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We study the regularity of the free boundary in the obstacle problem for the fractional Laplacian under the assumption that the obstacle φ satisfies Δφ ≤ 0 near the contact region. Our main result establishes that the free boundary consists of a set of regular points, which is known to be a (n-1)-dimensional C1,α manifold by the results obtained by Caffarelli, Salsa, and Silvestre (Invent. Math. 2008), and a set of singular points, which we prove to be contained in a union of k-dimensional C1-submanifold, k = 0, . . . , n-1. Such a complete result on the structure of the free boundary, proved by L. A. Caffarelli (Acta Math. 1977 and J. Fourier Anal. Appl. 1998), was known only in the case of the classical Laplacian, and it is new even for the Signorini problem (which corresponds to the particular case of the ½-fractional Laplacian). A key ingredient behind our results is the validity of a new non-degeneracy condition supBr(x0)(u-φ) ≥ cr2, valid at all free boundary points x0. [ABSTRACT FROM AUTHOR]

We study the positivity and regularity of solutions to the fractional porous medium equations [ABSTRACT FROM AUTHOR]

We study the obstacle problem for integro-differential operators of order 2 s, with $$s\in (0,1)$$ . Our main result establish that the free boundary is $$C^{1,\gamma }$$ and $$u\in C^{1,s}$$ near all regular points. Namely, we prove the following dichotomy at all free boundary points $$x_0\in \partial \{u=\varphi \}$$ : where d is the distance to the contact set $$\{u=\varphi \}$$ . Moreover, we show that the set of free boundary points $$x_0$$ satisfying (i) is open, and that the free boundary is $$C^{1,\gamma }$$ and $$u\in C^{1,s}$$ near those points. These results were only known for the fractional Laplacian [2], and are completely new for more general integro-differential operators. The methods we develop here are purely nonlocal, and do not rely on any monotonicity-type formula for the operator. Thanks to this, our techniques can be applied in the much more general context of fully nonlinear integro-differential operators: we establish similar regularity results for obstacle problems with convex operators. [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]

Abstract: The parabolic obstacle problem for the fractional Laplacian naturally arises in American option models when the asset prices are driven by pure‐jump Lévy processes. In this paper we study the regularity of the free boundary. Our main result establishes that, when s > 1 2, the free boundary is a C1,α graph in x and t near any regular free boundary point ( x 0 , t 0 ) ∈ ∂ { u > φ }. Furthermore, we also prove that solutions u are C1 + s in x and t near such points, with a precise expansion of the form u ( x , t ) − φ ( x ) = c 0 ( ( x − x 0 ) ⋅ e + κ ( t − t 0 ) ) + 1 + s + o ( | x − x 0 | 1 + s + α + | t − t 0 | 1 + s + α ) , with c 0 > 0 , e ∈ S n − 1, and a > 0. © 2018 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

In this paper we prove the Pohozaev identity for the semilinear Dirichlet problem $${(-\Delta)^s u =f(u)}$$ in $${\Omega, u\equiv0}$$ in $${{\mathbb R}^n\backslash\Omega}$$ . Here, $${s\in(0,1)}$$ , (−Δ) is the fractional Laplacian in $${\mathbb{R}^n}$$ , and Ω is a bounded C domain. To establish the identity we use, among other things, that if u is a bounded solution then $${u/\delta^s|_{\Omega}}$$ is C up to the boundary ∂Ω, where δ( x) = dist( x, ∂Ω). In the fractional Pohozaev identity, the function $${u/\delta^s|_{\partial\Omega}}$$ plays the role that ∂u/ ∂ν plays in the classical one. Surprisingly, from a nonlocal problem we obtain an identity with a boundary term (an integral over ∂Ω) which is completely local. As an application of our identity, we deduce the nonexistence of nontrivial solutions in star-shaped domains for supercritical nonlinearities. [ABSTRACT FROM AUTHOR]

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]

We establish sharp higher-order Hölder regularity estimates up to the boundary for solutions to equations of the form ∂tu-Lu=f(t,x) in I×Ω where I⊂R, Ω⊂Rn and f is Hölder continuous. The nonlocal operators L that we consider are those arising in stochastic processes with jumps, such as the fractional Laplacian (-Δ)s, s∈(0,1). Our main result establishes that, if f is Cγ is space and Cγ/2s in time, and Ω is a C2,γ domain, then u/ds is Cs+γ up to the boundary in space and u is C1+γ/2s up the boundary in time, where d is the distance to ∂Ω. This is the first higher order boundary regularity estimate for nonlocal parabolic equations, and is new even for the fractional Laplacian in C∞ domains. [ABSTRACT FROM AUTHOR]

We study the regularity of the free boundary in the obstacle for the p -Laplacian, min ⁡ { − Δ p u , u − φ } = 0 in Ω ⊂ R n . Here, Δ p u = div ( | ∇ u | p − 2 ∇ u ) , and p ∈ ( 1 , 2 ) ∪ ( 2 , ∞ ) . Near those free boundary points where ∇ φ ≠ 0 , the operator Δ p is uniformly elliptic and smooth, and hence the free boundary is well understood. However, when ∇ φ = 0 then Δ p is singular or degenerate, and nothing was known about the regularity of the free boundary at those points. Here we study the regularity of the free boundary where ∇ φ = 0 . On the one hand, for every p ≠ 2 we construct explicit global 2-homogeneous solutions to the p -Laplacian obstacle problem whose free boundaries have a corner at the origin. In particular, we show that the free boundary is in general not C 1 at points where ∇ φ = 0 . On the other hand, under the “concavity” assumption | ∇ φ | 2 − p Δ p φ < 0 , we show the free boundary is countably ( n − 1 ) -rectifiable and we prove a nondegeneracy property for u at all free boundary points. [ABSTRACT FROM AUTHOR]