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

For the two-phase membrane problem$$\mathrm\Delta\hspace{0.2em}\mathrm u\hspace{0.4em}=\hspace{0.4em}{\mathrm\lambda}_{+\mathrm\chi\{\mathrm u>0\}}\hspace{1em}-\hspace{1em}{\mathrm\lambda}_{-\mathrm\chi\{\mathrm u<0\}}\text{,}$$where λ+ and λ− are positive Lipschitz functions, and we prove in higher dimensions that the free boundary is in a neighborhood of each “branch point” the union of two C¹-graphs. The result is optimal in the sense that these graphs are in general not of class C^{1, Dini}, as shown by a counter-example. As an application we obtain a stability result with respect to perturbations of the boundary data.