The Parabolic Two-Phase Membrane Problem: Regularity in Higher Dimensions

For the parabolic obstacle-problem-like equation $$\Delta u - \partial_t u = \lambda_+ \chi_{\{u>0\}} - \lambda_- \chi_{\{u<0\}} ,$$ where $\lambda_+$ and $\lambda_-$ are positive Lipschitz functions, we prove in arbitrary finite dimension that the free boundary $\partial\{u>0\} \cup\partia...

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Bibliographic Details
Main Authors Shahgholian, Henrik, Uraltseva, Nina, Weiss, Georg S
Format Journal Article
LanguageEnglish
Published 20.12.2007
Subjects
Mathematics - Analysis of PDEs
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Summary:For the parabolic obstacle-problem-like equation $$\Delta u - \partial_t u = \lambda_+ \chi_{\{u>0\}} - \lambda_- \chi_{\{u<0\}} ,$$ where $\lambda_+$ and $\lambda_-$ are positive Lipschitz functions, we prove in arbitrary finite dimension that the free boundary $\partial\{u>0\} \cup\partial\{u<0\}$ is in a neighborhood of each ``branch point'' the union of two Lipschitz graphs that are continuously differentiable with respect to the space variables. The result extends the elliptic paper \cite{imrn} to the parabolic case. The result is optimal in the sense that the graphs are in general not better than Lipschitz, as shown by a counter-example.
DOI:10.48550/arxiv.0712.3411