Geodesic
An Extension of the Serrin's Lower Semicontinuity Theorem
Journal of convex analysis, Tome 9 (2002) no. 2, pp. 475-502 Cet article a éte moissonné depuis la source Heldermann Verlag

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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}.
Classification : 49J45, 35D05
Mots-clés : Lower semicontinuity, strong convergence in L1, convex functions, local Lipschitz continuity, local Hoelder continuity, calculus of variations 
@article{JCA_2002_9_2_JCA_2002_9_2_a9,
     author = {M. Gori and P. Marcellini},
     title = {An {Extension} of the {Serrin's} {Lower} {Semicontinuity} {Theorem}},
     journal = {Journal of convex analysis},
     pages = {475--502},
     year = {2002},
     volume = {9},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/JCA_2002_9_2_JCA_2002_9_2_a9/}
}
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M. Gori; P. Marcellini. An Extension of the Serrin's Lower Semicontinuity Theorem. Journal of convex analysis, Tome 9 (2002) no. 2, pp. 475-502. http://geodesic.mathdoc.fr/item/JCA_2002_9_2_JCA_2002_9_2_a9/