An unstable elliptic free boundary problem arising in solid combustion

摘要

We prove a regularity result for the unstable elliptic free boundary problem Delta u = -chi{u>0} (0.1) 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 vertical bar({u>0})over bar, u vertical bar({u<0})over bar 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.