Regularity of Almost-Minimizers of H\"older-Coefficient Surface Energies

TL;DR

This paper investigates the regularity properties of almost-minimizers of anisotropic surface energies with H"older continuous coefficients, establishing local differentiability and singular set estimates using an excess-decay approach.

Contribution

It proves almost-minimizers are locally H"older continuously differentiable at regular points and provides dimension estimates for their singular sets, extending regularity theory in anisotropic surface energies.

Findings

Almost-minimizers are locally H"older continuously differentiable at regular points.

Dimension estimates are provided for the singular set of almost-minimizers.

The proof employs an excess-decay type argument within the framework of sets of finite perimeter.

Abstract

We study almost-minimizers of anisotropic surface energies defined by a H\"older continuous matrix of coefficients acting on the unit normal direction to the surface. In this generalization of the Plateau problem, we prove almost-minimizers are locally H\"older continuously differentiable at regular points and give dimension estimates for the size of the singular set. We work in the framework of sets of locally finite perimeter and our proof follows an excess-decay type argument.