An Unstable Elliptic Free Boundary Problem arising in Solid Combustion

We prove a regularity result for the unstable elliptic free boundary problem $\mathrm{\Delta }u=-{\chi }_{\{u>0\}}$ related to traveling waves in a problem arising in solid combustion. The maximal solution and every local minimizer of the energy are regular, that is, $\{u=0\}$ is locally an analytic surface and $u|{\bar{{u>0}}}, u|{\bar{{u<0}}}$ are locally analytic functions. Moreover we prove a partial regularity result for solutions that are non-degenerate of second order: here $\{u=0\}$ is analytic up to a closed set of Hausdorff dimension $n-2$. We discuss possible singularities.