An Unstable Elliptic Free Boundary Problem arising in Solid Combustion

Details An Unstable Elliptic Free Boundary Problem arising in Solid Combustion An Unstable Elliptic Free Boundary Problem arising in Solid Combustion

2005-07-15

arxiv.org/abs/math/0507315v1

Mathematics

Analysis of PDEs

Scientific paper

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.

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