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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

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 $s\in\mathbb{N}$.

Received:
Feb 11, 2003
Published:
Feb 11, 2003
MSC Codes:
35E99, 49J45, 74B20
Keywords:
a-quasiconvexity, gap, non-standard growth conditions, lower semicontinuity, sobolev embedding theorem, radon-nikodym decomposition theorem

Related publications

inJournal
2004 Repository Open Access
Irene Fonseca, Giovanni Leoni and Stefan Müller

A-quasiconvexity : weak-star convergence and the gap

In: Annales de l'Institut Henri Poincaré / C, 21 (2004) 2, pp. 209-236