Nonlocal minimal surfaces

TL;DR

This paper explores nonlocal minimal surfaces by replacing the BV norm with the fractional Sobolev norm, establishing regularity results for minimizers with flat boundaries.

Contribution

It introduces a nonlocal analogue of de Giorgi's minimal surface theory using the $H^\sigma$ norm and proves regularity of minimizers under flatness conditions.

Findings

Minimizers with sufficiently flat boundaries are smooth hypersurfaces.

The study extends classical minimal surface theory to a nonlocal fractional setting.

Abstract

The de Giorgi theory for minimal surfaces consists in studying sets whose indicator function is a (local) minimum of the BV norm. In this paper we replace the BV norm by the $H^\sigma$ norm, with $\sigma<1/2$, and try to understand what the minimisers look like. Parallel to the de Giorgi theory we prove that, if the boundary of a minimiser is sufficiently flat in the unit ball, then it is a smooth piece of hypersurface.