Parabolic boundary Harnack inequalities with right-hand side. (English) Zbl 07922984
Summary: 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.
MSC:
| 35Bxx | Qualitative properties of solutions to partial differential equations |
| 35Jxx | Elliptic equations and elliptic systems |
| 35Kxx | Parabolic equations and parabolic systems |
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