Work

Parabolic Boundary Harnack Inequalities with Right-Hand Side

Year: 2024

Type: article

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 $$f \in L^q$$ 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.

Source: Archive for Rational Mechanics and Analysis

DOI: 10.1007/s00205-024-02017-4

Author: Clara Torres-Latorre

Institution: Universitat de Barcelona

Language: English

Topic: Nonlinear Partial Differential Equations

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $2,890

Funders: European Research Council, Agencia Estatal de Investigación, H2020 European Research Council