Regularity of the free boundary in a two-phase semilinear problem in two dimensions Erik LindgrenA. Petrosyan 35R35regularity of the free boundarytwo-phase semilinear problemquenching problemmonotonicity formulaAlexandrov reflection-comparison 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. Indiana University Mathematics Journal 2008 text pdf 10.1512/iumj.2008.57.3433 10.1512/iumj.2008.57.3433 en Indiana Univ. Math. J. 57 (2008) 3397 - 3418 state-of-the-art mathematics http://iumj.org/access/