Unstable Free Boundary Problems in Solid Combustion

We discuss unstable free boundary problems arising in SHS/gasless combustion and smoldering combustion. We focus on a result obtained in collaboration with Régis Monneau (CERMICS, Paris) concerning the stationary equation $$\mathrm{\Delta }u=-{\chi }_{\{u>0\}}$$ arising as first order approximation in solid combustion: some partial regularity holds for maximal solutions of this equation. In particular, in two space dimensions the interface consists locally of a finite number of $C1$-arcs meeting in quadruple junctions.