Optimal regularity of the thin obstacle problem by an epiperimetric inequality

The key point to prove the optimal C-1,C-1/2 regularity of the thin obstacle problem is that the frequency at a point of the free boundary x(0) is an element of Gamma (u), say N-x0 (0(+), u), satisfies the lower bound N-x0 (0(+), u) >= 3 2. In this paper, we show an alternative method to prove this estimate, using an epiperimetric inequality for negative energies W3/2. It allows to say that there are not lambda-homogeneous global solutions with lambda is an element of(1, 3/2), and by this frequency gap, we obtain the desired lower bound, thus a new self-contained proof of the optimal regularity.