Regional Fractional Laplacian Research Articles

Liouville theorem of the regional fractional Lane–Emden equations

The purpose of this paper is to study the Liouville property for the Lane–Emden equation involving the regional fractional Laplacian ( − Δ ) Ω s u + Vu = h 1 u p + h 2 in Ω , u = 0 on ∂Ω , where s ∈ ( 0 , 1 ) , p>0, h 1 , h 2 are nonnegative functions and Ω ⊂ R N − 1 × [ 0 , + ∞ ) with N ≥ 2 , is an unbounded domain satisfying Ω t := { x ′ ∈ R N − 1 : ( x ′ , t ) ∈ Ω } with t ≥ 0 having an increasing monotonicity, that is, Ω t ⊂ Ω t ′ for t ′ ≥ t . The potential V ( x ′ , t ) decays as t → + ∞ . The properties of the limit domain Ω ∞ := lim t → ∞ Ω t in R N − 1 play an important role to obtain the nonexistence of positive solutions for semilinear elliptic equations with the regional fractional Laplacian. For s ∈ ( 0 , 1 2 ] , we provide a surprising nonexistence result if Ω ∞ is bounded. This particular phenomenon occurs because of the peculiar properties of the regional fractional Laplacian with the order s ∈ ( 0 , 1 2 ] .

Global dynamics of a diffusive SARS-CoV-2 model with antiviral treatment and fractional Laplacian operator

In this paper, we propose and investigate the global dynamics of a SARS-CoV-2 infection model with diffusion and antiviral treatment. The proposed model takes into account the two modes of transmission (virus-to-cell and cell-to-cell), the lytic and nonlytic immune responses. The diffusion into the model is formulated by the regional fractional Laplacian operator. Furthermore, the global asymptotic stability of equilibria is rigorously established by means of a new proposed method constructing Lyapunov functions for a class of partial differential equations (PDEs) with regional fractional Laplacian operator. The proposed method is applied to the classical reaction-diffusion equations with normal diffusion.

Hydrodynamic limit for a boundary driven super-diffusive symmetric exclusion

We study the hydrodynamic limit for symmetric exclusion processes with heavy-tailed long jumps and in contact with infinitely extended reservoirs. We show how the corresponding hydrodynamic equations are affected by the parameters defining the model. The hydrodynamic equations are characterized by a class of super-diffusive operators that are given by the regional fractional Laplacian with some additional reaction terms and various boundary conditions. Here we answer to all the questions left open in Bernardin et al. (2021) and we prove a conjecture stated in that same article.

The Neumann problem for the fractional Laplacian: regularity up to the boundary

We study the regularity up to the boundary of solutions to the Neumann problem for the fractional Laplacian. We prove that if $u$ is a weak solution of $(-\Delta)^s u=f$ in $\Omega$, $\mathcal N_s u=0$ in $\Omega^c$, then $u$ is $C^\alpha$ up tp the boundary for some $\alpha>0$. Moreover, in case $s>\frac12$, we then show that $u\in C^{2s-1+\alpha}(\overline\Omega)$. To prove these results we need, among other things, a delicate Moser iteration on the boundary with some logarithmic corrections. Our methods allow us to treat as well the Neumann problem for the regional fractional Laplacian, and we establish the same boundary regularity result. Prior to our results, the interior regularity for these Neumann problems was well understood, but near the boundary even the continuity of solutions was open.

A mixed operator approach to peridynamics

<abstract><p>In the present paper we propose a model describing the nonlocal behavior of an elastic body using a peridynamical approach. Indeed, peridynamics is a suitable framework for problems where discontinuities appear naturally, such as fractures, dislocations, or, in general, multiscale materials. In particular, the regional fractional Laplacian is used as the nonlocal operator. Moreover, a combination of the fractional and classical Laplacian operators is used to obtain a better description of the phenomenological response in elasticity. We consider models with linear and nonlinear perturbations. In the linear case, we prove the existence and uniqueness of the solution, while in the nonlinear case the existence of at least two nontrivial solutions of opposite sign is proved. The linear and nonlinear problems are also solved by a numerical approach which estimates the regional fractional Laplacian by means of its singular integral representation. In both cases, a numerical estimation of the solutions is obtained, using in the nonlinear case an approach involving a random variation of an initial guess of the solution. Moreover, in the linear case a parametric analysis is made in order to study the effects of the parameters involved in the model, such as the order of the fractional Laplacian and the mixture law between local and nonlocal behavior.</p></abstract>

On the $ s $-derivative of weak solutions of the Poisson problem involving the regional fractional Laplacian

In this paper, we analyze the $ s $-dependence of the solution $ u_s $ to the free Poisson problem $ (-\Delta)^s_{\Omega}u_s = f $ in an open bounded set $ \Omega\subset\mathbb{R}^N $. Precisely, we show that the solution map $ (0, 1)\rightarrow L^2(\Omega) $, $ s\mapsto u_s $ is continuously differentiable. Moreover, when $ f = \lambda_su_s $, we also analyze the one-sided differentiability of the first nontrivial eigenvalue of $ (-\Delta)^s_{\Omega} $ regarded as a function of $ s\in(0, 1) $.

Existence results for nonlocal problems governed by the regional fractional Laplacian

The aim of the present paper is to study existence results of minimizers of the critical fractional Sobolev constant on bounded domains. Under some values of the fractional parameter we show that the best constant is achieved. If moreover the underlying domain is a ball, we obtain positive radial minimizers for all possible values of the fractional parameter in higher dimension, while we impose a positive mass condition in low dimension.

The Eigenvalue Problem for the Regional Fractional Laplacian in the Small Order Limit

In this note, we study the asymptotic behavior of eigenvalues and eigenfunctions of the regional fractional Laplacian (−Δ)Ωs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(-{\\Delta })^{s}_{\\Omega }$\\end{document} as s→0+.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$s\\rightarrow 0^{+}.$\\end{document} Our analysis leads to a study of the regional logarithmic Laplacian, which arises as a formal derivative of regional fractional Laplacians at s = 0.

Singular solutions for fractional parabolic boundary value problems

The standard problem for the classical heat equation posed in a bounded domain Omega of {mathbb {R}}^n is the initial and boundary value problem. If the Laplace operator is replaced by a version of the fractional Laplacian, the initial and boundary value problem can still be solved on the condition that the non-zero boundary data must be singular, i.e., the solution u(t, x) blows up as x approaches partial Omega in a definite way. In this paper we construct a theory of existence and uniqueness of solutions of the parabolic problem with singular data taken in a very precise sense, and also admitting initial data and a forcing term. When the boundary data are zero we recover the standard fractional heat semigroup. A general class of integro-differential operators may replace the classical fractional Laplacian operators, thus enlarging the scope of the work. As further results on the spectral theory of the fractional heat semigroup, we show that a one-sided Weyl-type law holds in the general class, which was previously known for the restricted and spectral fractional Laplacians, but is new for the censored (or regional) fractional Laplacian. This yields bounds on the fractional heat kernel.

A Hopf lemma for the regional fractional Laplacian

We provide a Hopf boundary lemma for the regional fractional Laplacian (-Delta )^s_{Omega }, with Omega subset mathbb {R}^N a bounded open set. More precisely, given u a pointwise or weak super-solution of the equation (-Delta )^s_{Omega }u=c(x)u in Omega, we show that the ratio u(x)/(text {dist}(x,partial Omega ))^{2s-1} is strictly positive as x approaches the boundary partial Omega of Omega. We also prove a strong maximum principle for distributional super-solutions.

Global Schauder theory for minimizers of the Hs(Ω) energy

We study the regularity of minimizers of the functional E(u):=[u]Hs(Ω)2+∫Ωfu. This corresponds to understanding solutions for the regional fractional Laplacian in Ω⊂RN. More precisely, we are interested on the global (up to the boundary) regularity of solutions, both in the case of free minimizers in Hs(Ω) (i.e., Neumann problem), or in the case of Dirichlet condition u∈H0s(Ω) when s>12.Our main result establishes the existence of a constant αs∈(0,1−s) satisfying 2s+αs>1 such that for all α∈(0,αs) the solution u∈C2s+α(Ω‾) in the Neumann case, and u/δ2s−1∈C1+α(Ω‾) in the Dirichlet case. Here, δ is the distance to ∂Ω. We also show the optimality of our result: these estimates fail for α>αs, even when f and ∂Ω are C∞.

On a Rayleigh-Faber-Krahn Inequality for the Regional Fractional Laplacian

We study a Rayleigh-Faber-Krahn inequality for regional fractional Laplacian operators. In particular, we show that there exists a compactly supported nonnegative Sobolev function $u_0$ that attains the infimum (which will be a positive real number) of the set \[ \left\{ \int\int_{\{u > 0\}\times\{u>0\}} \frac{|u(x) - u(y)|^2}{|x - y|^{n + 2 \sigma}}d x d y : u \in \mathring H^\sigma(\mathbb{R}^n), \int_{\mathbb{R}^n} u^2 = 1, |\{u > 0 \}| \leq 1\right\}. \] Unlike the corresponding problem for the usual fractional Laplacian, where the domain of the integration is $\mathbb{R}^n \times \mathbb{R}^n$, symmetrization techniques may not apply here. Our approach is instead based on the direct method and new a priori diameter estimates. We also present several remaining open questions concerning the regularity and shape of the minimizers, and the form of the Euler-Lagrange equations.

Fractional diffusion maps

In this paper, we extend the diffusion maps algorithm on a family of heat kernels that are either local (having exponential decay) or nonlocal (having polynomial decay), arising in various applications. For example, these kernels have been used as a regularizer in various supervised learning tasks for denoising images. Importantly, these heat kernels give rise to operators that include (but are not restricted to) the generators of the classical Laplacian associated to Brownian processes as well as the fractional Laplacian associated with β -stable Lévy processes. For local kernels, while the method is a version of the diffusion maps algorithm, we show that the applications with non-Gaussian local heat kernels approximate temporally rescaled Laplace-Beltrami operators. For the non-local heat kernels, we modify the diffusion maps algorithm to estimate fractional Laplacian operators. Here, the graph distance is used to approximate the geodesic distance with appropriate error bounds. While this approximation becomes numerically expensive as the number of data points increases, it produces an accurate operator estimation that is robust to the choice of the kernel bandwidth parameter value. In contrast, the local kernels are numerically more efficient but more sensitive to the choice of kernel bandwidth parameter value. In an application to estimate non-smooth regression functions, we find that using the nonlocal kernel as a regularizer produces a more robust and accurate estimate than using local kernels. For manifolds with boundary, we find that the proposed fractional diffusion maps framework implemented with non-local kernels approximates the regional fractional Laplacian.

Regularity Results for Nonlocal Evolution Venttsel’ Problems

Abstract We consider parabolic nonlocal Venttsel’ problems in polygonal and piecewise smooth two-dimensional domains and study existence, uniqueness and regularity in (anisotropic) weighted Sobolev spaces of the solution. The nonlocal term can be regarded as a regional fractional Laplacian on the boundary. The regularity results deeply rely on a priori estimates, obtained via the so-called Munchhausen trick, and sophisticated extension theorem for anisotropic weighted Sobolev spaces.

Convergence of fractional diffusion processes in extension domains

We study the asymptotic behavior of anomalous fractional diffusion processes in bad domains via the convergence of the associated energy forms. We introduce the associated Robin–Venttsel’ problems for the regional fractional Laplacian. We provide a suitable notion of fractional normal derivative on irregular sets via a fractional Green formula as well as existence and uniqueness results for the solution of the Robin–Venttsel’ problem by a semigroup approach. Submarkovianity and ultracontractivity properties of the associated semigroup are proved.

Compactness of solutions to nonlocal elliptic equations

We show that all nonnegative solutions of the critical semilinear elliptic equation involving the regional fractional Laplacian are locally universally bounded. This strongly contrasts with the standard fractional Laplacian case. Secondly, we consider the fractional critical elliptic equations with nonnegative potentials. We prove compactness of solutions provided the potentials only have non-degenerate zeros. Corresponding to Schoen's Weyl tensor vanishing conjecture for the Yamabe equation on manifolds, we establish a Laplacian vanishing rate of the potentials at blow-up points of solutions.

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

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)ραis a variational version of the regional fractional Laplacian, whose range of scope is a ball with radius ρ(x) > 0, ρ,Q,Kare competing functions.

The Dirichlet elliptic problem involving regional fractional Laplacian

In this paper, we study the solutions of elliptic equations involving regional fractional Laplacian (E) (−Δ)Ωαu=f in a bounded regular domain Ω in RN(N≥2) with C2 boundary ∂Ω, subject to Dirichlet boundary g on ∂Ω, where α∈(12,1) and the operator (−Δ)Ωα denotes the regional fractional Laplacian. We prove that when g ≡ 0, problem (E) admits a unique weak solution under the hypotheses that f ∈ L2(Ω), f ∈ L1(Ω, ρβdx), and f∈M(Ω,ρβ), where ρ(x) = dist(x, ∂Ω), β = 2α – 1, and M(Ω,ρβ) is a space of all Radon measures ν satisfying ∫Ωρβd|ν| < + ∞. Finally, we provide an integration by parts formula for the classical solution of (E) with boundary data g.

A comparative study on nonlocal diffusion operators related to the fractional Laplacian

In this paper, we study four nonlocal diffusion operators, including the fractional Laplacian, spectral fractional Laplacian, regional fractional Laplacian, and peridynamic operator. These operators represent the infinitesimal generators of different stochastic processes, and especially their differences on a bounded domain are significant. We provide extensive numerical experiments to understand and compare their differences. We find that these four operators collapse to the classical Laplace operator as \alpha \to 2. The eigenvalues and eigenfunctions of these four operators are different, and the k-th (for k \in N) eigenvalue of the spectral fractional Laplacian is always larger than those of the fractional Laplacian and regional fractional Laplacian. For any \alpha \in (0, 2), the peridynamic operator can provide a good approximation to the fractional Laplacian, if the horizon size \delta is sufficiently large. We find that the solution of the peridynamic model converges to that of the fractional Laplacian model at a rate of O(\delta^{-\alpha}). In contrast, although the regional fractional Laplacian can be used to approximate the fractional Laplacian as \alpha \to 2, it generally provides inconsistent result from that of the fractional Laplacian if \alpha \ll 2. Moreover, some conjectures are made from our numerical results, which could contribute to the mathematics analysis on these operators.

An Eigenvalue Problem for Nonlocal Equations

In this paper we study the existence of a positive weak solution for a class of nonlocal equations under Dirichlet boundary conditions and involving the regional fractional Laplacian operator...Our result extends to the fractional setting some theorems obtained recently for ordinary and classical elliptic equations, as well as some characterization properties proved for differential problems involving different elliptic operators. With respect to these cases studied in literature, the nonlocal one considered here presents some additional difficulties, so that a careful analysis of the fractional spaces involved is necessary, as well as some nonlocal L^q estimates, recently proved in the nonlocal framework.