Direct epiperimetric inequalities for the thin obstacle problem and applications

For the thin obstacle problem, we prove by a new direct method that in any dimension the Weiss' energies with frequency $\frac32$ and $2m$, for $m\in \mathbb N$, satisfy an epiperimetric inequality, in the latter case of logarithmic type. In particular, at difference from the classical statements, we do not assume any a priori closeness to a special class of homogeneous functions. In dimension $2$, we also prove the epiperimetric inequality at any free boundary point. As a first application, we improve the set of admissible frequencies for blow ups, previously known to be $\lambda \in \{\frac32\} \cup [2,\infty)$, and we classify the global $\lambda$-homogeneous minimizers, with $\lambda\in [\frac32,2+c]\cup\bigcup_{m\in \mathbb N}(2m-c_m^-,2m+c_m^+)$, showing as a consequence that the frequencies $\frac32$ and $2m$ are isolated. Secondly, we give a short and self-contained proof of the regularity of the free boundary previously obtained by Athanasopoulos-Caffarelli-Salsa (Amer. J. Math., 130(2) (2008), 485-498) for regular points and Garofalo-Petrosyan (Invent. Math., 177(2) (2009), 415-461) for singular points, by means of an epiperimetric inequality of logarithmic type which applies for the first time also at all singular points of thin-obstacle free boundaries. In particular we improve the $C^1$ regularity of the singular set with frequency $2m$ by an explicit logarithmic modulus of continuity.