A strict maximum principle for nonlocal minimal surfaces

TL;DR

This paper establishes a strict maximum principle for fractional nonlocal minimal surfaces, showing that two such surfaces sharing a boundary point and one contained in the other must be identical, using advanced regularity and blow-up analysis.

Contribution

It introduces a novel strict maximum principle for nonlocal minimal surfaces, extending classical results to the fractional setting with new regularity and blow-up techniques.

Findings

Proves the strict maximum principle for nonlocal minimal surfaces.

Develops a specialized blow-up and regularity analysis for irregular contact points.

Utilizes a fractional Harnack inequality to support the main result.

Abstract

In the setting of fractional minimal surfaces, we prove that if two nonlocal minimal sets are one included in the other and share a common boundary point, then they must necessarily coincide. This strict maximum principle is not obvious, since the surfaces may touch at an irregular point, therefore a suitable blow-up analysis must be combined with a bespoke regularity theory to obtain this result. For the classical case, an analogous result was proved by Leon Simon. Our proof also relies on a Harnack Inequality for nonlocal minimal surfaces that has been recently introduced by Xavier Cabr\'e and Matteo Cozzi and which can be seen as a fractional counterpart of a classical result by Enrico Bombieri and Enrico Giusti. In our setting, an additional difficulty comes from the analysis of the corresponding nonlocal integral equation on a hypersurface, which presents a remainder whose sign and fine properties need to be carefully addressed.