Optimal regularity of the thin obstacle problem by an epiperimetric inequality

TL;DR

This paper introduces a new proof for the optimal regularity of the thin obstacle problem by employing an epiperimetric inequality, establishing a frequency gap that ensures the solution's regularity.

Contribution

It provides an alternative, self-contained proof of the optimal regularity using an epiperimetric inequality to establish a frequency gap.

Findings

Proves a lower bound on the frequency at free boundary points.

Shows the non-existence of certain homogeneous solutions with intermediate frequencies.

Establishes a new method for proving regularity in the thin obstacle problem.

Abstract

The key point to prove the optimal $C^{1,\frac12}$ regularity of the thin obstacle problem is that the frequency at a point of the free boundary $x_0\in\Gamma(u)$, say $N^{x_0}(0^+,u)$, satisfies the lower bound $N^{x_0}(0^+,u)\ge\frac32$. In this paper we show an alternative method to prove this estimate, using an epiperimetric inequality for negative energies $W_\frac32$. It allows to say that there are not $\lambda-$homogeneous global solutions with $\lambda\in (1,\frac32)$, and by this frequancy gap, we obtain the desired lower bound, thus a new self contained proof of the optimal regularity.