*Accepted Paper*

**Inserted:** 5 oct 2017

**Last Updated:** 27 nov 2018

**Journal:** Comm. Pure Appl. Math.

**Year:** 2017

**Abstract:**

We introduce a new logarithmic epiperimetric inequality for the $2m$-Weiss' energy in any dimension and we recover with a simple direct approach the usual epiperimetric inequality for the $\frac32$-Weiss' energy. In particular, even in the latter case, at difference from the classical statements, we do not assume any a-priori closeness to a special class of homogeneous function. In dimension $2$, we also prove for the first time the classical epiperimetric inequality for the $(2m-\frac12)$-Weiss' energy, thus covering all the admissible energies.

As a first application, we classify the global $\lambda$-homogeneous minimizers of the thin obstacle problem, 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 and thus improving on the previously known results. Moreover, we give an example of a new family of $(2m-\frac12)$-homogeneous minimizers in dimension higher than $2$.

Secondly, we give a short and self-contained proof of the regularity of the free boundary of the thin obstacle problem, previously obtained by Athanasopoulos-Caffarelli-Salsa for regular points and Garofalo-Petrosyan for singular points. In particular we improve the $C^1$ regularity of the singular set with frequency $2m$ by an explicit logarithmic modulus of continuity.

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