C^{2,\alpha}$ regularity of flat free boundaries for the thin one-phase problem

TL;DR

This paper establishes $C^{2,eta}$ regularity for flat free boundaries in the thin one-phase problem, which models a fractional Laplacian-based free boundary scenario, advancing understanding of boundary smoothness in lower dimensions.

Contribution

It proves $C^{2,eta}$ regularity for flat free boundaries in the thin one-phase problem, a significant step in free boundary regularity theory for fractional Laplacian models.

Findings

Proves $C^{2,eta}$ regularity for flat free boundaries.

Connects free boundary regularity to fractional Laplacian models.

Advances understanding of boundary smoothness in lower-dimensional free boundary problems.

Abstract

We prove $C^{2,\alpha}$ regularity of sufficiently flat free boundaries, for the thin one-phase problem in which the free boundary occurs on a lower dimensional subspace. This problem appears also as a model of a one-phase free boundary problem in the context of the fractional Laplacian $(-\Delta)^{1/2}$.