  We prove a regularity result for the unstable elliptic free boundary problem
$\Delta u = -\chi_{\{u>0\}}$ 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.
