A note on Meyers' Theorem in Wk,1

Irene Fonseca
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213

Giovanni Leoni
Dipartimento di Scienze e Tecnologie Avanzate
Università del Piemonte
Orientale, Alessandria, Italy 15100

Jan Malý
Department of Mathematical Analysis
Faculty of Mathematics and Physics
Charles University
Sokolovská 83,186 75 Praha 8, Czech Republic

Roberto Paroni
Dipartimento di Ingegneria Civile
Università degli Studi di Udine
Udine, Italy 33100

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, 1 -quasiconvexity or k-polyconvexity 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, such as when $f(x,u(x),\cdots,\nabla^{k}u(x))=h(x)g(\nabla^{k}u(x)).$


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