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Regularity of relaxed minimizers of quasiconvex variational integrals with \((p, q)\)-growth. (English) Zbl 1173.49032
Summary: We consider autonomous integrals
\[ F[u]:=\int_\Omega f(Du) \,dx \quad \text{for }u:\mathbb R^n\supset\Omega\to \mathbb R^N \]
in the multidimensional calculus of variations, where the integrand \(f\) is a strictly quasiconvex \(C ^2\)-function satisfying the \((p,q)\)-growth conditions
\[ \gamma|A|^p \leqq f(A)\leqq\Gamma (1+|A|^q)\quad \text{for every }A\in\mathbb R^{nN} \]
with exponents \({1 < p\leqq q < \infty}\).
We examine the Lebesgue-Serrin extension
\[ {\mathcal F}_{\text{loc}}[u]:= \inf\left\{\liminf_{k\to\infty} F[u_k]: W^{1,q}_{\text{loc}}\ni u_k {\mathop \rightharpoonup \limits_{k\to\infty}} u\text{ weakly in }W^{1,p}\right\} \]
of \(F\) and establish an existence result for minimizers of \({\mathcal F}_{\text{loc}}\). 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.

MSC:
49N60 Regularity of solutions in optimal control
49J10 Existence theories for free problems in two or more independent variables
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