This paper deals with the fractional Sobolev spaces W s;p . We analyze the relations among some of their possible denitions and their role in the trace theory. We prove continuous and compact embeddings, investigating the problem of the extension domains and other regularity results. Most of the results we present here are probably well known to the experts, but we believe that our proofs are original and we do not make use of any interpolation techniques nor pass through the theory of Besov spaces. We also present some counterexamples in non-Lipschitz domains.

/pdf/hitchhiker-s-guide-to-the-fractional-sobolev-spaces-irm2sao95g.pdf

Hitchhiker's guide to the fractional Sobolev spaces

Given a function ϕ and s ∈ (0, 1), we will study the solutions of the following obstacle problem: • u ≥ ϕ in R n , • (−� ) s u ≥ 0i nR n , • (−� ) s u(x) = 0 for those x such that u(x )>ϕ (x), • lim|x|→+∞ u(x) = 0. We show that when ϕ is C 1,s or smoother, the solution u is in the space C 1,α for every α< s. In the case where the contact set {u = ϕ} is convex, we prove the optimal regularity result u ∈ C 1,s . When ϕ is only C 1,β for a β< s, we prove that our solution u is C 1,α for every α< β. c � 2006 Wiley Periodicals, Inc.

/pdf/regularity-of-the-obstacle-problem-for-a-fractional-power-of-1j9n4p62qu.pdf

Regularity of the obstacle problem for a fractional power of the laplace operator

A new collection of real world applications of fractional calculus in science and engineering

Motivated by the critical dissipative quasi-geostrophic equation, we prove that drift-diffusion equations with L 2 initial data and minimal assumptions on the drift are locally Holder continuous. As an application we show that solutions of the quasi-geostrophic equation with initial L 2 data and critical diffusion (-Δ) 1/2 are locally smooth for any space dimension.

http://annals.math.princeton.edu/wp-content/uploads/annals-v171-n3-p10-p.pdf

Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation

/pdf/the-dirichlet-problem-for-the-fractional-laplacian-1w4qdfqmwi.pdf

The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary

The operator square root of the Laplacian (−△) 1/2 can be obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition. In this paper we obtain similar characterizations for general fractional powers of the Laplacian and other integro-differential operators. From those characterizations we derive some properties of these integro-differential equations from purely local arguments in the extension problems.

/pdf/an-extension-problem-related-to-the-fractional-laplacian-409vqt2buh.pdf

An Extension Problem Related to the Fractional Laplacian

We consider nonlinear integro-differential equations, like the ones that arise from stochastic control problems with purely jump Levy processes. We obtain a nonlocal version of the ABP estimate, Harnack inequality, and interior C 1,� regularity for general fully nonlinear integro- differential equations. Our estimates remain uniform as the degree of the equation approaches two, so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations.

/pdf/regularity-theory-for-fully-nonlinear-integro-differential-3e4eszd9bu.pdf

Regularity theory for fully nonlinear integro‐differential equations

We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem. We write an equivalent characterization as a thin obstacle problem. In this way we are able to apply local type arguments to obtain sharp regularity estimates for the solution and study the regularity of the free boundary.

/pdf/regularity-estimates-for-the-solution-and-the-free-boundary-o5t0g6mems.pdf

Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian

We prove general uniqueness results for radial solutions of linear and nonlinear equations involving the fractional Laplacian (−Δ)^s with s ∊ (0,1) for any space dimensions N ≥ 1. By extending a monotonicity formula found by Cabre and Sire , we show that the linear equation 
(−Δ)^s u + Vu=0 in R^N
has at most one radial and bounded solution vanishing at infinity, provided that the potential V is radial and nondecreasing. In particular, this result implies that all radial eigenvalues of the corresponding fractional Schrodinger operator H = (−Δ)^s + V are simple. Furthermore, by combining these findings on linear equations with topological bounds for a related problem on the upper half-space R^N_+^(+1), we show uniqueness and nondegeneracy of ground state solutions for the nonlinear equation 
(−Δ)^s Q + Q- │Q│^ɑ Q=0 in R^N 
for arbitrary space dimensions N ≥ 1 and all admissible exponents α > 0. This generalizes the nondegeneracy and uniqueness result for dimension N = 1 recently obtained by the first two authors and, in particular, the uniqueness result for solitary waves of the Benjamin-Ono equation found by Amick and Toland.

/pdf/uniqueness-of-radial-solutions-for-the-fractional-laplacian-3qutbu7q5f.pdf

Luis Silvestre

Papers

An Extension Problem Related to the Fractional Laplacian

Regularity of the obstacle problem for a fractional power of the laplace operator

Regularity theory for fully nonlinear integro‐differential equations

Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian

Uniqueness of Radial Solutions for the Fractional Laplacian