Boundary Harnack estimates in slit domains and applications to thin free boundary problems

TL;DR

This paper establishes a higher order boundary Harnack inequality for harmonic functions in slit domains, leading to smoothness results for free boundaries in the Signorini problem.

Contribution

It introduces a higher order boundary Harnack inequality in slit domains and applies it to prove smoothness of free boundaries in the Signorini problem.

Findings

Higher order boundary Harnack inequality proved

Smoothness ($C^^$) of free boundary near non-degenerate points established

Enhanced understanding of free boundary regularity in thin obstacle problems

Abstract

We provide a higher order boundary Harnack inequality for harmonic functions in slit domains. As a corollary we obtain the $C^\infty$ regularity of the free boundary in the Signorini problem near non-degenerate points.