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MiS Preprint
12/2003
$\mathcal{A}$-quasiconvexity: weak-star convergence and the gap
Irene Fonseca, Stefan Müller and Giovanni Leoni
Abstract
Lower semicontinuity results with respect to weak-$\ast$ convergence in the sense of measures and with respect to weak convergence in $L^{p}$\ are obtained for functionals \[ v\in L^{1}(\Omega;\mathbb{R}^{m})\mapsto\int_{\Omega}f(x,v(x))\,dx, \] where admissible sequences $\{v_{n}\}$ satisfy a first order system of PDEs $\mathcal{A}v_{n}=0$. We suppose that $\mathcal{A}$ has constant rank, $f$\ is $\mathcal{A}$-quasiconvex and satisfies the non standard growth conditions \[ \frac{1}{C}(|v|^{p}-1)\leq f(v)\leq C(1+|v|^{q}) \] with $q\in\lbrack p,pN/(N-1))$ for $p\leq N-1$, $q\in\lbrack p,p+1)$ for $p>N-1.$ In particular, our results generalize earlier work where $\mathcal{A}v=0$ reduced to $v=\nabla^{s}u$ for some $s\in\mathbb{N}$.
