Partial regularity for minimizers of discontinuous quasiconvex integrals with general growth

TL;DR

This paper establishes partial Hölder continuity for minimizers of quasiconvex functionals with discontinuous integrands under general growth conditions, extending regularity results to more complex variational problems.

Contribution

It proves partial regularity for minimizers of quasiconvex integrals with discontinuous integrands satisfying VMO conditions and general growth, broadening the scope of regularity theory.

Findings

Proves partial Hölder continuity of minimizers.

Handles integrands with discontinuities in the spatial variable.

Extends regularity results to general growth conditions.

Abstract

We prove the partial H\"older continuity for minimizers of quasiconvex functionals \[ \mathcal{F}({\bf u}) \colon =\int_{\Omega} f(x,{\bf u},D{\bf u})\,\mathrm{d}x, \] where $f$ satisfies a uniform VMO condition with respect to the $x$-variable and is continuous with respect to ${\bf u}$. The growth condition with respect to the gradient variable is assumed a general one.