An Epiperimetric Inequality for the Thin Obstacle problem

TL;DR

This paper establishes an epiperimetric inequality for the thin obstacle problem, enabling analysis of solution convergence rates and free boundary regularity, extending Weiss's classical results to a new context.

Contribution

It introduces a novel epiperimetric inequality specifically for the thin obstacle problem, advancing understanding of free boundary regularity and solution behavior.

Findings

Proves an epiperimetric inequality for the thin obstacle problem.

Provides tools to analyze convergence rates of rescaled solutions.

Enhances understanding of free boundary regularity.

Abstract

We prove an epiperimetric inequality for the thin obstacle problem, extending the pioneering results by Weiss on the classical obstacle problem (Invent. Math. 138 (1999), no. 1, 23-50). This inequality provides the means to study the rate of converge of the rescaled solutions to their limits, as well as the regularity properties of the free boundary.