The Two-Phase Membrane Problem--Regularity of the Free Boundaries in Higher Dimensions

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For the two-phase membrane problem \Delta u=\lambda_{+}\chi_{\{u>0\}}-\lambda_{-}\chi_{\{u<0\}}, where \lambda_{+} and \lambda_{-} are positive Lipschitz functions, we prove in higher dimensions that the free boundary is in a neighborhood of each "branch point'; the union of two C^{1}-graphs. The result is optimal in the sense that these graphs are in general not of class C^{1,\mbox{Dini}}, as shown by a counter-example. As application we obtain a stability result with respect to perturbations of the boundary data.

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