Regularity for non-local almost minimal boundaries and applications

TL;DR

This paper develops a theory for non-local almost minimal boundaries, proving their smoothness under flatness conditions and applying these results to non-local curvature problems and obstacle problems.

Contribution

It introduces a non-local version of almost minimal boundaries and extends regularity results to this new setting, generalizing classical geometric measure theory.

Findings

Flat non-local almost minimal boundaries are smooth.

Established $C^{1,eta}$ regularity for sets with prescribed non-local mean curvature.

Proved regularity of solutions to non-local obstacle problems.

Abstract

We introduce a notion of non-local almost minimal boundaries similar to that introduced by Almgren in geometric measure theory. Extending methods developed recently for non-local minimal surfaces we prove that flat non-local almost minimal boundaries are smooth. This can be viewed as a non-local version of the Almgren-De Giorgi-Tamanini regularity theory. The main result has several applications, among these $C^{1,\alpha}$ regularity for sets with prescribed non-local mean curvature in $L^p$ and regularity of solutions to non-local obstacle problems.