Direct Epiperimetric Inequalities for the Thin Obstacle Problem and Applications

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 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 λ∈[]3/2,2+c∪⋃m∈ℕ‍(),2m−cm−2m+cm+, 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. © 2019 Wiley Periodicals, Inc.

An epiperimetric inequality approach to the parabolic Signorini problem

“…The main steps in the proof are discrete decay estimates for the Weiss energy W κ , κ = 3/2 and κ = 2m, which can be viewed as parabolic epiperimetric inequalities. Epiperimetric inequalities were introduced by G. Weiss [13] to the classical obstacle problem and they continue to be a subject of intense research interest in the elliptic setting [5,6,7,10,11]. We briefly comment about our results and the related literature.…”

“…Note that in Theorem 2 we obtain a logarithmic decay (in ln(−t)) of the L 2 norm instead of an exponential decay as in Theorem 1. A polynomial decay rate of this kind towards the asymptotic solutions was obtained originally in the elliptic case [5,6,7]. Moreover, in the classical obstacle problem it was shown, that there is in general no exponential decay rate at singular points, cf.…”

“…with the Signorini conditioñ u ≥ 0, ∂ nũ (y) ≤ 0,ũ∂ nũ = 0 on {y n = 0}. (5) Proof. The proof follows from a direct computation.…”

“…The logarithmic epiperimetric inequality was recently established for the elliptic obstacle and thin obstacle problems [5,6,7], which inspired our paper. Very different from the proofs in [5,6,11] and the existing epiperimetric inequality approach for the parabolic problem (where they reduce the parabolic problem to a stationary problem by treating the time derivative as inhomogeneity), our approach is purely dynamical and does not rely on the (almost) minimality. Hence it can possibly be generalized to study the asymptotics around critical points instead of energy minimizers for a broader class of free boundary problems.…”

“…The non-compactness brings additional difficulties and we could not get the polynomial decay rate towards general blow-ups at singular points as in the elliptic setting (cf. [5,6]). It is unlikely that the above decay rate is optimal, thus it is a very interesting open problem to improve the decay rate and further explore the optimality.…”