An epiperimetric inequality approach to the regularity of the free boundary in the Signorini problem with variable coefficients

TL;DR

This paper proves the $C^{1,eta}$ regularity of the free boundary in the Signorini problem with variable coefficients, using new monotonicity formulas and epiperimetric inequalities.

Contribution

It introduces novel monotonicity formulas and an epiperimetric inequality to establish boundary regularity for variable coefficient elliptic operators.

Findings

Established $C^{1,eta}$ regularity of the free boundary

Developed new monotonicity formula

Proved a new epiperimetric inequality

Abstract

In this paper we establish the $C^{1,\beta}$ regularity of the regular part of the free boundary in the Signorini problem for elliptic operators with variable Lipschitz coefficients. This work is a continuation of the recent paper [GSVG14], where two of us established the interior optimal regularity of the solution. Two of the central results of the present work are a new monotonicity formula and a new epiperimetric inequality.