JournalsifbVol. 14, No. 3pp. 307–342

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

  • Mark Allen

    Purdue University, West Lafayette, USA
  • Arshak Petrosyan

    Purdue University, West Lafayette, USA
A two-phase problem with a lower-dimensional free boundary cover

Abstract

For a bounded domain DRnD\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 uu. One of the main result states that the free boundaries

Γ+={u(,0)>0}andΓ={u(,0)<0}\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=3n=3 the free boundaries are C1C^1 regular on a dense subset.

Cite this article

Mark Allen, Arshak Petrosyan, A two-phase problem with a lower-dimensional free boundary. Interfaces Free Bound. 14 (2012), no. 3, pp. 307–342

DOI 10.4171/IFB/283