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 $\frac{3}{2}$ 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 $\lambda \in \{\frac{3}{2}\}\cup [2,\infty )$, and we classify the global $\lambda$-homogeneous minimizers, with $\lambda \in [\frac{3}{2},2+c]\cup {\bigcup }_{m\in \mathbb{N}}(2m-c_m^{-},2m+c_m^{+})$, showing as a consequence that the frequencies $\frac{3}{2}$ 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.

References in corpus (3)

• Regularity estimates for the solution and the free boundary to the obstacle problem for the fractional Laplacian
• Some New Monotonicity Formulas and the Singular Set in the Lower Dimensional Obstacle Problem
• The Variable Coefficient Thin Obstacle Problem: Higher Regularity

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• Integro-Differential Elliptic Equations
• Free boundary regularity for almost every solution to the Signorini problem
• Uniqueness of the blow-up at isolated singularities for the Alt-Caffarelli functional
• On the singular set in the thin obstacle problem: higher order blow-ups and the very thin obstacle problem
• Half-space solutions with $7/2$ frequency in the thin obstacle problem
• Free boundary partial regularity in the thin obstacle problem
• On the logarithmic epiperimetric inequality for the obstacle problem
• Fractional Parabolic Theory as a High-Dimensional Limit of Fractional Elliptic Theory
• Free boundary regularity in the multiple membrane problem in the plane