it}{local Lipschitz continuity}, which - we show with a specific counterexample - cannot be replaced, in general, by local \QTR{it}{H\"{o}lder continuity}.

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