The classical obstacle problem with H\"older continuous coefficients

TL;DR

This paper develops quasi-monotonicity formulas for classical obstacle problems with coefficients that are Dini and double-Dini continuous, leading to new regularity results for free boundaries under H"older continuity.

Contribution

It introduces Weiss' and Monneau's type quasi-monotonicity formulas for obstacle problems with Dini continuous coefficients, advancing free boundary regularity theory.

Findings

Established quasi-monotonicity formulas for Dini and double-Dini continuous coefficients.

Derived free boundary regularity results under H"older continuity assumptions.

Extended classical obstacle problem analysis to more general coefficient regularities.

Abstract

Weiss' and Monneau's type quasi-monotonicity formulas are established for quadratic energies having matrix of coefficients which are Dini, double-Dini continuity, respectively. Free boundary regularity for the corresponding classical obstacle problems under H\"older continuity assumptions is then deduced.