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

Henrik Shahgholian, Nina Uraltseva, Georg S. Weiss

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 arb…