Segregated configurations involving the square root of the laplacian and their free boundaries

TL;DR

This paper investigates the local structure and regularity of free boundaries in segregated critical configurations involving the square root of the laplacian, extending regularity results to anomalous diffusion cases.

Contribution

It develops an improvement of flatness theory and uses Almgren's monotonicity formula to establish partial regularity of the nodal set in anomalous diffusion scenarios.

Findings

Partial regularity of free boundaries up to a small dimensional set

Extension of known regularity results to the square root of the laplacian case

Development of new analytical tools for anomalous diffusion problems

Abstract

We study the local structure and the regularity of free boundaries of segregated critical configurations involving the square root of the laplacian. We develop an improvement of flatness theory and, as a consequence of this and Almgren's monotonicity formula, we obtain partial regularity (up to a small dimensional set) of the nodal set, thus extending the known resultsfor the standard diffusion to some anomalous case.