Ariel Salort, Eugenio Vecchi
In this note we obtain an asymptotic estimate for growth behavior of variational eigenvalues of the $p-$fractional eigenvalue problem on a smooth bounded domain with Dirichlet boundary condition.
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In this paper we analyze possible extensions of the classical Steklov eigenvalue problem to the fractional setting. In particular, we find a nonlocal eigenvalue problem of fractional type that approximate, when taking a suitable limit, the classical Steklov eigenvalue problem.
We prove an asymptotic estimate for the growth of variational eigenvalues of fractional p-Laplacian eigenvalue problems on a smooth bounded domain.
We consider the eigenvalue problem for the restricted fractional Laplacian in a bounded domain with homogeneous Dirichlet boundary conditions. We introduce the notion of fractional capacity for compact subsets, with the property that the eigenvalues are not affected by the removal of zero fractional capacity sets. Given a simple eigenvalue, we remove from the domain a family of compact sets which are concentrating to a set of zero fractional capacity and we detect the asymptotic expansion of the eigenvalue variation; this expansion depends on the eigenfunction associated to the limit eigenvalu…
In this paper, we give some properties and remarks of the new fractional Sobolev spaces with variable exponents. We also study the eigenvalue problem involving the new fractional $p(\cdot)$-Laplacian.
We study a non-local eigenvalue problem related to the fractional Sobolev spaces for large values of p and derive the limit equation as p goes to infinity. Its viscosity solutions have many interesting properties and the eigenvalues exhibit a strange behaviour. Keywords: eigenvalue, non-local equation, non-linear equation
We discuss some basic properties of the eigenfunctions of a class of nonlocal operators whose model is the fractional p-Laplacian.
We prove that a linear fractional operator with an asymptotically constant lower order term in the whole space admits eigenvalues.
In this work we study the homogenization for eigenvalues of the fractional $p-$Laplace in a bounded domain both with Dirichlet and Neumann conditions. We obtain the convergence of eigenvalues and the explicit order of the convergence rates.
In this paper, we show the existence of a sequence of eigenvalues for a Dirichlet problem involving two mixed fractional operators with different orders. We provide lower and upper bounds for the sum of the eigenvalues. Applications of mixed fractional operators with different orders include medicine, plasma physics, and population dynamics.
We study a nonlinear, nonlocal eigenvalue problem driven by the fractional p-Laplacian with an indefinite, singular weight chosen in an optimal class. We prove the existence of an unbounded sequence of positive variational eigenvalues and alternative characterizations of the first and second eigenvalues. Then, by means of such characterizations, we prove strict decreasing monotonicity of such eigenvalues with respect to the weight function.
In this note, we study the asymptotic behavior of eigenvalues and eigenfunctions of the regional fractional Laplacian $(-Δ)^s$ as $ s \to 0^+$. Our analysis leads to a study of the regional logarithmic Laplacian, which arises as a formal derivative of regional fractional Laplacians at $s = 0$.
We establish an upper bound of the sum of the eigenvalues for the Dirichlet problem of the fractional Laplacian. Our result is obtained by a subtle computation of the Rayleigh quotient for specific functions.
Eigenvalue problems in fractional anisotropic Orlicz-Sobolev spaces
本文引入分数阶各向异性Orlicz-Sobolev空间，利用变分方法研究含椭圆型非局部积分微分算子的分数阶各向异性问题，揭示了各向异性系数增长率竞争对特征值集合存在性与非存在性的关键影响。
We find interpretation using optimal mass transport theory for eigenvalue problems obtained as limits of the eigenvalue problems for the fractional $p-$Laplacian operators as $p\to +\infty$. We deal both with Dirichlet and Neumann boundary conditions.
By virtue of $Γ-$convergence arguments, we investigate the stability of variational eigenvalues associated with a given topological index for the fractional $p$-Laplacian operator, in the singular limit as the nonlocal operator converges to the $p$-Laplacian. We also obtain the convergence of the corresponding normalized eigenfunctions in a suitable fractional norm.
Generic properties of eigenvalues of the fractional Laplacian
该研究针对分数阶拉普拉斯算子的Dirichlet特征值问题，证明光滑有界域经任意小扰动后其所有特征值均为单的，推得区间上特征值全为单，并得出非局部算子特征值在参数通有选择下也为单的结论。
In the present work we study existence of sequences of variational eigenvalues to non-local non-standard growth problems ruled by the fractional $g-$Laplacian operator with different boundary conditions (Dirichlet, Neumann and Robin). Due to the non-homogeneous nature of the operator several drawbacks must be overcome, leading to some results that contrast with the case of power functions.
Simplicity of eigenvalues of the fractional Laplace operator in an interval
本文为区间上分数阶拉普拉斯算子特征值的单重性提供简短证明，该结果由Fall等人近期证明。
We provide fundamental properties of the first eigenpair for fractional $p$-Laplacian eigenvalue problems under singular weights, which is related to Hardy type inequality, and also show that the second eigenvalue is well-defined. We obtain a-priori bounds and the continuity of solutions to problems with such singular weights with some additional assumptions. Moreover, applying the above results, we show a global bifurcation emenating from the first eigenvalue, the Fredholm alternative for non-resonant problems, and obtain the existence of infinitely many solutions for some nonlinear problems …
We develop some properties of the $p-$Neumann derivative for the fractional $p-$Laplacian in bounded domains with general $p>1$. In particular, we prove the existence of a diverging sequence of eigenvalues and we introduce the evolution problem associated to such operators, studying the basic properties of solutions. Finally, we study a nonlinear problem with source in absence of the Ambrosetti-Rabinowitz condition.
Let $Ω\subset\mathbb{R}^N$, $N\geq 2$, be an open bounded connected set. We consider the fractional weighted eigenvalue problem $(-Δ)^s u =λρu$ in $Ω$ with homogeneous Dirichlet boundary condition, where $(-Δ)^s$, $s\in (0,1)$, is the fractional Laplacian operator, $λ\in \mathbb{R}$ and $ ρ\in L^\infty(Ω)$. We study weak* continuity, convexity and Gâteaux differentiability of the map $ρ\mapsto1/λ_1(ρ)$, where $λ_1(ρ)$ is the first positive eigenvalue. Moreover, denoting by $\mathcal{G}(ρ_0)$ the class of rearrangements of $ρ_0$, we prove the existence of a minimizer of $λ_1(ρ)$ when $ρ$ varies…
We study the eigenvalue problem for the $g-$Laplacian operator in fractional order Orlicz-Sobolev spaces, where $g=G'$ and neither $G$ nor its conjugated function satisfy the $Δ_2$ condition. Our main result is the existence of a nontrivial solution to such a problem; this is achieved by first showing that the corresponding minimization problem has a solution and then applying a generalized Lagrange multiplier theorem to get the existence of an eigenvalue. Further, we prove closedness of the spectrum and some properties of the eigenvalues and, as an application, we show existence for a class o…
Eigenvalue estimates for the fractional Laplacian on lattice subgraphs
本文研究图子图上带 Dirichlet 边界条件的分数阶 Laplace 算子，并针对格子图建立前 k 个特征值之和的上下界估计，推广了 Li-Yau 与 Kröger 的经典结论。
First Eigenvalue and Torsional Rigidity: Isoperimetric Inequalities for the Fractional Laplacian
本文研究了分数阶拉普拉斯算子在等周不等式中的性质，证明了在具有相同扭刚度的集合中，球体是第一特征值最小的集合，并建立了相关反Holder不等式。
In this paper we analyze an eigenvalue problem related to the nonlocal $p-$laplace operator plus a potential. After reviewing some elementary properties of the first eigenvalue of these operators (existence, positivity of associated eigenfunctions, simplicity and isolation) we investigate the dependance of the first eigenvalue on the potential function and establish the existence of some {\em optimal} potentials in some admissible classes.
In this paper we study an optimal shape design problem for the first eigenvalue of the fractional $p-$laplacian with mixed boundary conditions. The optimization variable is the set where the Dirichlet condition is imposed (that is restricted to have measure equal than a prescribed quantity, $α$). We show existence of an optimal design and analyze the asymptotic behavior when the fractional parameter $s\uparrow 1$ obtaining asymptotic bounds that are independent of $α$.
On Fractional Anisotropic Musielak-Sobolev Spaces with Applications to Nonlocal Eigenvalue Problems
本文引入并研究分数阶各向异性Musielak-Sobolev空间，建立其基本性质与嵌入结果，将其应用于非局部各向异性特征值问题并证明特征值存在性，扩展并统一了相关现有模型。
Lower and Upper Bounds for Sums of Eigenvalues of the Fractional-Logarithmic Laplacian
本文研究分数对数拉普拉斯算子特征值之和的上下界，克服了其径向变量低频行为导致的非单调性挑战。
In this work we study the asymptotic behavior of the first non-zero Neumann $p-$fractional eigenvalue $λ_1(s,p)$ as $s\to 1^-$ and as $p\to\infty.$ We show that there exists a constant $\mathcal{K}$ such that $\mathcal{K}(1-s)λ_1(s,p)$ goes to the first non-zero Neumann eigenvalue of the $p-$Laplacian. While in the limit case $p\to \infty,$ we prove that $λ_1(1,s)^{1/p}$ goes to an eigenvalue of the Hölder $\infty-$Laplacian.
In this article, we study the asymptotics of Dirichlet eigenvalues and eigenfunctions of the fractional Laplacian $(-Δ)^s$ in bounded open Lipschitz sets in the small order limit $s \to 0^+$. While it is easy to see that all eigenvalues converge to $1$ as $s \to 0^+$, we show that the first order correction in these asymptotics is given by the eigenvalues of the logarithmic Laplacian operator, i.e., the singular integral operator with symbol $2\log|ξ|$. By this we generalize a result of Chen and the third author which was restricted to the principal eigenvalue. Moreover, we show that $L^2$-nor…
The evolution problem associated with the fractional first eigenvalue
本文研究与第一分数阶特征值相关的演化问题，在粘性解框架下证明齐次 Dirichlet 问题适定，并利用第一特征值给出解的指数衰减界。
The purpose of this article is to establish new lower bounds for the sums of powers of eigenvalues of the Dirichlet fractional Laplacian operator $(-Δ)^{α/2}|_Ω$ restricted to a bounded domain $Ω\subset{\mathbb R}^d$ with $d=2,$ $1\leq α\leq 2$ and $d\geq 3,$ $0< α\le 2$. Our main result yields a sharper lower bound, in the sense of Weyl asymptotics, for the Berezin-Li-Yau type inequality improving the previous result in [36]. Furthermore, we give a result improving the bounds for analogous elliptic operators in [19].
A note on the persistence of multiplicity of eigenvalues of fractional Laplacian under perturbations
本文研究分数拉普拉斯算子特征值重数在扰动下的保持性，给出了保持重数的条件，并证明重数2的特征值对应的保持扰动集为余维2的光滑流形。
Lower bounds for fractional Orlicz-type eigenvalues
本文在 Young 函数 $A$ 及其互补函数均不满足 $Δ_2$ 条件的假设下，利用区域几何与 $A$ 的增长性质，建立了分数阶 $A-$Laplacian 特征值和临界值的精确下界。
In this article, we study the eigenvalue of nonlinear $p-$fractional Hardy operator \begin{align*} (-Δ)_p^αu - μ\frac{|u|^{p-2}u}{|x|^{pα}} = λV(x) |u|^{p-2}u \; \text{in}\; Ω, \quad u = 0 \; \mbox{in}\; \mathbb{R}^n \setminusΩ, \end{align*} where $n>pα$, $p\geq2$, $α\in(0,1)$, $0\leq μ<C_{n,α,p}$ and $Ω$ is a domain in $\mathbb{R}^n$ with Lipschitz boundary containing $0$. In particular, $Ω=\mathbb{R}^n$ is admitted. The weight function $V$ may change sign and may have singular points. We also show that the least positive eigenvalue is simple and it is unique associated to a non-negative eige…
We study the positive principal eigenvalue of a weighted problem associated with the Neumann spectral fractional Laplacian. This analysis is related to the investigation of the survival threshold in population dynamics. Our main result concerns the optimization of such threshold with respect to the fractional order $s\in(0,1]$, the case $s=1$ corresponding to the standard Neumann Laplacian: when the habitat is not too fragmented, the principal positive eigenvalue can not have local minima for $0<s<1$. As a consequence, the best strategy for survival is either following the diffusion with $s=1$…
In this paper we analyze the asymptotic behavior of several fractional eigenvalue problems by means of Gamma-convergence methods. This method allows us to treat different eigenvalue problems under a unified framework. We are able to recover some known results for the behavior of the eigenvalues of the $p-$fractional laplacian when the fractional parameter $s$ goes to 1, and to extend some known results for the behavior of the same eigenvalue problem when $p$ goes to $\infty$. Finally we analyze other eigenvalue problems not previously covered in the literature.
Let $Ω$ be a bounded open domain on the Euclidean space $\mathbb{R}^{n}$ and $\mathbb{Q}_{+}$ be the set of all positive rational numbers. In 2017, Chen and Zeng investigated the eigenvalues with higher order of the fractional Laplacian $\left.(-Δ)^{s}\right|_Ω$ for $s>0$ and $s \in \mathbb{Q}_{+}$, and they obtained a universal inequality of Yang type(\emph{ Universal inequality and upper bounds of eigenvalues for non-integer poly-Laplacian on a bounded domain, Calculus of Variations and Partial Differential Equations, (2017) \textbf{56}:131}). In the spirit of Chen and Zeng's work, we study …
We consider the eigenvalue problem for the fractional $p \& q-$Laplacian \begin{equation} \left\{\begin{aligned} (- Δ)_p^{s}\, u + μ(- Δ)_q^{s}\, u+ |u|^{p-2}u+μ|u|^{q-2}u=λ V(x)|u|^{p-2}u\quad & \text{in } Ω\\ u=0\quad& \text{in}\quad\R^N\backslashΩ, \end{aligned}\right. \end{equation} where $Ω$ is an open bounded, and possibly disconnected domain, $λ\in\R$, $1<q<p<\frac{N}{s}$, $μ>0$ with a weight function in $L^\infty(Ω)$ that is allowed no change sign. We show that the problem has a continuous spectrum. Moreover, our result reveals a discontinuity property for the spectrum as the parameter…
Three non-zero solutions of a Neumann eigenvalue problems involving the fractional p-Laplacian
本文研究含分数阶p-Laplacian的非局部Neumann特征值问题，证明存在正特征值开区间，使问题在该区间内至少有三个非零解。
Minimizing Eigenvalues of the Fractional Laplacian
本文研究了分数拉普拉斯算子特征值最小化问题，揭示了极值解自由边界的独特行为，并给出了存在性、正则性和非退化性等主要贡献。
Ljusternik-Schnirelmann eigenvalues for the fractional $m-$Laplacian without the $Δ_2$ condition
本文研究分数阶m-拉普拉斯算子的特征值问题，在放宽其原函数的Δ₂条件约束下，证明存在趋于无穷的特征值序列，推进了非局部扩散模型的相关理论研究。
We analyze the behavior of the eigenvalues of the following non local mixed problem $\left\{ \begin{array}{rcll} (-Δ)^{s} u &=& λ_1(D) \ u &\innΩ,\\ u&=&0&\inn D,\\ \mathcal{N}_{s}u&=&0&\inn N. \end{array}\right $ Our goal is to construct different sequences of problems by modifying the configuration of the sets $D$ and $N$, and to provide sufficient and necessary conditions on the size and the location of these sets in order to obtain sequences of eigenvalues that in the limit recover the eigenvalues of the Dirichlet or Neumann problem. We will see that the non locality plays a crucial role h…
An eigenvalue problem for a nonlocal quasilinear anisotropic equation in fractional Orlicz Sobolev spaces without the $Δ_2$--condition
本文研究了一类非局部非标准生长扩散问题的特征值问题，不假设Young函数的Δ₂条件，证明了特征对序列(uk,λk)趋向于(0,+∞)的存在性。
In this article we consider a homogeneous eigenvalue problem ruled by the fractional $g-$Laplacian operator whose Euler-Lagrange equation is obtained by minimization of a quotient involving Luxemburg norms. We prove existence of an infinite sequence of variational eigenvalues and study its behavior as the fractional parameter $s\uparrow 1$ among other stability results.
We prove a fractional Pohozaev type identity in a generalized framework and discuss its applications. Specifically, we shall consider applications to nonexistence of solutions in the case of supercritical semilinear Dirichlet problems and regarding a Hadamard formula for the derivative of Dirichlet eigenvalues of the fractional Laplacian with respect to domain deformations. We also derive the simplicity of radial eigenvalues in the case of radial bounded domains and apply the Hadamard formula to this case.
Eigenvalues of nonlinear $(p,q)$-fractional Laplace operators under nonlocal Neumann conditions
研究非局部 $(p,q)$-分数阶 Laplace 算子在齐次非局部 Neumann 条件下的特征值问题，完整刻画其谱集并证明特征函数全局有界。
We study the eigenvalue problem for a system of fractional $p-$Laplacians, that is, $$ \begin{cases} (-Δ_p)^r u = λ\dfracαp|u|^{α-2}u|v|^β &\text{in } Ω,\vspace{.1cm} (-Δ_p)^s u = λ\dfracβp|u|^α|v|^{β-2}v &\text{in } Ω, u=v=0 &\text{in }Ω^c=\R^N\setminusΩ. \end{cases} $$ We show that there is a first (smallest) eigenvalue that is simple and has associated eigen-pairs composed of positive and bounded functions. Moreover, there is a sequence of eigenvalues $λ_n$ such that $λ_n\to\infty$ as $n\to\infty$. In addition, we study the limit as $p\to \infty$ of the first eigenvalue, $λ_{1,p}$, and we o…
We add a divergence-free drift with increasing magnitude to the fractional Laplacian on a bounded smooth domain, and discuss the behavior of the principal eigenvalue for the Dirichlet problem. The eigenvalue remains bounded if and only if the drift has non-trivial first integrals in the domain of the quadratic form of the fractional Laplacian.
We review some recent results on eigenvalues of fractional Laplacians and fractional Schrödinger operators. We discuss, in particular, Lieb-Thirring inequalities and their generalizations, as well as semi-classical asymptotics.
Eigenvalue for a problem involving the fractional (p,q)-Laplacian operator and nonlinearity with a singular and a supercritical Sobolev growth
本文通过变分法与上下解方法，研究分数阶 (p,q)-Laplacian 奇异椭圆特征值问题，涵盖超临界 Sobolev 增长，并建立解的 L∞ 范数估计。
We study a Dirichlet-type boundary value problem for a pseudo-differential equation driven by the fractional Laplacian, with a non-linear reaction term which is resonant at infinity between two non-principal eigenvalues: for such equation we prove existence of a non-trivial solution. Under further assumptions on the behavior of the reaction at zero, we detect at least three non-trivial solutions (one positive, one negative, and one of undetermined sign). All results are based on the properties of weighted fractional eigenvalues, and on Morse theory.
Compact Embeddings, Eigenvalue Problems, and subelliptic Brezis-Nirenberg equations involving singularity on stratified Lie groups
本文研究分层李群上分数阶$p$-次拉普拉斯算子的特征值问题，证明第一特征值的正性、简单性和孤立性，并利用Nehari流形技术证明一类奇异次椭圆问题至少存在两个弱解，同时通过Moser迭代建立正解的有界性。
Non-degeneracy of the bubble in a fractional and singular 1D Liouville equation
本文研究了分数和奇异性1D Liouville方程在实数轴上的非退化性，通过共形变换将线性化方程转化为有界域上的Steklov特征值问题，并证明了相应特征值的简单性。
We study the asymptotic behavior of the solutions of a spectral problem for the Laplacian in a domain with rapidly oscillating boundary. We consider the case where the eigenvalue of the limit problem is multiple. We construct the leading terms of the asymptotic expansions for the eigenelements and verify the asymptotics.
We significantly expand the number of functions whose image under the fractional Laplace operator can be computed explicitly. In particular, we show that the fractional Laplace operator maps Meijer G-functions of |x|^2, or generalized hypergeometric functions of -|x|^2, multiplied by a solid harmonic polynomial, into the same class of functions. As one important application of this result, we produce a complete system of eigenfunctions of the operator (1-|x|^2)_+^{alpha/2} (-Delta)^{alpha/2} with the Dirichlet boundary conditions outside of the unit ball. The latter result will be used to esti…
By exploiting a variational technique based upon projecting over the Pohozaev manifold, we prove existence of positive solutions for a class of nonlinear fractional Schrodinger equations having a nonhomogenous nonautonomous asymptotically linear nonlinearity.
In this article we study eigenvalues and minimizers of a fractional non-standard growth problem. We prove several properties on this quantities and their corresponding eigenfunctions.
This article deals with the existence and non-existence of positive solutions for the eigenvalue problem driven by nonhomogeneous fractional $p\& q$ Laplacian operator with indefinite weights $$\left(-Δ_p\right)^αu + \left(-Δ_q\right)^βu \,= λ\left[a(x) \left|u\right|^{p-2}u + b(x) \left|u\right|^{q-2}u \right]\quad\quad\textrm{in $Ø$},$$ where $Ø$ is a smooth bounded domain in $\R^N$ extended by zero outside. When $Ø=\R^N$ and $b\equiv0$, we further show that there exists a continuous family of the eigenvalue if $1<q<p<q^*_β=\frac{Nq}{N-qβ}$ and $0\leq a\in L^{\left(\frac{q_β^*}{s}\right)'}\l…
In this note we provide a second-order asymptotic expansion of the fractional perimeter Per$_s(E)$, as $s\to 1^-$, in terms of the local perimeter and of a higher order nonlocal functional.
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