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Publication 00-CNA-07

A note on Meyers' Theorem in Wk,1

Irene Fonseca, Giovanni Leoni, and Roberto Paroni

Abstract:

Lower semicontinuity properties of multiple-integrals

\begin{displaymath}u\in W^{k,1}(\Omega;\mathbb{R} ^{d})\mapsto\int_{\Omega}f(x,u(x),\cdots ,\nabla^{k}u(x))\,dx \end{displaymath}

are studied when f grows at most linearly with respect to the highest order derivative, $\nabla^{k}u,$ and admissible $W^{k,1}(\Omega;\mathbb{R} ^{d})$sequences converge strongly in $W^{k-1,1}(\Omega;\mathbb{R} ^{d}).$ It is shown that under certain continuity assumptions on f, convexity or 1 -quasiconvexity of $\xi\longmapsto f(x_{0},u(x_{0}),\cdots,\nabla^{k-1}% u(x_{0}),\xi)$ ensure lower semicontinuity. The case where $f(x_{0}% ,u(x_{0}),\cdots,\nabla^{k-1}u(x_{0}),\cdot)$ is k-quasiconvex remains open except in some very particular cases, as an example when $f(x,u(x),\cdots ,\nabla^{k}u(x))=h(x)g(\nabla^{k}u(x)).$

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