The key point to prove the optimal C1,21β regularity of the thin
obstacle problem is that the frequency at a point of the free boundary
x0ββΞ(u), say Nx0β(0+,u), satisfies the lower bound
Nx0β(0+,u)β₯23β.
In this paper we show an alternative method to prove this estimate, using an
epiperimetric inequality for negative energies W23ββ. It allows to say
that there are not Ξ»βhomogeneous global solutions with Ξ»β(1,23β), and by this frequancy gap, we obtain the desired lower bound,
thus a new self contained proof of the optimal regularity