Partial regularity for a class of anisotropic variational integrals with convex hull property

Article type: Research Article

Authors: Bildhauer, M. | Fuchs, M.

Affiliations: Universität des Saarlandes, Fachbereich 6.1, Mathematik, Postfach 151150, 66041 Saarbrücken, Germany E‐mail: [email protected]‐sb.de, [email protected]‐sb.de

Note: [] Corresponding author.

Abstract: We consider integrands f :$\mathbb{R}$nN→$\mathbb{R}$ which are of lower (upper) growth rate s≥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 :Ω→$\mathbb{R}$N of the energy ∫Ωf(∇u) dx respect a closed convex set K⊂$\mathbb{R}$N, then so does u on the whole domain. We show partial C1,α‐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 Acerbi and Fusco [2] do not apply, in particular, we give some extensions to the subquadratic case.

Keywords: regularity, minimizers, anisotropic growth

Journal: Asymptotic Analysis, vol. 32, no. 3-4, pp. 293-315, 2002