Published Paper

Inserted: 22 feb 2012
Last Updated: 5 mar 2012

Journal: Proc. R. Soc. Edinb., Sect. A, Math.
Volume: 139
Number: 3
Pages: 595-621
Year: 2009
Links: Link to the published version

Abstract:

We consider strictly quasiconvex 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 multi-dimensional calculus of variations. For the $C^2$-integrand $f\colon\mathbb{R}^{Nn}\to\mathbb{R}$ we impose $(p,q)$-growth conditions \[\gamma\lvert\xi\rvert^p\le f(\xi)\le\Gamma(1+\lvert\xi\rvert^q) \qquad\text{for all }\xi \in\mathbb{R}^{Nn}\] with $\gamma, \Gamma >0$ and $1 < p\le q < \min\big\{p+\frac1n,\frac{2n-1}{2n-2}p\big\}$. Under these assumptions we prove partial $C^{1,\alpha}_{\rm loc}$-regularity for strong local minimizers of $F$ and the associated relaxed functional $\cal F$.

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