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

Abstract

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.