A logarithmic epiperimetric inequality for the obstacle problem

TL;DR

This paper establishes a new logarithmic epiperimetric inequality for the obstacle problem, providing explicit regularity results at singular points and advancing understanding of the problem's structure.

Contribution

It introduces a novel logarithmic epiperimetric inequality at singular points, improving regularity estimates and answering a longstanding question by Weiss.

Findings

Proved epiperimetric inequality at regular and singular points

Introduced logarithmic epiperimetric inequality for singular points

Achieved explicit logarithmic regularity of the singular set

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.