An epiperimetric inequality for odd frequencies in the thin obstacle problem

TL;DR

This paper establishes an epiperimetric inequality for odd frequencies in the thin obstacle problem, leading to new insights on convergence rates and regularity of contact sets.

Contribution

It introduces the first epiperimetric inequality for odd frequencies in the thin obstacle problem and applies it to analyze solution regularity and convergence.

Findings

Rate of convergence of blow-up sequences at odd frequency points

Regularity results for the contact set strata

Recovery of the frequency gap for odd frequencies

Abstract

We prove for the first time an epiperimetric inequality for the thin obstacle Weiss' energy with odd frequencies and we apply it to solutions to the thin obstacle problem with general $C^{k,\gamma}$. In particular, we obtain the rate of convergence of the blow-up sequences at points of odd frequencies and the regularity of the strata of the corresponding contact set. We also recover the frequency gap for odd frequencies obtained by Savin and Yu.