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 ∈ L q for q > n + 2 . In the case of the heat equation, we also show the optimal C 1 - ε regularity of the quotient. As a corollary, we obtain a new way to prove that flat Lipschitz free boundaries are C 1 , α in the parabolic obstacle problem and in the parabolic Signorini problem.