The $C^{1,\alpha}$ boundary Harnack principle in a slit domain and its application to the Signorini problem

TL;DR

This paper establishes the $C^{2,eta}$ regularity of free boundaries in the Signorini problem with variable coefficients, utilizing a $C^{1,eta}$ boundary Harnack inequality in slit domains and analyzing degenerate elliptic equations.

Contribution

It introduces a novel approach to prove boundary regularity in the Signorini problem using a boundary Harnack inequality and Schauder estimates for degenerate elliptic equations.

Findings

Proved $C^{2,eta}$ regularity of free boundary in Signorini problem.

Developed a $C^{1,eta}$ boundary Harnack inequality in slit domains.

Established $C^{1,eta}$ Schauder estimates for degenerate elliptic equations.

Abstract

We prove the $C^{2,\alpha}$ regularity of the free boundary in the Signorini problem with variable coefficients. We use a $C^{1,\alpha}$ boundary Harnack inequality in slit domain. The key method is to study a non-standard degenerate elliptic equation and obtain a $C^{1,\alpha}$ Schauder estimate.