A solution to parabolic system with the fractional Laplacian

The existence of a solution to the parabolic system with the fractional Laplacian （-△） α/2, α 〉 0 is proven, this solution decays at different rates along different time sequences going to infinity. As an application, the existence of a solution to the generalized Navier-Stokes equations is proven, which decays at different rates along different time sequences going to infinity. The generalized Navier-Stokes equations are the equations resulting from replacing -△ in the Navier-Stokes equations by （-△）^m, m〉 0. At last, a similar result for 3-D incompressible anisotropic Navier-Stokes system is obtained.     

Variational inequalities for the spectral fractional Laplacian

In this paper we study obstacle problems for the Navier (spectral) fractional Laplacian (?Δ_Ω) ^s of order s ∈ (0,1) in a bounded domain Ω ? R ⁿ .     

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

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.     

An extension problem for the CR fractional Laplacian

We show that the conformally invariant fractional powers of the sub-Laplacian on the Heisenberg group are given in terms of the scattering operator for an extension problem to the Siegel upper halfspace. Remarkably, this extension problem is different from the one studied, among others, by Caffarelli and Silvestre. We also prove an energy identity that yields a sharp trace Sobolev embedding.     

Symmetry results for systems involving fractional Laplacian

In this paper we investigate symmetry results for positive solutions of systems involving the fractional Laplacian (1) $\left\{ \begin{gathered} ( - \Delta )^{\alpha _1 } u_1 (x) = f_1 (u_2 (x)),x \in \mathbb{R}^\mathbb{N} , \hfill \\ ( - \Delta )^{\alpha _2 } u_2 (x) = f_2 (u_1 (x)),x \in \mathbb{R}^\mathbb{N} , \hfill \\ \lim _{|x| \to \infty } u_1 (x) = \lim _{|x| \to \infty } u_2 (x) = 0 \hfill \\ \end{gathered} \right. $ where N ≥ 2 and α ₁, α ₂ ∈ (0, 1). We prove symmetry properties by the method of moving planes.     

Bounding the gap between extremal Laplacian eigenvalues of graphs

Let G be a graph whose Laplacian eigenvalues are 0 = λ₁ ? λ₂ ? ? ? λ_n. We investigate the gap (expressed either as a difference or as a ratio) between the extremal non-trivial Laplacian eigenvalues of a connected graph (that is λ_n and λ₂). This gap is closely related to the average density of cuts in a graph. We focus here on the problem of bounding the gap from below.     

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

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 (−Δ)^su=g${(-\mathrm{\Delta })}^{s}u=g$ in Ω  , u≡0$u\equiv 0$ in Rⁿ\Ω${\mathbb{R}}^n\backslash \Omega$, for some s∈(0,1)$s\in (0,1)$ and g∈L^∞(Ω)$g\in L^{\infty }(\Omega )$, then u   is C^s(Rⁿ)$C^s({\mathbb{R}}^n)$ and u/δ^s_|Ω$u/{\delta }^s{|}_{\Omega }$ is C^α$C^{\alpha }$ up to the boundary ∂Ω   for some α∈(0,1)$\alpha \in (0,1)$, where δ(x)=dist(x,∂Ω)$\delta (x)=\mathrm{dist}(x,\partial \Omega )$. For this, we develop a fractional analog of the Krylov boundary Harnack method.     

The spectral collocation method for efficiently solving PDEs with fractional Laplacian

We derive a spectral collocation approximation to the fractional Laplacian operator based on the Riemann-Liouville fractional derivative operators on a bounded domain Ω = [a, b]. Corresponding matrix representations of (?△) ^α/2 for α ∈ (0,1) and α ∈ (1,2) are obtained. A space-fractional advection-dispersion equation is then solved to investigate the numerical performance of this method under various choices of parameters. It turns out that the proposed method has high accuracy and is efficient for solving these space-fractional advection-dispersion equations when the forcing term is smooth.     

奇异分数阶Laplacian方程正弱解的存在性及正则性

因为奇异项使得分数阶Laplacian方程没有变分结构,所以临界点理论不能直接使用,成为研究此类方程弱解存在性的本质困难.本文首次运用闭锥上的临界点理论,得到奇异分数阶Laplacian方程的正弱解及其正则性.而且,此方法适用于其他奇异分数阶问题.     

New extremal domains for the first eigenvalue of the Laplacian in flat tori

We prove the existence of nontrivial compact extremal domains for the first eigenvalue of the Laplacian in manifolds ${\mathbb{R}^{n}\times \mathbb{R}{/}T\, \mathbb{Z}}$ with flat metric, for some T > 0. These domains are close to the cylinder-type domain ${B_1 \times \mathbb{R}{/}T\, \mathbb{Z}}$ , where B ₁ is the unit ball in ${\mathbb{R}^{n}}$ , they are invariant by rotation with respect to the vertical axe, and are not invariant by vertical translations. Such domains can be extended by periodicity to nontrivial and noncompact domains in Euclidean spaces whose first eigenfunction of the Laplacian with 0 Dirichlet boundary condition has also constant Neumann data at the boundary.     

Time-dependent gradient perturbations of fractional Laplacian

We construct a fundamental solution of the equation ${\partial_t - \Delta^{\alpha/2} - b(\cdot, \cdot) \cdot\nabla_{x} = 0}We construct a fundamental solution of the equation ?_t - D^a/2 - b(·, ·) ·?_x = 0{\partial_t - \Delta^{\alpha/2} - b(\cdot, \cdot) \cdot\nabla_{x} = 0} for a ? (1, 2){\alpha \in (1, 2)} and b satisfying a certain integral space-time condition. We also show it has α-stable upper and lower bounds.     

Decay of mass for nonlinear equation with fractional Laplacian

The large time behavior of non-negative solutions to the reaction–diffusion equation ?_t u=-(-D)^a/2u - u^p{\partial_t u=-(-\Delta)^{\alpha/2}u - u^p}, ${(\alpha\in(0,2], \;p > 1)}${(\alpha\in(0,2], \;p > 1)} posed on \mathbbR^N{\mathbb{R}^N} and supplemented with an integrable initial condition is studied. We show that the anomalous diffusion term determines the large time asymptotics for p > 1 + α/N, while nonlinear effects win if p ≤ 1 + α/N.     

Decay of mass for nonlinear equation with fractional Laplacian

The large time behavior of non-negative solutions to the reaction–diffusion equation ${\partial_t u=-(-\Delta)^{\alpha/2}u - u^p}$ , ${(\alpha\in(0,2], \;p > 1)}$ posed on ${\mathbb{R}^N}$ and supplemented with an integrable initial condition is studied. We show that the anomalous diffusion term determines the large time asymptotics for p > 1 + α/N, while nonlinear effects win if p ≤ 1 + α/N.     

Interior regularity results for zeroth order operators approaching the fractional Laplacian

In this article we are interested in interior regularity results for the solution \({\mu _ \in } \in C(\bar \Omega )\) of the Dirichlet problem

$$\{ _{\mu = 0in{\Omega ^c},}^{{I_ \in }(\mu ) = {f_ \in }in\Omega }$$

where Ω is a bounded, open set and \({f_ \in } \in C(\bar \Omega )\) for all ? ∈ (0, 1). For some σ ∈ (0, 2) fixed, the operator \(\mathcal{I}_{\in}\) is explicitly given by

$${I_ \in }(\mu ,x) = \int_{{R^N}} {\frac{{[\mu (x + z) - \mu (x)]dz}}{{{ \in ^{N + \sigma }} + |z{|^{N + \sigma }}}}} ,$$

which is an approximation of the well-known fractional Laplacian of order σ, as ? tends to zero. The purpose of this article is to understand how the interior regularity of u? evolves as ? approaches zero. We establish that u_? has a modulus of continuity which depends on the modulus of f_?, which becomes the expected Hölder profile for fractional problems, as ? → 0. This analysis includes the case when f? deteriorates its modulus of continuity as ? → 0.

     

Regularity in a one-phase free boundary problem for the fractional Laplacian

For a one-phase free boundary problem involving a fractional Laplacian, we prove that “flat free boundaries” are C^1,α$C^{1,\alpha }$. We recover the regularity results of Caffarelli for viscosity solutions of the classical Bernoulli-type free boundary problem with the standard Laplacian.