Separation of a Lower Dimensional Free Boundary in a Two Phase Problem

TL;DR

This paper investigates a two phase free boundary problem involving the fractional Laplacian, demonstrating that the positive and negative free boundaries cannot intersect, which advances understanding of the problem's geometric properties.

Contribution

It establishes a novel result that the two free boundaries in a fractional Laplacian two phase problem are separated, providing new insights into their local behavior.

Findings

The positive and negative free boundaries do not touch.

The result applies to local properties of the free boundary.

Advances understanding of fractional Laplacian free boundary problems.

Abstract

This paper studies local properties of a two phase free boundary problem for the fractional Laplacian. The main result states that the two free boundaries of the positive and negativity sets cannot touch.