Optimal regularity of the thin obstacle problem by an epiperimetric inequality

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 frequency gap, we obtain the desired lower bound, thus a new self-contained proof of the optimal regularity.