Francesca Colasuonno, Marco Squassina
In the framework of variable exponent Sobolev spaces, we prove that the variational eigenvalues defined by inf sup procedures of Rayleigh ratios for the Luxemburg norms are all stable under uniform convergence of the exponents.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this paper, we give some stability estimates for the Faber-Krahn inequality relative to the eigenvalues of Hessian operators
A highly nonlinear eigenvalue problem is studied in a Sobolev space with variable exponent. The Euler-Lagrange equation for the minimization of a Rayleigh quotient of two Luxemburg norms is derived. The asymptotic case with a "variable infinity" is treated. Local uniqueness is proved for the viscosity solutions.
A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle
该文研究带内部固定球形障碍物的第一Steklov-拉普拉斯特征值问题，证明内球和体积固定时球壳在近球形集合中局部最大化第一特征值。
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.
A recent area of interest is the development and study of eigenvalue problems arising in scattering theory that may provide potential target signatures for use in nondestructive testing of materials. We consider a generalization of the electromagnetic Stekloff eigenvalue problem that depends upon a smoothing parameter, for which we establish two main results that were previously unavailable for this type of eigenvalue problem. First, we use the theory of trace class operators to prove that infinitely many eigenvalues exist for a sufficiently high degree of smoothing, even for an absorbing medi…
Symmetry-breaking and local stability of a two-phase eigenvalue problem in optimal insulation
研究具有罗宾边界条件的拉普拉斯算子两相特征值问题的第一特征值，通过固定内球主诺伊曼特征值，刻画球在体积约束下成为局部最小值的条件。
Nonlinear variable exponent Picone identity and $p(x)$-sub-Laplacian first eigenvalue for general vector fields
本文建立$p(x)$-次拉普拉斯的广义非线性变指数Picone恒等式，并据此证明其第一非平凡Dirichlet特征值的唯一性、简单性、单调性与孤立性，同时推导出变指数Hardy型不等式与Caccioppoli估计。
Sharp stability on the second Robin eigenvalue with negative boundary parameters
本文研究了负边界参数下第二Robin特征值的稳定性，通过构造近球形域族证明了等周不等式中不对称度指数的尖锐性。
In this paper we study the Steklov-Dirichlet eigenvalues $λ_k(Ω,Γ_S)$, where $Ω\subset \mathbb{R}^d$ is a domain and $Γ_S\subset \partial Ω$ is the subset of the boundary in which we impose the Steklov conditions. After a first discussion about the regularity properties of the Steklov-Dirichlet eigenfunctions we obtain a stability result for the eigenvalues. We study the optimization problem under a measure constraint on the set $Γ_S$, we prove the existence of a minimizer and the non-existence of a maximizer. In the plane we prove a continuity result for the eigenvalues imposing a bound on th…
We discuss stability of the first eigenvalue of the 1-Laplacian under perturbations of the domain.
The principal eigenvalue for linear elliptic operator has been known to be one of very useful tools to investigate many important partial differential equations. Due to the pioneering works of Berestycki et al. \cite{BCV1,BCV2}, the study of qualitative properties for the principal eigenvalue of nonlocal operators has attracted a lot of attention of the community from theory to application (For examples \cite{LL22-1,LLS22,LZ1,LZ2,SLLY,DDF,XLR}). In this paper, motivated from the study of mathematical modeling the dynamics of infectious diseases in \cite{NV1, ZZLD, WD}, we analyze the asymptoti…
We consider the nonlinear eigenvalue problem $-{\rm div}(|\nabla u|^{p(x)-2}\nabla u)=λ|u|^{q(x)-2}u$ in $Ω$, $u=0$ on $\partialΩ$, where $Ω$ is a bounded open set in $\RR^N$ with smooth boundary and $p$, $q$ are continuous functions on $\barΩ$ such that $1<\inf\_Ωq< \inf\_Ωp<\sup\_Ωq$, $\sup\_Ωp<N$, and $q(x)<Np(x)/(N-p(x))$ for all $x\in\barΩ$. The main result of this paper establishes that any $λ>0$ sufficiently small is an eigenvalue of the above nonhomogeneous quasilinear problem. The proof relies on simple variational arguments based on Ekeland's variational principle.
In this note we analyze how perturbations of a ball $\mathfrak{B}_r \subset \mathbb{R}^n$ behaves in terms of their first (non-trivial) Neumann and Dirichlet $\infty-$eigenvalues when a volume constraint $\\mathscr{L}^n(Ω) = \mathscr{L}^n(\mathfrak{B}_r)$ is imposed. Our main result states that $Ω$ is uniformly close to a ball when it has first Neumann and Dirichlet eigenvalues close to the ones for the ball of the same volume $\mathfrak{B}_r$. In fact, we show that, if $$ |λ_{1,\infty}^D(Ω) - λ_{1,\infty}^D(\mathfrak{B}_r)| = δ_1 \quad \text{and} \quad |λ_{1,\infty}^N(Ω) - λ_{1,\infty}^N(\mat…
On the stability of eigenvalues of varying bilinear forms in abstract Hilbertian settings and applications
本文发展抽象希尔伯特空间下变双线性形式特征值的谱扰动理论，量化收敛速率并给出扰动特征值渐近展开首项的变分刻画，还将其应用于拉普拉斯相关算子的特征值问题研究。
In recent decades qualitative inverse scattering methods with eigenvalues as target signatures received much attention. To understand those methods a knowledge on the properties of the related eigenvalue problems is essential. However, even the existence of eigenvalues for such (nonselfadjoint) problems is a challenging question and existing results for absorbing media are usually established under unrealistic assumptions or a smoothing of the eigenvalue problem. We present a technique to prove the existence of infinitely many eigenvalues for such problems under realistic assumptions. In parti…
Stability and eigenvalue bounds for micropolar shear flows
本文研究微极流体平行剪切流的线性稳定性，推导Orr-Sommerfeld方程，并建立二维扰动的特征值界、波速界及稳定的充分条件。
We prove stability estimates for the isoperimetric inequalities for the first and the second nonzero Laplace eigenvalues on surfaces, both globally and in a fixed conformal class. We employ the notion of eigenvalues of measures and show that if a normalized eigenvalue is close to its maximal value, the corresponding measure must be close in the Sobolev space $W^{-1,2}$ to the set of maximizing measures. In particular, this implies a qualitative stability result: metrics almost maximizing the normalized eigenvalue must be $W^{-1,2}$-close to a maximal metric. Following this approach, we prove s…
In this paper, we study the eigenvalue problem for the Monge-Ampère operator on general bounded convex domains. We prove the existence, uniqueness and variational characterization of the Monge-Ampère eigenvalue. The convex Monge-Ampère eigenfunctions are shown to be unique up to positive multiplicative constants. Our results are the singular counterpart of previous results by P-L. Lions and K. Tso in the smooth, uniformly convex setting. Moreover, we prove the stability of the Monge-Ampère eigenvalue with respect to the Hausdorff convergence of the domains. This stability property makes it pos…
We first establish a sharp relation between the order of vanishing of a Dirichlet eigenfunction at a point and the leading term of the asymptotic expansion of the Dirichlet eigenvalue variation, as a removed compact set concentrates at that point. Then we apply this spectral stability result to the study of the asymptotic behaviour of eigenvalues of Aharonov-Bohm operators with two colliding poles moving on an axis of symmetry of the domain.
Quantitative stability control of the full spectrum of the Dirichlet Laplacian by the second eigenvalue
本文研究有限测度开集 Dirichlet 拉普拉斯算子全谱在第二特征值接近参考集时的定量稳定性，对任意 k 和 Ω 给出指数为 1/(d+1) 的控制估计，并在 λ_k(Ω)≥λ_k(Θ) 时改进为精确指数 1/2。
Stability of extremal domains for the first eigenvalue of the Laplacian operator
本文计算了一般黎曼流形上第一Dirichlet特征值的二阶变分并建立稳定性判据，分类了球面中的稳定极值区域，并在特定曲率或体积条件下给出了紧致曲面上稳定区域的拓扑界。
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.
A stability result for the first Robin-Neumann eigenvalue: A double perturbation approach
该研究针对带Robin外边界、Neumann内边界的拉普拉斯算子第一特征值问题，引入新型混合不对称性，建立了等周不等式的定量版本，得到首个内外边界同时扰动的稳定性结果。
We study the eigenvalue problem for the Neumann-Laplace operator in conformal regular planar domains $Ω\subset\mathbb{C}$. Conformal regular domains support the Poincaré inequality and this allows us to estimate the variation of the eigenvalues of the Neumann Laplacian upon domain perturbation via energy type integrals. Boundaries of such domains can have any Hausdorff dimension between one and two.
In this paper we establish the orbital stability of periodic traveling waves for a general class of dispersive equations. We use the Implicit Function Theorem to guarantee the existence of smooth solutions depending of the corresponding wave speed. Essentially, our method establishes that if the linearized operator has only one negative eigenvalue which is simple and zero is a simple eigenvalue the orbital stability is determined provided that a convenient condition about the average of the wave is satisfied. We use our approach to prove the orbital stability of periodic dnoidal waves associat…
We consider a class of nonlinear Dirichlet problems involving the $p(x)$--Laplace operator. Our framework is based on the theory of Sobolev spaces with variable exponent and we establish the existence of a weak solution in such a space. The proof relies on the Mountain Pass Theorem.
There is a family of potentials that minimize the lowest eigenvalue of a Schrödinger eigenvalue under the constraint of a given L^p norm of the potential. We give effective estimates for the amount by which the eigenvalue increases when the potential is not one of these optimal potentials. Our results are analogous to those for the isoperimetric problem and the Sobolev inequality. We also prove a stability estimate for Hölder's inequality, which we believe to be new.
We study general semilinear scalar-field equations on the real line with variable coefficients in the linear terms. These coefficients are uniformly small, but slowly decaying, perturbations of a constant-coefficient operator. We are motivated by the question of how these perturbations of the equation may change the stability properties of kink solutions (one-dimensional topological solitons). We prove existence of a stationary kink solution in our setting, and perform a detailed spectral analysis of the corresponding linearized operator, based on perturbing the linearized operator around the …
We study the point spectrum of the linearization at a solitary wave solution $ϕ_ω(x)e^{-\mathrm{i}ωt}$ to the nonlinear Dirac equation in $\mathbb{R}^n$, $n\ge 1$, with the nonlinear term given by $f(ψ^*βψ)βψ$ (known as the Soler model). We focus on the spectral stability, that is, the absence of eigenvalues with nonzero real part, in the non-relativistic limit $ω\lesssim m$, in the case when $f\in C^1(\mathbb{R}\setminus\{0\})$, $f(τ)=|τ|^k+O(|τ|^K)$ for $τ\to 0$, with $0<k<K$. For $n\ge 1$, we prove the spectral stability of small amplitude solitary waves ($ω\lesssim m$) for the charge-subcr…
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 study an eigenvalue problem for functions in R^N and we find sufficient conditions for the existence of the fundamental eigenvalue. This result can be applied to the study of the orbital stability of the standing waves of the nonlinear Schroedinger equation.
We consider the Bresse model with three control boundary conditions. We prove the exponential stability of the system using the semigroup theory of linear operators and a result obtained by Prüss.
In this paper, we prove the exponential stability of the solution of the nonlinear dissipative Schrödinger equation on a star-shaped network and where the damping is localized on one branch and at the infinity.
The goal of this paper is to study the Dirichlet eigenvalues of bounded domains $Ω\subset Ω'$. With a local spectral stability requirement on $Ω$, we show that the difference of the Dirichlet eigenvalues of $Ω'$ and $Ω$ is explicitly controlled from above in terms of the first eigenvalue of $Ω'\setminus\barΩ$ and of geometric constants depending on the inner domain $Ω$. In particular, $Ω'$ can be an arbitrary bounded domain.
Eigenvalue stability of Hermitian and normal matrices
研究Hermitian和正常矩阵特征值映射的连续性，证明其在Sobolev空间中的非均匀连续性，并探讨其在奇异值、矩阵条件数等应用中的稳定性。
We prove stability results associated with upper bounds for the first eigenvalue of certain second order differential operators of divergence-type on hypersurfaces of the Euclidean space. We deduce some applications to $r$-stability as well as to almost-Einstein hypersurfaces.
We consider general second order uniformly elliptic operators subject to homogeneous boundary conditions on open sets $ϕ(Ω)$ parametrized by Lipschitz homeomorphisms $ϕ$ defined on a fixed reference domain $Ω$. Given two open sets $ϕ(Ω)$, $\tilde ϕ(Ω)$ we estimate the variation of resolvents, eigenvalues and eigenfunctions via the Sobolev norm $\|\tilde ϕ-ϕ\|_{W^{1,p}(Ω)}$ for finite values of $p$, under natural summability conditions on eigenfunctions and their gradients. We prove that such conditions are satisfied for a wide class of operators and open sets, including open sets with Lipschit…
This paper is devoted to providing quantitative bounds on the location of eigenvalues, both discrete and embedded, of non self-adjoint Lamé operators of elasticity $-Δ^\ast + V$ in terms of suitable norms of the potential $V$. In particular, this allows to get sufficient conditions on the size of the potential such that the point spectrum of the perturbed operator remains empty. In three dimensions we show full spectral stability under suitable form-subordinated perturbations: we prove that the spectrum is purely continuous and coincides with the non negative semi-axis as in the free case.
We propose an iterative method to find pointwise growth exponential growth rates in linear problems posed on essentially one-dimensional domains. Such pointwise growth rates capture pointwise stability and instability in extended systems and arise as spectral values of a family of matrices that depends analytically on a spectral parameter, obtained via a scattering-type problem. Different from methods in the literature that rely on computing determinants of this nonlinear matrix pencil, we propose and analyze an inverse power method that allows one to locate robustly the closest spectral value…
Pollicott-Ruelle resonances for chaotic flows are the characteristic frequencies of correlations. They are typically defined as eigenvalues of the generator of the flow acting on specially designed functional spaces. We show that these resonances can be computed as viscosity limits of eigenvalues of second order elliptic operators. These eigenvalues are the characteristic frequencies of correlations for a stochastically perturbed flow.
We revisit the existence and stability of the critical front in the extended Fisher-KPP equation, refining earlier results of Rottschäfer and Wayne [28] which establish stability of fronts without identifying a precise decay rate. We verify that the front is marginally spectrally stable: while the essential spectrum touches the imaginary axis at the origin, there are no unstable eigenvalues and no eigenvalue (or resonance) embedded in the essential spectrum at the origin. Together with the recent work of Avery and Scheel [3], this implies nonlinear stability of the critical front with sharp $t…
We determine the nonlinear stability of shock-fronted travelling waves arising in a reaction-nonlinear diffusion PDE, subject to a fourth-order spatial derivative term multiplied by a small parameter $\varepsilon$ that models {\it nonlocal regularization}. Motivated by the authors' recent stability analysis of shock-fronted travelling waves under viscous relaxation, our numerical analysis is guided by the observation that there is a fast-slow decomposition of the associated eigenvalue problem for the linearised operator. In particular, we observe an astonishing reduction of the complex four-di…
We prove a stability estimate related to the multi-dimensional Borg-Levinson theorem of determining a potential from spectral data: the Dirichlet eigenvalues and the normal derivatives of the eigenfunctions on the boundary of a bounded domain. The estimate is of Hölder type, and we allow finitely many eigenvalues and normal derivatives to be unknown. We also show that if the spectral data is known asymptotically only, up to $O(k^{-a})$ with $a>>1$, then we still have Hölder stability.
We numerically study solitary waves in the coupled nonlinear Schrödinger equations. We detect pitchfork bifurcations of the fundamental solitary wave and compute eigenvalues and eigenfunctions of the corresponding eigenvalue problems to determine the spectral stability of solitary waves born at the pitchfork bifurcations. Our numerical results demonstrate the theoretical ones which the authors obtained recently. We also compute generalized eigenfunctions associated with the zero eigenvalue for the bifurcated solitary wave exhibiting a saddle-node bifurcation, and show that it does not change i…
In a first part of this paper we investigate the continuity (stability) of the spectrum of a class of non-local Schrödinger operators on varying the potentials. By imposing conditions of different strength on the convergence of the sequence of potentials, we give either direct proofs to show the strong or norm resolvent convergence of the so-obtained sequence of non-local Schrödinger operators, or via $Γ$-convergence of the related positive forms for more rough potentials. In a second part we use these results to show via a sequence of suitably constructed approximants that the ground states o…
Computing the first eigenpair for problems with variable exponents
本文研究了可变指数本征值问题中第一个本征对的计算，比较了均匀定义的第一个本征值与文献中非均匀定义的区别，并突显了对称破缺现象。
We study front solutions of a system that models combustion in highly hydraulically resistant porous media. The spectral stability of the fronts is tackled by a combination of energy estimates and numerical Evans function computations. Our results suggest that there is a parameter regime for which there are no unstable eigenvalues. We use recent works about partially parabolic systems to prove that in the absence of unstable eigenvalues the fronts are convectively stable.
We investigate spectral stability of vortex solutions of the Gross-Pitaevskii equation, a mean-field approximation for Bose-Einstein condensates (BEC) in an effectively two-dimensional axisymmetric harmonic trap. We study eigenvalues of the linearization both rigorously and through computation of the Evans function, a sensitive and robust technique whose use we justify mathematically. The absence of unstable eigenvalues is justified a posteriori through use of the Krein signature of purely imaginary eigenvalues, which also can be used to significantly reduce computational effort. In particular…
The dependence on the domain is studied for the Dirichlet eigenvalues of an elliptic operator considered in bounded domains. Their proximity is measured by a norm of the difference of two orthogonal projectors corresponding to the reference domain and the perturbed one; this allows to compare domains that have non-smooth boundaries and different topology. The main result is an asymptotic formula in which the remainder is evaluated in terms of this quantity. As an application, the stability of eigenvalues is estimated by virtue of integrals of squares of the gradients of eigenfunctions for elli…
In this paper, we study the long time behavior of nonlinear quantum walks when the initial data is small in $l^2$. In particular, we study the case where the linear part of the quantum walk evolution operator has exactly two eigenvalues and show that the solution decomposed into nonlinear bound states bifurcating from the eigenvalues and scattering waves.
We consider a spectral stability estimate by Burenkov and Lamberti concerning the variation of the eigenvalues of second order uniformly elliptic operators on variable open sets in the N-dimensional euclidean space, and we prove that it is sharp for any dimension N. This is done by studying the eigenvalue problem for the Dirichlet Laplacian on special open sets inscribed in suitable spherical cones.
This work is concerned with two optimisation problems that we tackle from a qualitative perspective. The first one deals with quantitative inequalities for spectral optimisation problems for Schrödinger operators in general domains, the second one deals with the turnpike property for optimal bilinear control problems. In the first part of this article, we prove, under mild technical assumptions, quantitative inequalities for the optimisation of the first eigenvalue of $-Δ-V$ with Dirichlet boundary conditions with respect to the potential $V$, under $L^\infty$ and $L^1$ constraints. This is do…
We study the boundary value problem $-{\rm div}((|\nabla u|^{p\_1(x) -2}+|\nabla u|^{p\_2(x)-2})\nabla u)=f(x,u)$ in $Ω$, $u=0$ on $\partialΩ$, where $Ω$ is a smooth bounded domain in $\RR^N$. We focus on the cases when $f\_\pm (x,u)=\pm(-λ|u|^{m(x)-2}u+|u|^{q(x)-2}u)$, where $m(x):=\max\{p\_1(x),p\_2(x)\} < q(x) < \frac{N\cdot m(x)}{N-m(x)}$ for any $x\in\barΩ$. In the first case we show the existence of infinitely many weak solutions for any $λ>0$. In the second case we prove that if $λ$ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponen…
We study the point spectrum of the nonlinear Dirac equation in any spatial dimension, linearized at one of the solitary wave solutions. We prove that, in any dimension, the linearized equation has no embedded eigenvalues in the part of the essential spectrum beyond the embedded thresholds. We then prove that the birth of point eigenvalues with nonzero real part (the ones which lead to linear instability) from the essential spectrum is only possible from the embedded eigenvalues or thresholds, and therefore can not take place beyond the embedded thresholds. We also prove that "in the nonrelativ…
A common challenge to proving asymptotic stability of solitary waves is understanding the spectrum of the operator associated with the linearized flow. The existence of eigenvalues can inhibit the dispersive estimates key to proving stability. Following the work of Marzuola & Simpson, we prove the absence of embedded eigenvalues for a collection of nonlinear Schrodinger equations, including some one and three dimensional supercritical equations, and the three dimensional cubic-quintic equation. Our results also rule out nonzero eigenvalues within the spectral gap and, in 3D, endpoint resonance…
On absence of embedded eigenvalues and stability of BGK waves
该研究针对一维静电Vlasov-Poisson系统的BGK波平衡态，证明其线性化算子在本质谱内无嵌入本征值，推导了非定量朗道阻尼，提出基于能量的方法处理共振相互作用，方法可推广至含椭圆、双曲临界点的谱问题。
Quantitative spectral stability for Aharonov-Bohm operators with many coalescing poles
本文研究多极点合并时 Aharonov–Bohm 算子单重特征值的定量稳定性，利用规范变换、裂纹 Laplace 问题与爆破分析给出渐近展开及符号判定。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。