2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/74036For the two-phase membrane problem $ Δu = {λ_+\over 2} χ_{\{u>0\}} - {λ_-\over 2} χ_{\{u<0\}} ,$ where $λ_+> 0$ and $λ_->0 ,$ we prove in two dimensions that the free boundary is in a neighborhood of each ``branch point'' the union of two $C^1$-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.Analysis of PDEstext