A logarithmic epiperimetric inequality for the obstacle problem

Abstract

For the general obstacle problem, we prove by direct methods an epiperimetric inequality at regular and singular points, thus answering a question of Weiss (Invent. Math., 138 (1999), 23-50). In particular at singular points we introduce a new tool, which we call logarithmic epiperimetric inequality, which yields an explicit logarithmic modulus of continuity on the C 1 regularity of the singular set, thus improving previous results of Caffarelli and Monneau [4,2,11].

Direct Epiperimetric Inequalities for the Thin Obstacle Problem and Applications

“…Our direct approach allows us to obtain a new logarithmic epiperimetric inequality for the family of energies W 2m , m 2 N, in any dimension. This, together with [5], is the first instance in the literature (even in the context of minimal surfaces) of an epiperimetric inequality of logarithmic type, and the first instance in the context of the lower-dimensional obstacle problems where an epiperimetric inequality for singular points has a direct proof. This result allows us to prove a complete and self-contained regularity result for Sing.u/ and improve the known results by giving an explicit modulus of continuity.…”

“…This result can be proved as a standard application of our various epiperimetric inequalities and the almost minimality of the blow-ups at a point of the free boundary, which follows from the regularity of the obstacle (see, for instance, [5]). In particular, it provides an improvement in the regularity of S 2m , 2m < l, from C 1 to C 1;log of the results of [3,11].…”

An epiperimetric inequality approach to the parabolic Signorini problem

“…As direct consequences of Lemma 2.4, we infer the following properties: (5), (4) which satisfies (6). Then, the function…”

“…A direct computation as in Lemma 2.4 gives that τ → W (ũ(τ )) satisfies the almost monotone decreasing property: for any 0 < τ a < τ b < ∞ W (ũ(τ b )) − W (ũ(τ a )) ≤ −2R Relying on this almost monotonicity, we seek to prove the following contraction result: Proposition 5.1. Letũ be a solution to (50) with κ = 3/2 andũ satisfies (6). Then there exists a universal constant c 0 ∈ (0, 1/4) such that W (ũ(τ + 1)) ≤ (1 − c 0 )W (ũ(τ )) + 2e −τ /2 M 2 f for any τ > 0.…”

Regularity of Interfaces in Phase Transitions via Obstacle Problems - Fields Medal Lecture

“…Still, one could have hoped to extend his argument to higher dimensions. This was achieved by Colombo, Spolaor, and Velichkov [11]. There, the authors introduced a quantitative argument to avoid a compactness step in Weiss' proof.…”