The two-phase fractional obstacle problem

Abstract

We study minimizers of the functional $${\int }_{B_1^{+}}|\nabla u{|}^2{x}_n^{a}dx+2{\int }_{B_1^{\prime }}({\lambda }_{+}{u}^{+}+{\lambda }_{-}{u}^{-})d{x}^{\prime },$$ for $a\in (-1,1)$. The problem arises in connection with heat flow with control on the boundary. It can also be seen as a non-local analogue of the, by now well studied, two-phase obstacle problem. Moreover, when $u$ does not change signs this is equivalent to the fractional obstacle problem. Our main results are the optimal regularity of the minimizer and the separation of the two free boundaries ${\Gamma }^{+}={\partial }^{\prime }\{u(\cdot ,0)>0\}$ and ${\Gamma }^{-}={\partial }^{\prime }\{u(\cdot ,0)<0\}$ when $a\ge 0$.

27 pages