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Journal of Convex Analysis 09 (2002), No. 2, 475--502
Copyright Heldermann Verlag 2002

An Extension of the Serrin's Lower Semicontinuity Theorem

Michele Gori
Dip. di Matematica, UniversitÓ di Pisa, Via Buonarroti 2, 56127 Pisa, Italy

Paolo Marcellini
Dip. di Matematica, UniversitÓ di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy


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 \textit{continuity} and \textit{convexity} property of the integrand $f(x,s,\xi )$, only assuming a mild (more precisely, \textit{local}) condition on the independent variable $x\in \Bbb{R}^{n}$, say \textit{local Lipschitz continuity}, which - we show with a specific counterexample - cannot be replaced, in general, by local \textit{H\"{o}lder continuity}.

Keywords: Lower semicontinuity, strong convergence in L1, convex functions, local Lipschitz continuity, local Hoelder continuity, calculus of variations.

MSC: 49J45; 35D05

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