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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
gori@mail.dm.unipi.it

Paolo Marcellini
Dip. di Matematica, UniversitÓ di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy
marcell@math.unifi.it



[Abstract-pdf]

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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