# Regularity of relaxed minimizers of quasiconvex variational integrals with $(p,q)$-growth

created by schmidt on 22 Feb 2012
modified on 05 Mar 2012

[BibTeX]

Published Paper

Inserted: 22 feb 2012
Last Updated: 5 mar 2012

Journal: Arch. Ration. Mech. Anal.
Volume: 193
Number: 2
Pages: 311-337
Year: 2009
We consider autonomous integrals $F[u]:=\int_\Omega f(Du)\,{\rm d}x \qquad\text{for }u\colon\mathbb{R}^n\supset\Omega\to\mathbb{R}^N$ in the multidimensional calculus of variations, where the integrand $f$ is a strictly quasiconvex $C^2$-function satisfying the $(p,q)$-growth conditions $\gamma\lvert A\rvert^p\le f(A)\le\Gamma(1+\lvert A\rvert^q)\qquad\text{for every }A\in\mathbb{R}^{nN}$ with exponents $1 < p \le q < \infty$.
We examine the Lebesgue-Serrin extension ${\mathscr F}_{\rm loc}[u]:=\inf\left\{\liminf_{k\to\infty}F[u_k]\,:\, W^{1,q}_{\rm loc}\ni u_k\underset{k\to\infty}{-\!\!\!-\!\!\!\!\rightharpoonup}u\text{ weakly in }W^{1,p}\right\}$ of $F$ and establish an existence result for minimizers of ${\mathscr F}_{\rm loc}$. Furthermore, we prove a corresponding partial $C^{1,\alpha}$-regularity theorem for $q < p+\frac{\min\{2,p\}}{2n}$, which is the first regularity result for this class of integrands.