Direct epiperimetric inequalities for the thin obstacle problem and applications

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 3/2‐Weiss energy. In particular, even in the latter case, unlike the classical statements, we do not assume any a priori closeness to a special class of homogeneous functions. In dimension 2, we also prove for the first time the classical epiperimetric inequality for the (2m − 1/2)‐Weiss energy, thus covering all the admissible energies. As a first application, we classify the global λ‐homogeneous minimizers of the thin obstacle problem, with [formula presented], showing as a consequence that the frequencies 3/2 and 2m are isolated and thus improving on the previously known results. Moreover, we give an example of a new family of (2m − 1/2)‐homogeneous minimizers in dimension higher than 2. Second, 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, and Salsa (2008) for regular points and Garofalo and Petrosyan (2009) for singular points. In particular, we improve the C1 regularity of the singular set with frequency 2m by an explicit logarithmic modulus of continuity.