New boundary Harnack inequalities with right hand side

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.