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We prove new boundary Harnack inequalities in Lipschitz domains for equationswith a right hand side. Our main result applies to non-divergence formoperators with bounded measurable coefficients and to divergence form operatorswith 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 thefully nonlinear thin obstacle problem.

New boundary Harnack inequalities with right hand side