The two-phase membrane problem—An intersection-comparison approach to the regularity at branch points

For the two-phase membrane problemΔu=λ+2χ{u>0}-λ-2χ{u<0},where λ+>0λ+>0 and λ->0, we prove in two dimensions that the free boundary is in a neighborhood of each “branch point” the union of two C1C1-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)(n-1)-dimensional Hausdorff measure.