An unstable elliptic free boundary problem arising in solid combustion

Abstract

We prove a regularity result for the unstable elliptic free boundary problem ${\displaystyle \Delta u=-{\chi }_{\{u>0\}}}$ 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 $u|{}_{\stackrel{̲}{\{u>0\}}},u|{}_{\stackrel{̲}{\{u<0\}}}$ are locally analytic functions. Moreover, we prove a partial regularity result for solutions that are nondegenerate of second order. Here $\{u=0\}$ is analytic up to a closed set of Hausdorff dimension $n-2$. We discuss possible singularities