The fractional unstable obstacle problem

We study a model for combustion on a boundary. Specifically, we study certain generalized solutions of the equation \[ (-\Delta)^s u = \chi_{\{u>c\}} \] for $0<s<1$ and an arbitrary constant $c$. Our main object of study is the free boundary $\partial\{u>c\}$. We study the behavior of the free boundary and prove an upper bound for the Hausdorff dimension of the singular set. We also show that when $s\leq 1/2$ certain symmetric solutions are stable; however, when $s>1/2$ these solutions are not stable and therefore not minimizers of the corresponding functional.