On non-local almost minimal sets and an application to the non-local Massari's Problem

TL;DR

This paper investigates non-local minimal sets with prescribed fractional mean curvature, establishing existence, regularity, and phenomena like stickiness, extending classical minimal surface theory to a fractional, non-local context.

Contribution

It introduces a non-local version of Massari's problem, proving existence and regularity results for fractional almost minimal sets and analyzing stickiness phenomena.

Findings

Existence of non-local minimal sets with prescribed fractional curvature.

Regularity results for these non-local minimal sets.

Identification of stickiness phenomena in the non-local setting.

Abstract

We consider a fractional Plateau's problem dealing with sets with prescribed non-local mean curvature. This problem can be seen as a non-local counterpart of the classical Massari's Problem. We obtain existence and regularity results, relying on a suitable version of the non-local theory for almost minimal sets. In this framework, the fractional curvature term in the energy functional can be interpreted as a perturbation of the fractional perimeter. In addition, we also discuss stickiness phenomena for non-local almost minimal sets.