Calculus of Variations and Geometric Measure Theory
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T. Schmidt

Regularity theorems for degenerate quasiconvex energies with $(p,q)$-growth

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


Published Paper

Inserted: 22 feb 2012
Last Updated: 5 mar 2012

Journal: Adv. Calc. Var.
Volume: 1
Number: 3
Pages: 241-270
Year: 2008
Links: Link to the published version


We study 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 function satisfying the $(p,q)$-growth conditions \[\gamma\lvert\xi\rvert^p\le f(\xi)\le\Gamma(1+\lvert\xi\rvert^q)\] with exponents $1 < p\le q < p+\frac{\min\{2,p\}}{2n}$. Imposing the additional assumption that $f$ resembles the degenerate behavior of the $p$-energy density, we establish a partial $C^{1,\alpha}$-regularity theorem for $F$-minimizers and a similar theorem for minimizers of a relaxed functional.

Our results cover the model case of polyconvex integrands \[f(\xi):=\frac1p \lvert\xi\rvert^p +h(\det\,\xi)\,,\] where $h$ is a smooth convex function with $\frac qn$-growth.


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