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We study the singular part of the free boundary in the obstacle problem for the fractional Laplacian, min{(-Δ)su, u - φ} = 0 in Rn, for general obstacles φ. Our main result establishes the complete structure and regularity of the singular set. To prove it, we construct new monotonicity formulas of Monneau-type that extend those in those of Garofalo-Petrosyan to all s ∈ (0, 1). [ABSTRACT FROM AUTHOR]

We introduce a new Neumann problem for the fractional Laplacian arising from a simple probabilistic consideration, and we discuss the basic properties of this model. We can consider both elliptic and parabolic equations in any domain. In addition, we formulate problems with nonhomogeneous Neumann conditions, and also with mixed Dirichlet and Neumann conditions, all of them having a clear probabilistic interpretation. We prove that solutions to the fractional heat equation with homogeneous Neumann conditions have the following natural properties: conservation of mass inside Ω, decreasing energy, and convergence to a constant as t→∞. Moreover, for the elliptic case we give the variational formulation of the problem, and establish existence of solutions. We also study the limit properties and the boundary behavior induced by this nonlocal Neumann condition. For concreteness, one may think that our nonlocal analogue of the classical Neumann condition ∂νu = 0 on ∂Ω consists in the nonlocal prescription ∫Ωu(x) − u(y)/∣x − y∣n+2s dy = 0 for x ∈ Rn \ Ω. We made an effort to keep all the arguments at the simplest possible technical level, in order to clarify the connections between the different scientific fields that are naturally involved in the problem, and make the paper accessible also to a wide, non-specialistic public (for this scope, we also tried to use and compare different concepts and notations in a somehow more unified way). [ABSTRACT FROM AUTHOR]

We study the interior regularity of solutions to the Dirichlet problem L u = g in Ω, u = 0 in R n ∖ Ω , for anisotropic operators of fractional type L u ( x ) = ∫ 0 + ∞ d ρ ∫ S n − 1 d a ( ω ) 2 u ( x ) − u ( x + ρ ω ) − u ( x − ρ ω ) ρ 1 + 2 s . Here, a is any measure on S n − 1 (a prototype example for L is given by the sum of one-dimensional fractional Laplacians in fixed, given directions). When a ∈ C ∞ ( S n − 1 ) and g is C ∞ ( Ω ) , solutions are known to be C ∞ inside Ω (but not up to the boundary). However, when a is a general measure, or even when a is L ∞ ( S n − 1 ) , solutions are only known to be C 3 s inside Ω. We prove here that, for general measures a , solutions are C 1 + 3 s − ϵ inside Ω for all ϵ > 0 whenever Ω is convex. When a ∈ L ∞ ( S n − 1 ) , we show that the same holds in all C 1 , 1 domains. In particular, solutions always possess a classical first derivative. The assumptions on the domain are sharp, since if the domain is not convex and the measure a is singular, we construct an explicit counterexample for which u is not C 3 s + ϵ for any ϵ > 0 – even if g and Ω are C ∞ . [ABSTRACT FROM AUTHOR]

We prove nonexistence of nontrivial bounded solutions to some nonlinear problems involving nonlocal operators of the form These operators are infinitesimal generators of symmetric Lévy processes. Our results apply to even kernelsKsatisfying thatK(y)|y|n+σis nondecreasing along rays from the origin, for some σ ∈ (0, 2) in caseaij ≡ 0 and for σ = 2 in case that (aij) is a positive definite symmetric matrix. Our nonexistence results concern Dirichlet problems forLin star-shaped domains with critical and supercritical nonlinearities (where the criticality condition is in relation tonand σ). We also establish nonexistence of bounded solutions to semilinear equations involving other nonlocal operators such as the higher order fractional Laplacian (− Δ)s(heres > 1) or the fractionalp-Laplacian. All these nonexistence results follow from a general variational inequality in the spirit of a classical identity by Pucci and Serrin. [ABSTRACT FROM AUTHOR]

Abstract: We study the problem in a bounded domain , where λ is a positive parameter. More precisely, we study the regularity of the extremal solution to this problem. Our main result yields the boundedness of the extremal solution in dimensions for all whenever Ω is, for every , convex in the -direction and symmetric with respect to . The same holds if and , or if and . These results are new even in the unit ball . [Copyright &y& Elsevier]

Abstract: We study the regularity up to the boundary of solutions to the Dirichlet problem for the fractional Laplacian. We prove that if u is a solution of in Ω, in , for some and , then u is and is up to the boundary ∂Ω for some , where . For this, we develop a fractional analog of the Krylov boundary Harnack method. Moreover, under further regularity assumptions on g we obtain higher order Hölder estimates for u and . Namely, the norms of u and in the sets are controlled by and , respectively. These regularity results are crucial tools in our proof of the Pohozaev identity for the fractional Laplacian (Ros-Oton and Serra, 2012 [19,20]). [Copyright &y& Elsevier]