Monotonicity formulas for obstacle problems with Lipschitz coefficients

TL;DR

This paper establishes quasi-monotonicity formulas for obstacle problems with Lipschitz coefficients, enabling comprehensive analysis of free-boundary regularity using classical approaches.

Contribution

It introduces new quasi-monotonicity formulas for obstacle problems with Lipschitz coefficients, advancing the understanding of free-boundary regularity.

Findings

Derived quasi-monotonicity formulas for obstacle problems

Achieved full regularity analysis of free-boundary points

Extended classical methods to Lipschitz coefficient settings

Abstract

We prove quasi-monotonicity formulas for classical obstacle-type problems with energies being the sum of a quadratic form with Lipschitz coefficients, and a H\"older continuous linear term. With the help of those formulas we are able to carry out the full analysis of the regularity of free-boundary points following the approaches by Caffarelli, Weiss and Monneau.