Direct epiperimetric inequalities for the thin obstacle problem and applications
Abstract
For the thin obstacle problem, we prove by a new direct method that in any dimension the Weiss' energies with frequency 32 and 2m, for m∈ 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 λ ∈ \32\ [2,∞), and we classify the global λ-homogeneous minimizers, with λ∈ [32,2+c]m∈ N(2m-cm-,2m+cm+), showing as a consequence that the frequencies 32 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 C1 regularity of the singular set with frequency 2m by an explicit logarithmic modulus of continuity.