article

Title: Regularity of the free boundary in a two-phase semilinear problem in two dimensions

Authors: Erik Lindgren and Arshak Petrosyan

Issue: Volume 57 (2008), Issue 7, 3397-3418

Abstract:

$We study minimizers of the energy functional \[ \int_{D}(|\nabla u|^2 + 2(\lambda_{+}(u^{+})^p + \lambda_{-}(u^{-})^p))\,\mathrm{d}x \] for $p \in (0,1)$ without any sign restriction on the function $u$. The main result states that in dimension two the free boundaries $\Gamma^{+} = \partial\{u>0\} \cap D$ and $\Gamma^{-} = \partial\{u<0\} \cap D$ are $C^1$ regular.$