For a bounded domain $D\subset \R^n$, we study minimizers of the energy functional 

$$
\int_{D}{|\nabla u|^2}\,dx + \int_{D \cap (\R^{n-1} \times \{0\} )}{\lambda^+ \chi_{ \{u > 0\} } + \lambda^- \chi_{ \{u<0\} }}\, d\H^{n-1},
$$

 without any sign restriction on the function $u$. One of the main result states that the free boundaries 

$$
\Gamma^+ = \partial \{u(\cdot,0) > 0\}\quad \text{and}\quad \Gamma^- = \partial \{u(\cdot, 0) < 0\}
$$

 never touch. Moreover, using Alexandrov-type reflection technique, we can show that in dimension $n=3$ the free boundaries are $C^1$ regular on a dense subset.

A two-phase problem with a lower-dimensional free boundary

Mark Allen

Arshak Petrosyan

Abstract

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