An Unstable Elliptic Free Boundary Problem arising in Solid Combustion

TL;DR

This paper proves regularity results for an unstable elliptic free boundary problem related to solid combustion, showing that solutions and free boundaries are analytic under certain conditions and characterizing singularities.

Contribution

It establishes the regularity and analyticity of solutions and free boundaries for an unstable elliptic free boundary problem, including partial regularity results and discussion of singularities.

Findings

Maximal solutions and local minimizers have analytic free boundaries.

Solutions non-degenerate of second order have analytic free boundaries except on a set of Hausdorff dimension n-2.

The paper discusses possible singularities in the free boundary.

Abstract

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.