New boundary Harnack inequalities with right hand side

TL;DR

This paper establishes new boundary Harnack inequalities for elliptic equations with right hand side in Lipschitz domains, leading to regularity results for free boundaries in obstacle problems.

Contribution

It introduces sharp boundary Harnack inequalities for non-divergence and divergence form operators with right hand side in L^q, q > n, extending previous results.

Findings

Boundary Harnack inequalities proven for equations with right hand side

Regularity of free boundary in obstacle problems derived

Results are sharp and applicable to various elliptic operators

Abstract

We prove new boundary Harnack inequalities in Lipschitz domains for equations with a right hand side. Our main result applies to non-divergence form operators with bounded measurable coefficients and to divergence form operators with continuous coefficients, whereas the right hand side is in $L^q$ with $q > n$. Our approach is based on the scaling and comparison arguments of \cite{DS20}, and we show that all our assumptions are sharp. As a consequence of our results, we deduce the $\mathcal{C}^{1,\alpha}$ regularity of the free boundary in the fully nonlinear obstacle problem and the fully nonlinear thin obstacle problem.