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 .     

The extremal solution for the fractional Laplacian

We study the extremal solution for the problem \((-\Delta )^s u=\lambda f(u)\) in \(\Omega \) , \(u\equiv 0\) in \(\mathbb R ^n\setminus \Omega \) , where \(\lambda >0\) is a parameter and \(s\in (0,1)\) . We extend some well known results for the extremal solution when the operator is the Laplacian to this nonlocal case. For general convex nonlinearities we prove that the extremal solution is bounded in dimensions \(n<4s\) . We also show that, for exponential and power-like nonlinearities, the extremal solution is bounded whenever \(n<10s\) . In the limit \(s\uparrow 1\) , \(n<10\) is optimal. In addition, we show that the extremal solution is \(H^s(\mathbb R ^n)\) in any dimension whenever the domain is convex. To obtain some of these results we need \(L^q\) estimates for solutions to the linear Dirichlet problem for the fractional Laplacian with \(L^p\) data. We prove optimal \(L^q\) and \(C^\beta \) estimates, depending on the value of \(p\) . These estimates follow from classical embedding results for the Riesz potential in \(\mathbb R ^n\) . Finally, to prove the \(H^s\) regularity of the extremal solution we need an \(L^\infty \) estimate near the boundary of convex domains, which we obtain via the moving planes method. For it, we use a maximum principle in small domains for integro-differential operators with decreasing kernels.     

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.     

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.     

Asymptotically compatible schemes for the approximation of fractional Laplacian and related nonlocal diffusion problems on bounded domains

Approximations of solutions of fractional Laplacian equations on bounded domains are considered. Such equations allow global interactions between points separated by arbitrarily large distances. Two approximations are introduced. First, interactions are localized so that only points less than some specified distance, referred to as the interaction radius, are allowed to interact. The resulting truncated problem is a special case of a more general nonlocal diffusion problem. The second approximation is the spatial discretization of the related nonlocal diffusion problem. A recently developed abstract framework for asymptotically compatible schemes is applied to prove convergence results for solutions of the truncated and discretized problem to the solutions of the fractional Laplacian problems. Intermediate results also provide new convergence results for the nonlocal diffusion problem. Special attention is paid to limiting behaviors as the interaction radius increases and the spatial grid size decreases, regardless of how these parameters may or may not be dependent. In particular, we show that conforming Galerkin finite element approximations of the nonlocal diffusion equation are always asymptotically compatible schemes for the corresponding fractional Laplacian model as the interaction radius increases and the grid size decreases. The results are developed with minimal regularity assumptions on the solution and are applicable to general domains and general geometric meshes with no restriction on the space dimension and with data that are only required to be square integrable. Furthermore, our results also solve an open conjecture given in the literature about the convergence of numerical solutions on a fixed mesh as the interaction radius increases.     

Maximality of the signless Laplacian energy

We study the problem of determining the graph with $n$ vertices having largest signless Laplacian energy. We conjecture it is the complete split graph whose independent set has (roughly) $2n∕3$ vertices. We show that the conjecture is true for several classes of graphs. In particular, the conjecture holds for the set of all complete split graphs of order $n$, for trees, for unicyclic and bicyclic graphs. We also give conditions on the number of edges, number of cycles and number of small eigenvalues so the graph satisfies the conjecture.     

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.     

Weak convergence to the fractional Brownian sheet in Besov spaces

In this paper we study the problem of the approximation in law of the fractional Brownian sheet in the topology of the anisotropic Besov spaces. We prove the convergence in law of two families of processes to the fractional Brownian sheet: the first family is constructed from a Poisson procces in the plane and the second family is defined by the partial sums of two sequences of real independent fractional brownian motions.     

Left-inverses of fractional Laplacian and sparse stochastic processes

The fractional Laplacian (-\triangle)^g/2(-\triangle)^{\gamma/2} commutes with the primary coordination transformations in the Euclidean space ℝ ^d : dilation, translation and rotation, and has tight link to splines, fractals and stable Levy processes. For 0 < γ < d, its inverse is the classical Riesz potential I _γ which is dilation-invariant and translation-invariant. In this work, we investigate the functional properties (continuity, decay and invertibility) of an extended class of differential operators that share those invariance properties. In particular, we extend the definition of the classical Riesz potential I _γ to any non-integer number γ larger than d and show that it is the unique left-inverse of the fractional Laplacian (-\triangle)^g/2(-\triangle)^{\gamma/2} which is dilation-invariant and translation-invariant. We observe that, for any 1 ≤ p ≤ ∞ and γ ≥ d(1 − 1/p), there exists a Schwartz function f such that I _γ f is not p-integrable. We then introduce the new unique left-inverse I _{γ, p} of the fractional Laplacian (-\triangle)^g/2(-\triangle)^{\gamma/2} with the property that I _{γ, p} is dilation-invariant (but not translation-invariant) and that I _{γ, p} f is p-integrable for any Schwartz function f. We finally apply that linear operator I _{γ, p} with p = 1 to solve the stochastic partial differential equation (-\triangle)^g/2 F = w(-\triangle)^{\gamma/2} \Phi=w with white Poisson noise as its driving term w.     

Weak convergence to the fractional Brownian sheet using martingale differences

We establish an invariance principle for the fractional Brownian sheet, starting from discrete random fields constructed from two-parameter strong martingales. This is an approximation in law of the fractional Brownian sheet in Skorohord space in the plane.     

Bounds for the signless Laplacian energy

The energy of a graph G is the sum of the absolute values of the eigenvalues of the adjacency matrix of G. The Laplacian (respectively, the signless Laplacian) energy of G is the sum of the absolute values of the differences between the eigenvalues of the Laplacian (respectively, signless Laplacian) matrix and the arithmetic mean of the vertex degrees of the graph. In this paper, among some results which relate these energies, we point out some bounds to them using the energy of the line graph of G. Most of these bounds are valid for both energies, Laplacian and signless Laplacian. However, we present two new upper bounds on the signless Laplacian which are not upper bounds for the Laplacian energy.     

Laplacian energy of a graph

Let G be a graph with n vertices and m edges. Let λ₁, λ₂, … , λ_n be the eigenvalues of the adjacency matrix of G, and let μ₁, μ₂, … , μ_n be the eigenvalues of the Laplacian matrix of G. An earlier much studied quantity is the energy of the graph G. We now define and investigate the Laplacian energy as . There is a great deal of analogy between the properties of E(G) and LE(G), but also some significant differences.     

Collapsing to Riemannian manifolds with boundary and the convergence of the eigenvalues of the Laplacian

We prove that the eigenvalues of the Laplacian acting on functions converge to those of the limit manifold for a special collapsing family of closed Riemannian manifolds without curvature bounds. The proof uses L ²-analysis.Dedicated to Professor Hajime Urakawa on his sixtieth birthday.The author is partially supported by the Grant-in-Aid for Scientific Research No. 16740026 of the Japan Society for the Promotion of Science.     

Weak convergence to fractional Brownian motion in Brownian scenery

 Kesten and Spitzer have shown that certain random walks in random sceneries converge to stable processes in random sceneries. In this paper, we consider certain random walks in sceneries defined using stationary Gaussian sequence, and show their convergence towards a certain self-similar process that we call fractional Brownian motion in Brownian scenery. Received: 17 April 2002 / Revised version: 11 October 2002 / Published online: 15 April 2003 Research supported by NSFC (10131040). Mathematics Subject Classification (2002): 60J55, 60J15, 60J65 Key words or phrases: Weak convergence – Random walk in random scenery – Local time – Fractional Brownian motion in Brownian scenery     

On the Laplacian energy of a graph

In this paper we consider the energy of a simple graph with respect to its Laplacian eigenvalues, and prove some basic properties of this energy. In particular, we find the minimal value of this energy in the class of all connected graphs on n vertices (n = 1, 2, ...). Besides, we consider the class of all connected graphs whose Laplacian energy is uniformly bounded by a constant α ⩾ 4, and completely describe this class in the case α = 40.