The Two-Phase Fractional Obstacle Problem

We study minimizers of the functional ∫ B 1 |∇u|xn dx+ 2 ∫ B′ 1 (λ+u + + λ−u −) dx′, for a ∈ (−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+ = ∂′{u(·, 0) > 0} and Γ− = ∂′{u(·, 0) < 0} when a ≥ 0.