Optimal regularity of the thin obstacle problem by an epiperimetric inequality

Abstract

The key point to prove the optimal $C^{1,\frac{1}{2}}$ 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 $N^{x_0}(0^{+},u)\ge \frac{3}{2}$. In this paper we show an alternative method to prove this estimate, using an epiperimetric inequality for negative energies $W_{\frac{3}{2}}$. It allows to say that there are not $\lambda -$homogeneous global solutions with $\lambda \in (1,\frac{3}{2})$, and by this frequancy gap, we obtain the desired lower bound, thus a new self contained proof of the optimal regularity.