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Partial regularity of minimizers of higher order integrals with \((p, q)\)-growth. (English) Zbl 1248.49053
Summary: We consider higher order functionals of the form \[ F[u] = \int_{\Omega} f(D^m u)dx \text{ for } u: \mathbb R^{n} \subset \Omega \to \mathbb R^{N} \] where the integrand \(f:\bigodot^m(\mathbb R^{n},\mathbb R^{N})\to\mathbb{R}\), \(m \geq 1\) is strictly quasiconvex and satisfies a non-standard growth condition. More precisely we assume that \(f\) fulfills the \((p,q)\)-growth condition \[ \gamma|A|^p\leq f(A)\leq L(1+|A|^q) \text{ for all } A \in \bigodot^m(\mathbb R^{n},\mathbb R^{N}), \] with \(\gamma , L > 0\) and \(1 < p \leq q < min\{p + {1 \over 2}, {{2n-1} \over {2n-2}}p \}\). We study minimizers of the functional \(F[.]\) and prove a partial \(C_{loc} ^{m,\alpha}\)-regularity result.
MSC:
49N60 Regularity of solutions in optimal control
49N99 Miscellaneous topics in calculus of variations and optimal control
49J45 Methods involving semicontinuity and convergence; relaxation
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