# Interfaces and Free Boundaries

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**Volume 14, Issue 3, 2012, pp. 307–342**

**DOI: 10.4171/IFB/283**

Published online: 2012-11-07

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

Mark Allen^{[1]}and Arshak Petrosyan

^{[2]}(1) Purdue University, West Lafayette, USA

(2) Purdue University, West Lafayette, USA

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.

*Keywords: *Two-phase free boundary problem, lower-dimensional free boundary, separation of phases, regularity of the free boundary, monotonicity formula, Alexandrov reflection technique, Steiner symmetrization

Allen Mark, Petrosyan Arshak: A two-phase problem with a lower-dimensional free boundary. *Interfaces Free Bound.* 14 (2012), 307-342. doi: 10.4171/IFB/283