New boundary Harnack inequalities with right hand side

Abstract

We prove new boundary Harnack inequalities in Lipschitz domai Our main result applies to non-divergence form operators with bc divergence form operators with continuous coefficients, whereas the approach is based on the scaling and comparison arguments of [13] are sharp. As a consequence of our results, we deduce the $\mathcal{C}^{1, \alpha}$ regularity of obstacle problem and the fully nonlinear thin obstacle problem.