Quasi-monotonicity formulas for classical obstacle problems with Sobolev coefficients and applications

TL;DR

This paper develops quasi-monotonicity formulas for classical obstacle problems with Sobolev coefficients, enabling advanced free boundary analysis, and extends existing mathematical tools to more general settings.

Contribution

It introduces Weiss' and Monneau's type formulas for obstacle problems with Sobolev space coefficients, broadening the scope of free boundary analysis techniques.

Findings

Established quasi-monotonicity formulas for Sobolev coefficient matrices

Applied formulas to analyze free boundaries in obstacle problems

Extended classical obstacle problem analysis to Sobolev coefficient settings

Abstract

We establish Weiss' and Monneau's type quasi-monotonicity formulas for quadratic energies having matrix of coefficients in a Sobolev space $W^{1,p}$, $p>n$, and provide an application to the corresponding free boundary analysis for the related classical obstacle problems.