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Publication 02-CNA-18

${\cal A}$-quasiconvexity: weak star convergence and the gap

Irene Fonseca
Department of Mathematical Sciences
Carnegie MEllon Unviersity
Pittsburgh, PA 15213
fonseca@andrew.cmu.edu

Giovannni Leoni
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
Giovanni@andrew.cmu.edu

Stefan Müller
Max-Planck Institut für Mathematik
in den Naturwissenschaften
Leipzig, Germany

Abstract Lower semicontinuity results with respect to weak-* convergence in the sense of measures and with respect to weak convergence in $L^p$ are obtained for f unctionals


\begin{displaymath}v \in L^1(\Omega;\mathbb{R}^m) \mapsto \int_{\Omega}f(x,v(x))dx, \end{displaymath}

where admissible sequences $\{v_n\}$ satisfy a first order system of PDEs ${\cal A}v_n=0$. We suppose that ${\cal A}$ has constant rank, $f$ is ${\cal A}$-quasiconvex and satisfies the nonstandard growth conditions


\begin{displaymath}\frac{1}{C} (\vert v\vert^p-1) \leq f(v) \leq C(1+\vert v\vert^q)\end{displaymath}

with $q \in [p,pN/(N-1))$ for $p \leq N-1,\ q \in [p,p +1)$ for $p > N-1$. In particular, our results generalize earlier work where ${\cal A}v =0$ reduced to $v = \nabla^su$ for some $s \in \mathbb{N}$.

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