Mathematical Research Letters

Volume 19 (2012)

Number 5

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

Pages: 1055 – 1074

DOI: http://dx.doi.org/10.4310/MRL.2012.v19.n5.a8

Author

Mark Allen (Department of Mathematics, Purdue University, West Lafayette, Indiana, U.S.A.)

Abstract

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., $\Gamma^+ \cap \Gamma^- = \emptyset$.

Full Text (PDF format)