Quasiconvex Bulk and Surface Energies with subquadratic growth

TL;DR

This paper proves partial Hölder continuity of the gradient for solutions to vectorial variational problems with quasiconvex bulk energies of subquadratic growth and anisotropic surface energies, expanding regularity results in this setting.

Contribution

It establishes regularity of solutions for a broad class of variational problems with minimal structural assumptions, including subquadratic growth and irregular surface energies.

Findings

Gradient of solutions is partially Hölder continuous.

Results apply to energies with subquadratic growth and anisotropic surface integrands.

No additional structure conditions are required on the bulk energy densities.

Abstract

We establish partial H\"older continuity of the gradient for equilibrium configurations of vectorial multidimensional variational problems, involving bulk and surface energies. The bulk energy densities are uniformly strictly quasiconvex functions with $p$-growth, $1<p< 2$, without any further structure conditions. The anisotropic surface energy is defined by means of an elliptic integrand $\Phi$ not necessarily regular.