The Two-Phase Membrane Problem -- an Intersection-Comparison Approach to the Regularity at Branch Points

Abstract

For the two-phase membrane problem $\Delta u=\frac{{\lambda }_{+}}{2}{\chi }_{\{u>0\}}-\frac{{\lambda }_{-}}{2}{\chi }_{\{u<0\}},$ where ${\lambda }_{+}>0$ and ${\lambda }_{-}>0,$ we prove in two dimensions that the free boundary is in a neighborhood of each ``branch point'' the union of two $C^1$-graphs. We also obtain a stability result with respect to perturbations of the boundary data. Our analysis uses an intersection-comparison approach based on the Aleksandrov reflection. In higher dimensions we show that the free boundary has finite $(n-1)$-dimensional Hausdorff measure.