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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 [DS20], and we show that all our assumptions are sharp.As a consequence of our results, we deduce the C 1,α regularity of the free boundary in the fully nonlinear obstacle problem and the fully nonlinear thin obstacle problem.

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We study the obstacle problem for parabolic operators of the type ∂ t + L, where L is an elliptic integro-differential operator of order 2s, such as (−∆) s , in the supercritical regime s ∈ (0, 1 2 ). The best result in this context was due to Caffarelli and Figalli, who established the C 1,s x regularity of solutions for the case L = (−∆) s , the same regularity as in the elliptic setting.Here we prove for the first time that solutions are actually more regular than in the elliptic case. More precisely, we show that they are C 1,1 in space and time, and that this is optimal. We also deduce the C 1,α regularity of the free boundary. Moreover, at all free boundary points (x 0 , t 0 ), we establish the following expansion:), with c 0 > 0, α > 0 and a ∈ R n .