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A parabolic two-phase obstacle-like equation. (English) Zbl 1168.35452
Summary: For the parabolic obstacle-problem-like equation \[ \Delta u - \partial _tu=\lambda _{+}\chi _{\{u>0\}} - \lambda _{ - }\chi _{\{u<0\}}, \] 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 case [the authors, Int. Math. Res. Not. 2007, No. 8, Article ID rnm026 (2007; Zbl 1175.35157)] to the parabolic case. There are substantial difficulties in the parabolic case due to the fact that the time derivative of the solution is in general not a continuous function. Our result is optimal in the sense that the graphs are in general not better than Lipschitz, as shown by a counter-example.

35R35 Free boundary problems for PDEs
35K10 Second-order parabolic equations
35B65 Smoothness and regularity of solutions to PDEs
Zbl 1175.35157
Full Text: DOI Link
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