Regularity Results for a free interface problem with H\"{o}lder coefficients

TL;DR

This paper investigates the regularity of free interfaces in variational problems with bulk and interface energies, allowing for general Dirichlet-type bulk energies with Hölder continuous coefficients, using the theory of b5;b5;-minimizers.

Contribution

It introduces regularity results for free interfaces under mild Hölder regularity assumptions on the coefficients, extending previous theories to more general variable-dependent energies.

Findings

Established regularity of free interfaces with Hölder continuous coefficients.

Extended the theory of b5;b5;-minimizers to problems with variable-dependent bulk energies.

Provided new insights into the structure of solutions under mild regularity conditions.

Abstract

We study a class of variational problems involving both bulk and interface energies. The bulk energy is of Dirichlet type albeit of very general form allowing the dependence from the unknown variable $u$ and the position $x$. We employ the regularity theory of $\Lambda$-minimizers to study the regularity of the free interface. The hallmark of the paper is the mild regularity assumption concerning the dependence of the coefficients with respect to $x$ and $u$ that is of H\"{o}lder type.