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.

EXISTENCE OF A WEAK SOLUTION FOR A CLASS OF FRACTIONAL LAPLACIAN EQUATIONS

2016 ◽  
Vol 102 (3) ◽  
pp. 392-404
Author(s):  
V. RAGHAVENDRA ◽  
RASMITA KAR
Keyword(s):  
Weak Solution ◽  
Fractional Laplacian ◽  
Nonlocal Problem ◽  
Bounded Subset ◽  
Lipschitz Boundary ◽  
Integrodifferential Operator ◽  
Image Position ◽  
Laplacian Equations ◽  
Open Bounded Subset ◽  
Fractional Type

We study the existence of a weak solution of a nonlocal problem$$\begin{eqnarray}\displaystyle & \displaystyle -{\mathcal{L}}_{K}u-\unicode[STIX]{x1D707}ug_{1}+h(u)g_{2}=f\quad \text{in }\unicode[STIX]{x1D6FA}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle u=0\quad \text{in }\mathbb{R}^{n}\setminus \unicode[STIX]{x1D6FA}, & \displaystyle \nonumber\end{eqnarray}$$where${\mathcal{L}}_{k}$is a general nonlocal integrodifferential operator of fractional type,$\unicode[STIX]{x1D707}$is a real parameter and$\unicode[STIX]{x1D6FA}$is an open bounded subset of$\mathbb{R}^{n}$($n>2s$, where$s\in (0,1)$is fixed) with Lipschitz boundary$\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$. Here$f,g_{1},g_{2}:\unicode[STIX]{x1D6FA}\rightarrow \mathbb{R}$and$h:\mathbb{R}\rightarrow \mathbb{R}$are functions satisfying suitable hypotheses.

Initial-boundary value problem for a fractional type degenerate heat equation

2018 ◽  
Vol 28 (06) ◽  
pp. 1199-1231
Author(s):  
Gerardo Huaroto ◽  
Wladimir Neves
Keyword(s):  
Heat Equation ◽  
Fractional Laplacian ◽  
Dirichlet Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlocal Diffusion ◽  
Laplacian Operator ◽  
Initial Boundary Value ◽  
Fractional Laplacian Operator ◽  
Fractional Type

In this paper, we study a fractional type degenerate heat equation posed in bounded domains. We show the existence of solutions for measurable and bounded non-negative initial data, and homogeneous Dirichlet boundary condition. The nonlocal diffusion effect relies on an inverse of the [Formula: see text]-fractional Laplacian operator, and the solvability is proved for any [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 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.

scholarly journals Stability of Nonlinear Dirichlet BVPs Governed by Fractional Laplacian

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Dorota Bors
Keyword(s):  
Optimal Control ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Boundary Data ◽  
Fractional Laplacian ◽  
Dirichlet Boundary ◽  
Sufficient Condition ◽  
Variational Structure

We consider a class of partial differential equations with the fractional Laplacian and the homogeneous Dirichlet boundary data. Some sufficient condition under which the solutions of the equations considered depend continuously on parameters is stated. The application of the results to some optimal control problem is presented. The methods applied in the paper make use of the variational structure of the problem.

Asymptotic behaviour for a non-local parabolic problem

2009 ◽  
Vol 20 (3) ◽  
pp. 247-267 ◽  
Author(s):  
LIU QILIN ◽  
LIANG FEI ◽  
LI YUXIANG
Keyword(s):  
Asymptotic Behaviour ◽  
Stationary Solution ◽  
Finite Time ◽  
Parabolic Problem ◽  
Dirichlet Boundary ◽  
Asymptotic Estimates ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Globally Asymptotically Stable ◽  
Non Local ◽  
Image Position

In this paper, we consider the asymptotic behaviour for the non-local parabolic problemwith a homogeneous Dirichlet boundary condition, where λ > 0,p> 0 andfis non-increasing. It is found that (a) for 0 <p≤ 1,u(x,t) is globally bounded and the unique stationary solution is globally asymptotically stable for any λ > 0; (b) for 1 <p< 2,u(x,t) is globally bounded for any λ > 0; (c) forp= 2, if 0 < λ < 2|∂Ω|2, thenu(x,t) is globally bounded; if λ = 2|∂Ω|2, there is no stationary solution andu(x,t) is a global solution andu(x,t) → ∞ ast→ ∞ for allx∈ Ω; if λ > 2|∂Ω|2, there is no stationary solution andu(x,t) blows up in finite time for allx∈ Ω; (d) forp> 2, there exists a λ* > 0 such that for λ > λ*, or for 0 < λ ≤ λ* andu0(x) sufficiently large,u(x,t) blows up in finite time. Moreover, some formal asymptotic estimates for the behaviour ofu(x,t) as it blows up are obtained forp≥ 2.

scholarly journals Nontrivial solutions of superlinear nonlocal problems

2016 ◽  
Vol 28 (6) ◽  
Author(s):  
Giovanni Molica Bisci ◽  
Dušan Repovš ◽  
Raffaella Servadei
Keyword(s):  
Weak Solutions ◽  
Boundary Data ◽  
Fractional Laplacian ◽  
Dirichlet Boundary ◽  
Existence Results ◽  
Nontrivial Solutions ◽  
Nonlocal Problems ◽  
Fountain Theorem ◽  
Nonlocal Equations ◽  
Laplacian Equations

AbstractWe study the question of the existence of infinitely many weak solutions for nonlocal equations of fractional Laplacian type with homogeneous Dirichlet boundary data, in presence of a superlinear term. Starting from the well-known Ambrosetti–Rabinowitz condition, we consider different growth assumptions on the nonlinearity, all of superlinear type. We obtain three different existence results in this setting by using the Fountain Theorem, which extend some classical results for semilinear Laplacian equations to the nonlocal fractional setting.

Existence results for Kirchhoff–type superlinear problems involving the fractional Laplacian

2018 ◽  
Vol 149 (04) ◽  
pp. 1061-1081 ◽  
Author(s):  
Zhang Binlin ◽  
Vicenţiu D. Rădulescu ◽  
Li Wang
Keyword(s):  
Fractional Laplacian ◽  
Differential Operators ◽  
Smooth Boundary ◽  
Critical Groups ◽  
Kirchhoff Type ◽  
Existence And Multiplicity ◽  
Non Local ◽  
Image Position ◽  
Open Bounded Subset ◽  
Fractional Laplace Operator

AbstractIn this paper, we study the existence and multiplicity of solutions for Kirchhoff-type superlinear problems involving non-local integro-differential operators. As a particular case, we consider the following Kirchhoff-type fractional Laplace equation:$$\matrix{ {\left\{ {\matrix{ {M\left( {\int\!\!\!\int\limits_{{\open R}^{2N}} {\displaystyle{{ \vert u(x)-u(y) \vert ^2} \over { \vert x-y \vert ^{N + 2s}}}} {\rm d}x{\rm d}y} \right){(-\Delta )}^su = f(x,u)\quad } \hfill &amp; {{\rm in }\Omega ,} \hfill \cr {u = 0\quad } \hfill &amp; {{\rm in }{\open R}^N{\rm \setminus }\Omega {\mkern 1mu} ,} \hfill \cr } } \right.} \hfill \cr } $$where ( − Δ)sis the fractional Laplace operator,s∈ (0, 1),N&gt; 2s, Ω is an open bounded subset of ℝNwith smooth boundary ∂Ω,$M:{\open R}_0^ + \to {\open R}^ + $is a continuous function satisfying certain assumptions, andf(x,u) is superlinear at infinity. By computing the critical groups at zero and at infinity, we obtain the existence of non-trivial solutions for the above problem via Morse theory. To the best of our knowledge, our results are new in the study of Kirchhoff–type Laplacian problems.

Toward High-Resolution Low-Voltage SEM

1988 ◽  
Vol 46 ◽  
pp. 218-219
Author(s):  
Zhifeng Shao
Keyword(s):  
Low Voltage ◽  
Spherical Aberration ◽  
Chromatic Aberration ◽  
Limiting Factor ◽  
Operating Voltage ◽  
Probe Radius ◽  
Non Local ◽  
Electron Microscopes ◽  
Image Position ◽  
Scanning Electron Microscopes

Recently, low voltage (≤5kV) scanning electron microscopes have become popular because of their unprecedented advantages, such as minimized charging effects and smaller specimen damage, etc. Perhaps the most important advantage of LVSEM is that they may be able to provide ultrahigh resolution since the interaction volume decreases when electron energy is reduced. It is obvious that no matter how low the operating voltage is, the resolution is always poorer than the probe radius. To achieve 10Å resolution at 5kV (including non-local effects), we would require a probe radius of 5∽6 Å. At low voltages, we can no longer ignore the effects of chromatic aberration because of the increased ratio δV/V. The 3rd order spherical aberration is another major limiting factor. The optimized aperture should be calculated as

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