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.
"Perturbed obstacle problems in Lipschitz domains: linear stability and nondegeneracy in measure." Rocky Mountain J. Math. 49 (5) 1407 - 1418, 2019. https://doi.org/10.1216/RMJ-2019-49-5-1407