On a scattering length for additive functionals and spectrum of fractional Laplacian with a non‐local perturbation

2019 ◽  
Vol 293 (2) ◽  
pp. 327-345 ◽  
Author(s):  
Daehong Kim ◽  
Masakuni Matsuura
Keyword(s):  
Fractional Laplacian ◽  
Scattering Length ◽  
Local Perturbation ◽  
Additive Functionals ◽  
Non Local

New Results from Old Investigation: A Note on Fractional M-Dimensional Differential Operators. The Fractional Laplacian

2015 ◽  
Vol 18 (2) ◽  
Author(s):  
Humberto Prado ◽  
Margarita Rivero ◽  
Juan J. Trujillo ◽  
M. Pilar Velasco
Keyword(s):  
Fractional Integral ◽  
Fractional Laplacian ◽  
Singular Integrals ◽  
Differential Operators ◽  
Riesz Potential ◽  
Fractional Power ◽  
Potential Operators ◽  
Relevant Role ◽  
Non Local ◽  
Fractional Power Of Operators

AbstractThe non local fractional Laplacian plays a relevant role when modeling the dynamics of many processes through complex media. From 1933 to 1949, within the framework of potential theory, the Hungarian mathematician Marcel Riesz discovered the well known Riesz potential operators, a generalization of the Riemann-Liouville fractional integral to dimension higher than one. The scope of this note is to highlight that in the above mentioned works, Riesz also gave the necessary tools to introduce several new definitions of the generalized coupled fractional Laplacian which can be applied to much wider domains of functions than those given in the literature, which are based in both the theory of fractional power of operators or in certain hyper-singular integrals. Moreover, we will introduce the corresponding fractional hyperbolic differential operator also called fractional Lorentzian Laplacian.

scholarly journals Some Remarks on the Duality Method for Integro-Differential Equations with Measure Data

2016 ◽  
Vol 16 (1) ◽  
pp. 115-124 ◽  
Author(s):  
Francesco Petitta
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Bounded Domain ◽  
Fractional Order ◽  
Radon Measure ◽  
Fractional Laplacian ◽  
Measure Data ◽  
Duality Method ◽  
Non Local ◽  
The One

AbstractWe deal with existence, uniqueness and regularity for solutions of the boundary value problem$\left\{\begin{aligned} \displaystyle\mathcal{L}^{s}u&\displaystyle=\mu&&% \displaystyle\text{in }\Omega,\\ \displaystyle u(x)&\displaystyle=0&&\displaystyle\text{on }\mathbb{R}^{n}% \backslash\Omega,\end{aligned}\right.$where Ω is a bounded domain of ${\mathbb{R}^{n}}$, μ is a bounded Radon measure on Ω, and ${\mathcal{L}^{s}}$ is a non-local operator of fractional order s whose kernel K is comparable with the one of the fractional Laplacian.

scholarly journals On the spectrum of two different fractional operators

2014 ◽  
Vol 144 (4) ◽  
pp. 831-855 ◽  
Author(s):  
Raffaella Servadei ◽  
Enrico Valdinoci
Keyword(s):  
Boundary Data ◽  
Fractional Laplacian ◽  
Dirichlet Boundary ◽  
Fractional Operators ◽  
Local Operators ◽  
Non Local ◽  
Definition Of ◽  
Image Position ◽  
The Laplace Operator ◽  
Fractional Type

In this paper we deal with two non-local operators that are both well known and widely studied in the literature in connection with elliptic problems of fractional type. More precisely, for a fixed s ∈ (0,1) we consider the integral definition of the fractional Laplacian given bywhere c(n, s) is a positive normalizing constant, and another fractional operator obtained via a spectral definition, that is,where ei, λi are the eigenfunctions and the eigenvalues of the Laplace operator −Δ in Ω with homogeneous Dirichlet boundary data, while ai represents the projection of u on the direction ei.The aim of this paper is to compare these two operators, with particular reference to their spectrum, in order to emphasize their differences.

scholarly journals On the asymptotic analysis of problems involving fractional Laplacian in cylindrical domains tending to infinity

2016 ◽  
Vol 19 (05) ◽  
pp. 1650035 ◽  
Author(s):  
Indranil Chowdhury ◽  
Prosenjit Roy
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Analysis ◽  
Fractional Laplacian ◽  
Elliptic Problems ◽  
Weighted Estimate ◽  
Dirichlet Problems ◽  
Nonlocal Equations ◽  
Cylindrical Domains ◽  
Non Local ◽  
Second Order Elliptic Problems

The paper is an attempt to investigate the issues of asymptotic analysis for problems involving fractional Laplacian where the domains tend to become unbounded in one-direction. Motivated from the pioneering work on second-order elliptic problems by Chipot and Rougirel in [On the asymptotic behaviour of the solution of elliptic problems in cylindrical domains becoming unbounded, Commun. Contemp. Math. 4(1) (2002) 15–44], where the force functions are considered on the cross-section of domains, we prove the non-local counterpart of their result.Recently in [Asymptotic behavior of elliptic nonlocal equations set in cylinders, Asymptot. Anal. 89(1–2) (2014) 21–35] Yeressian established a weighted estimate for solutions of non-local Dirichlet problems which exhibit the asymptotic behavior. The case when [Formula: see text] was also treated as an example to show how the weighted estimate might be used to achieve the asymptotic behavior. In this paper, we extend this result to each order between [Formula: see text] and [Formula: see text].

scholarly journals Semilinear equations for non-local operators: Beyond the fractional Laplacian

2021 ◽  
Vol 207 ◽  
pp. 112303
Author(s):  
Ivan Biočić ◽  
Zoran Vondraček ◽  
Vanja Wagner
Keyword(s):  
Fractional Laplacian ◽  
Semilinear Equations ◽  
Local Operators ◽  
Non Local

scholarly journals A non-local coupling model involving three fractional Laplacians

2021 ◽  
pp. 2150007
Author(s):  
A. Gárriz ◽  
L. I. Ignat
Keyword(s):  
Fractional Laplacian ◽  
Gradient Flow ◽  
Coupling Model ◽  
Energy Functional ◽  
Diffusion Problem ◽  
Coupling Mechanism ◽  
The Third ◽  
Non Local ◽  
Nash Inequalities ◽  
Laplacian Operators

In this paper, we study a non-local diffusion problem that involves three different fractional Laplacian operators acting on two domains. Each domain has an associated operator that governs the diffusion on it, and the third operator serves as a coupling mechanism between the two of them. The model proposed is the gradient flow of a non-local energy functional. In the first part of the paper, we provide results about existence of solutions and the conservation of mass. The second part encompasses results about the [Formula: see text] decay of the solutions. The third part is devoted to study, the asymptotic behavior of the solutions of the problem when the two domains are a ball and its complementary. Exterior fractional Sobolev and Nash inequalities of independent interest are also provided in Appendix A.

scholarly journals EXISTENCE AND CONCENTRATION OF SOLUTION FOR A NON-LOCAL REGIONAL SCHRÖDINGER EQUATION WITH COMPETING POTENTIALS

2018 ◽  
Vol 61 (2) ◽  
pp. 441-460 ◽  
Author(s):  
CLAUDIANOR O. ALVES ◽  
CÉSAR E. TORRES LEDESMA
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Fractional Laplacian ◽  
Positive Parameter ◽  
Concentration Phenomena ◽  
Formula Group ◽  
Non Local ◽  
Image Position

AbstractIn this paper, we study the existence and concentration phenomena of solutions for the following non-local regional Schrödinger equation $$\begin{equation*} \left\{ \begin{array}{l} \epsilon^{2\alpha}(-\Delta)_\rho^{\alpha} u + Q(x)u = K(x)|u|^{p-1}u,\;\;\mbox{in}\;\; \mathbb{R}^n,\\ u\in H^{\alpha}(\mathbb{R}^n) \end{array} \right. \end{equation*}$$ where ϵ is a positive parameter, 0 < α < 1, $1<p<\frac{n+2\alpha}{n-2\alpha}$, n > 2α; (−Δ)ρα is a variational version of the regional fractional Laplacian, whose range of scope is a ball with radius ρ(x) > 0, ρ, Q, K are competing functions.

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