Parabolic boundary Harnack inequalities with right-hand side

TL;DR

This paper establishes a boundary Harnack inequality for parabolic equations with non-zero right-hand sides in Lipschitz domains, extending regularity results and boundary behavior understanding for solutions of parabolic PDEs.

Contribution

It introduces a novel blow-up method to prove the parabolic boundary Harnack inequality with right-hand side, applicable to non-divergence form operators with measurable coefficients.

Findings

Proves boundary Harnack inequality with right-hand side in Lipschitz domains.

Shows optimal regularity of solution quotients for the heat equation.

Provides new regularity results for free boundaries in parabolic obstacle problems.

Abstract

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