The fractional unstable obstacle problem. (April 2020) - Allen, Mark; Smit Vega Garcia, Mariana
Abstract: We study a model for combustion on a boundary. Specifically, we study certain generalized solutions of the equation ( − Δ ) s u = χ { u > c } for 0 < s < 1 and an arbitrary constant c . Our main object of study is the free boundary ∂ { 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 ≤ 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.