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A note on Meyers' Theorem in

Authors:	Irene Fonseca Giovanni Leoni Jan Malý Roberto Paroni

Institution:	Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 ; Dipartimento di Scienze e Tecnologie Avanzate, Università del Piemonte Orientale, Alessandria, Italy 15100 ; Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83,186 75 Praha 8, Czech Republic ; 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 may grow 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 convexity, -quasiconvexity or -polyconvexity of $\begin{displaymath}\xi\mapsto f(x_{0},u(x_{0}),\cdots,\nabla^{k-1}u(x_{0}),\xi)\end{displaymath}$ ensures lower semicontinuity. The case where $f(x_{0},u(x_{0}),\cdots,\nabla^{k-1}u(x_{0}),\cdot)$ is -quasiconvex remains open except in some very particular cases, such as when $f(x,u(x),\cdots,\nabla^{k}u(x))=h(x)g(\nabla^{k}u(x)).$

Keywords:	$k$-quasiconvexity higher-order lower semicontinuity gradient truncation

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