Published Paper

Inserted: 22 feb 2012
Last Updated: 5 mar 2012

Journal: Calc. Var. Partial Differ. Equ.
Volume: 32
Number: 1
Pages: 1-24
Year: 2008
Links: Link to the published version

Abstract:

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 $W^{1,p}$-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 $1 < p\le q < \infty$.

Under these assumptions we establish an existence result for minimizers of $F$ in $W^{1,p}(\Omega;\mathbb{R}^N)$ provided $q<\frac{np}{n-1}$. We prove a corresponding partial $C^{1,\alpha}$-regularity theorem for $q < p+\frac{\min\{2,p\}}{2n}$. This is the first regularity result for autonomous quasiconvex integrals with $(p,q)$-growth.

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