Optimal regularity and nondegeneracy of a free boundary problem related to the fractional Laplacian

TL;DR

This paper investigates the optimal regularity and nondegeneracy properties of a free boundary problem associated with the fractional Laplacian, introducing a variant of the boundary Harnack inequality without boundary zero conditions.

Contribution

It establishes new regularity and nondegeneracy results for a fractional Laplacian free boundary problem and proves a boundary Harnack inequality variant without boundary zero assumptions.

Findings

Proved optimal regularity of solutions.

Established nondegeneracy of the free boundary.

Developed a boundary Harnack inequality variant.

Abstract

We discuss the optimal regularity and nondegeneracy of a free boundary problem related to the fractional Laplacian. This work is related to, but addresses a different problem from, recent work of Caffarelli, Roquejoffre, and Sire. A variant of the boundary Harnack inequality is also proved, where it is no longer required that the function be 0 along the boundary.