C^{2,α}$ regularity of flat free boundaries for the thin one-phase problem

We prove $C^{2,α}$ regularity of sufficiently flat free boundaries, for the thin one-phase problem in which the free boundary occurs on a lower dimensional subspace. This problem appears also as a model of a one-phase free boundary problem in the context of the fractional Laplacian $(-Δ)^{1/2}$.

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