Regularity of minimizers of some variational integrals with discontinuity

TL;DR

This paper establishes regularity properties for vector-valued minimizers of certain variational integrals with discontinuous integrands in the spatial variable, extending regularity theory to less regular settings.

Contribution

It proves regularity results for minimizers of variational integrals with discontinuous integrands in the vector-valued case, broadening the scope of existing regularity theory.

Findings

Regularity properties are established for minimizers with discontinuous integrands.

The results apply to integrals with polynomial growth of order p ≥ 2.

The work extends regularity theory to cases with less regular integrands.

Abstract

We prove regularity properties in the vector valued case for minimizers of variational integrals of the form $$A(u) = \int_\Omega A(x,u,Du) dx$$ where the integrand $A(x,u,Du)$ is not necessarily continuous respect to the variable $x,$ grows polinomially like $|\xi|^p,$ $p \geq 2.$