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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi> <mml:mo>∈</mml:mo> <mml:msup><mml:mi>L</mml:mi> <mml:mi>q</mml:mi></mml:msup> </mml:mrow> </mml:math> for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>q</mml:mi> <mml:mo>></mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn></mml:mrow> </mml:math> . In the case of the heat equation, we also show the optimal <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>C</mml:mi> <mml:mrow><mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>ε</mml:mi></mml:mrow> </mml:msup> </mml:math> regularity of the quotient. As a corollary, we obtain a new way to prove that flat Lipschitz free boundaries are <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>C</mml:mi> <mml:mrow><mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>α</mml:mi></mml:mrow> </mml:msup> </mml:math> in the parabolic obstacle problem and in the parabolic Signorini problem.