Regularity for non-local almost minimal boundaries and applications

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We introduce a notion of non-local almost minimal boundaries similar to that introduced by Almgren in geometric measure theory. Extending methods developed recently for non-local minimal surfaces we prove that flat non-local almost minimal boundaries are smooth. This can be viewed as a non-local version of the Almgren-De Giorgi-Tamanini regularity theory. The main result has several applications, among these C¹,α regularity for sets with prescribed non-local mean curvature in Lᵖ and regularity of solutions to non-local obstacle problems.