Parabolic Boundary Harnack Inequalities with Right-Hand Side

We prove the parabolic boundary Harnack inequality in parabolic flat Lipschitz domains by blow-up techniques, allowing, for the first time, a non-zero right-hand side. Our method allows us to treat solutions to equations driven by non-divergence form operators with bounded measurable coefficients, and a right-hand side \(f \in L^q\) $f\in L^q$ for \(q > n+2\) $q>n+2$ . In the case of the heat equation, we also show the optimal \(C^{1-\varepsilon }\) $C^{1-\epsilon }$ regularity of the quotient. As a corollary, we obtain a new way to prove that flat Lipschitz free boundaries are \(C^{1,\alpha }\) $C^{1,\alpha }$ in the parabolic obstacle problem and in the parabolic Signorini problem.