An epiperimetric inequality for the lower dimensional obstacle problem

TL;DR

This paper proves an epiperimetric inequality for the lower dimensional obstacle problem, extending Weiss's classical result, with implications for free-boundary regularity, using homogeneity and { extGamma}-convergence methods.

Contribution

It provides the first proof of an epiperimetric inequality in the lower dimensional obstacle setting, adapting previous approaches to this new context.

Findings

Establishes an epiperimetric inequality for the lower dimensional obstacle problem.

Demonstrates implications for free-boundary regularity.

Adapts homogeneity and { extGamma}-convergence techniques to this setting.

Abstract

In this paper we give a proof of an epiperimetric inequality in the setting of the lower dimensional obstacle problem. The inequality was introduced by Weiss (Invent. Math., 138 (1999), no. 1, 23-50) for the classical obstacle problem and has striking consequences concerning the regularity of the free-boundary. Our proof follows the approach of Focardi and Spadaro (Adv. Differential Equations 21 (2015), no 1-2, 153-200.) which uses an homogeneity approach and a {\Gamma}-convergence analysis.