Parabolic Boundary Harnack Inequalities with Right-Hand Side

Abstract

AbstractWe 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 ∈ 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 - ε 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 , α in the parabolic obstacle problem and in the parabolic Signorini problem.