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

Henrik Shahgholian·Georg S. Weiss
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Abstract

For 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.

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Henrik Shahgholian, Georg S. Weiss. 2005-02-01. The Two-Phase Membrane Problem -- an Intersection-Comparison Approach to the Regularity at Branch Points. https://arxiv.org/abs/math/0502015

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