Separation of a lower dimensional free boundary in a two-phase problem

We study minimizers of the energy functional\[\int_{D}{|x_n|^a |\nabla u|^2} + \int_{D \cap ({\mathbb R}^{n-1} \times \{0\} )}{\lambda^+ \chi_{ \{u > 0\} } + \lambda^- \chi_{ \{u<0\} }} \ d{\mathcal{H}}^{n-1}\]without any sign restriction on the function $u$. The main result states that the two free boundaries\[\Gamma^+ = \partial \{u(\ \cdot \ , 0) > 0\} \text{ and }\Gamma^- = \partial \{u(\ \cdot \ , 0) < 0\}\]cannot touch, i.e., $\Gamma^+ \cap \Gamma^- = \emptyset$.

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Published 15 March 2013