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A note on Meyers' Theorem in $W^{k,1}$

Authors: Irene Fonseca, Giovanni Leoni, Jan Malý and Roberto Paroni
Journal: Trans. Amer. Math. Soc. 354 (2002), 3723-3741
MSC (2000): Primary 49J45, 49Q20
Published electronically: April 30, 2002
MathSciNet review: 1911518
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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 $f$ 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 $f,$ convexity, $1$-quasiconvexity or $k$-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 $k$-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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Additional Information

Irene Fonseca
Affiliation: Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213

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

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

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

Keywords: $k$-quasiconvexity, higher-order lower semicontinuity, gradient truncation
Received by editor(s): April 1, 2001
Published electronically: April 30, 2002
Additional Notes: The research of I. Fonseca was partially supported by the National Science Foundation under Grant No. DMS–9731957.
The research of G. Leoni was partially supported by MURST, Project “Metodi Variazionali ed Equazioni Differenziali Non Lineari”, by the Italian CNR, through the strategic project “Metodi e modelli per la Matematica e l’Ingegneria”, and by GNAFA
The research of J. Malý was supported by CEZ MSM 113200007, grants GAČR 201/00/0768 and GAUK 170/99.
The authors wish to thank Guy Bouchitté for stimulating discussions on the subject of this work, and the Center for Nonlinear Analysis (NSF Grant No. DMS–9803791) for its support during the preparation of this paper.
Article copyright: © Copyright 2002 American Mathematical Society

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