The Classical Obstacle problem with coefficients in fractional Sobolev spaces

TL;DR

This paper develops quasi-monotonicity formulas for obstacle problems with fractional Sobolev coefficients, leading to new regularity results for free boundaries using classical approaches.

Contribution

It introduces quasi-monotonicity formulas for obstacle problems with fractional Sobolev coefficients, advancing free boundary regularity theory.

Findings

Established quasi-monotonicity formulas for the problem.

Proved regularity of free-boundary points.

Extended classical methods to fractional Sobolev coefficients.

Abstract

We prove quasi-monotonicity formulae for classical obstacle-type problems with quadratic energies with coefficients in fractional Sobolev spaces, and a linear term with a Dini-type continuity property. These formulae are used to obtain the regularity of free-boundary points following the approaches by Caffarelli, Monneau and Weiss.