An unstable elliptic free boundary problem arising in solid combustion

Abstract

We prove a regularity result for the unstable elliptic free boundary problem Δu=-χ{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|{u>0},u|{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