Perturbed Obstacle Problems in Lipschitz Domains: Linear Stability and Non-degeneracy in Measure

TL;DR

This paper establishes quantitative bounds on how contact sets in the obstacle problem change under perturbations of data in Lipschitz domains, demonstrating linear stability in measure.

Contribution

It provides the first linear stability estimates for contact sets in obstacle problems under general data perturbations in Lipschitz domains.

Findings

Lebesgue measure of contact set differences is linearly bounded by data perturbations

Quantitative bounds are derived for changes in contact sets

Results apply to general Lipschitz domains

Abstract

We consider the classical obstacle problem on bounded, connected Lipschitz domains $D \subset \mathbb{R}^n$. We derive quantitative bounds on the changes to contact sets under general perturbations to both the right hand side and the boundary data for obstacle problems. In particular, we show that the Lebesgue measure of the symmetric difference between two contact sets is linearly comparable to the $L^1$-norm of perturbations in the data.