  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.

Analysis of PDEs