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Partial regularity for a class of anisotropic variational integrals with convex hull property. (English) Zbl 1076.49018
Summary: We consider integrands \(f: \mathbb R^{nN}\to \mathbb R\) which are of lower (upper) growth rate \(s\geq 2\) \((q > s)\) and which satisfy an additional structural condition implying the convex hull property, i.e., if the boundary data of a minimizer \(u:\Omega\to\mathbb R^N\) of the energy \(\int_\Omega f(\nabla u)\, dx\) respect a closed convex set \(K\subset \mathbb R^N\), then so does \(u\) on the whole domain. We show partial \(C^{1,\alpha}\)-regularity of bounded local minimizers if \(q<\min\{s+2/3,sn/(n-2)\}\) and discuss cases in which the latter condition on the exponents can be improved. Moreover, we give examples of integrands which fit into our category and to which the results of A. Acerbi and N. Fusco [J. Differ. Equations 107, No. 1, 46–67 (1994; Zbl 0807.49010)] do not apply; in particular, we give some extensions to the subquadratic case.

49N60 Regularity of solutions in optimal control
49J45 Methods involving semicontinuity and convergence; relaxation
49J10 Existence theories for free problems in two or more independent variables