Search

The aim of this work is to study homogeneous stable solutions to the thin (or fractional) one-phase free boundary problem. The problem of classifying stable (or minimal) homogeneous solutions in dimensions ≥ 3 is completely open. In this context, axially symmetric solutions are expected to play the same role as Simons' cone in the classical theory of minimal surfaces, but even in this simpler case the problem is open. The goal of this paper is twofold. On the one hand, our first main contribution is to find, for the first time, the stability condition for the thin one-phase problem. Quite surprisingly, this requires the use of "large solutions" for the fractional Laplacian, which blow up on the free boundary. On the other hand, using our new stability condition, we show that any axially symmetric homogeneous stable solution in dimensions ≤ 5 is one-dimensional, independently of the parameter s ∈ (0,1). [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]

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]

We study the regularity of minimizers of the functional E (u) : = [ u ] H s (Ω) 2 + ∫ Ω f u. This corresponds to understanding solutions for the regional fractional Laplacian in Ω ⊂ R N. More precisely, we are interested on the global (up to the boundary) regularity of solutions, both in the case of free minimizers in H s (Ω) (i.e., Neumann problem), or in the case of Dirichlet condition u ∈ H 0 s (Ω) when s > 1 2. Our main result establishes the existence of a constant α s ∈ (0 , 1 − s) satisfying 2 s + α s > 1 such that for all α ∈ (0 , α s) the solution u ∈ C 2 s + α (Ω ‾) in the Neumann case, and u / δ 2 s − 1 ∈ C 1 + α (Ω ‾) in the Dirichlet case. Here, δ is the distance to ∂Ω. We also show the optimality of our result: these estimates fail for α > α s , even when f 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]

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 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]