The two-phase fractional obstacle problem

We study minimizers of the functional $\int_{B_1^+}|\nabla u|^2 x_n^a\,d x +2\int_{B_1'} (λ_+ u^++λ_- u^-)\,d x',$ 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 $Γ^+=\partial'\{u(\cdot,0)>0\}$ and $Γ^-=\partial'\{u(\cdot,0)<0\}$ when $a\geq 0$.

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