Stability results for some fully nonlinear eigenvalue estimates

带球形障碍物的Steklov拉普拉斯特征值问题的一个稳定性结果

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。

Stability of variational eigenvalues for the fractional p-Laplacian

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.

Generalized eigenvalues for fully tnonlinear singular or degenerate operators in the radial case

In this paper we extend some existence's results concerning the generalized eigenvalues for fully nonlinear operators singular or degenerate. We consider the radial case and we prove the existence of an infinite number of eigenvalues, simple and isolated. This completes the results obtained by the author with Isabeau Birindelli for the first eigenvalues in the radial case, and the results obtained for the Pucci's operator by Busca Esteban and Quaas and for the $p$-Laplace operator by Del Pino and Manasevich.

Existence and stability of electromagnetic Stekloff eigenvalues with a trace class modification

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…

A Neumann eigenvalue problem for fully nonlinear operators

In this paper we study the asymptotic behavior of the principal eigenvalues associated to the Pucci operator in bounded domain $Ω$ with Neumann/Robin boundary condition i.e. $\partial_n u=αu$ when $α$ tends to infinity. This study requires Lipschitz estimates up to the boundary that are interesting in their own rights.

Eigenvalue and Dirichlet problem for fully-nonlinear operators in non smooth domains

In this paper we study the maximum principle, the existence of eigenvalue and the existence of solution for the Dirichlet problem for operators which are fully-nonlinear, elliptic but presenting some singularity or degeneracy which are similar to those of the p-Laplacian, the novelty resides in the fact that we consider the equations in bounded domains which only satisfy the exterior cone condition.

A lower bound for the principal eigenvalue of fully nonlinear elliptic operators

In this article we present a new technique to obtain a lower bound for the principal Dirichlet eigenvalue of a fully nonlinear elliptic operator. We ilustrate the construction of an appropriate radial function required to obtain the bound in several examples. In particular we use our results to prove that $\lim_{p\to \infty}λ_{1,p}=λ_{1,\infty}=\left(\fracπ{2R}\right)^2$ where $λ_{1,p}$ and $λ_{1,\infty}$ are the principal eigenvalue for the homogeneous $p$-laplacian and the homogeneous infinity laplacian respectively.

最优绝缘中两相特征值问题的对称性破缺与局部稳定性

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。

第二Robin特征值的稳定性分析：负边界参数情况

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。

Steklov-Dirichlet spectrum: stability, optimization and continuity of eigenvalues

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…

通过Alexandroff-Bakelman-Pucci方法得到的完全非线性椭圆方程的特征值不等式

Eigenvalue Inequalities for Fully Nonlinear Elliptic Equations via the Alexandroff-Bakelman-Pucci Method

本文利用Alexandroff-Bakelman-Pucci方法建立Dirichlet或Robin特征值问题中椭圆微分算子的特征值不等式，推广至完全非线性椭圆方程，给出拉普拉斯算子L^p范数紧下界及解梯度的L^∞界与C^3估计。

Stability and Perturbations of the Domain for the First Eigenvalue of the 1-Laplacian

We discuss stability of the first eigenvalue of the 1-Laplacian under perturbations of the domain.

Stability and limiting properties of generalized principal eigenvalue for inhomogeneous nonlocal cooperative system

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…

The Brunn-Minkowski inequality for the principal eigenvalue of fully nonlinear homogeneous elliptic operators

We prove that the principal eigenvalue of any fully nonlinear homogeneous elliptic operator which fulfills a very simple convexity assumption satisfies a Brunn-Minkowski type inequality on the class of open bounded sets in $\mathbb{R}^n$ satisfying a uniform exterior sphere condition. In particular the result applies to the (possibly normalized) $p$-Laplacian, and to the minimal Pucci operator. The proof is inspired by the approach introduced by Colesanti for the principal frequency of the Laplacian within the class of convex domains, and relies on a generalization of the convex envelope metho…

Asymptotic Behavior in Degenerate Parabolic Fully Nonlinear equations and its application to Elliptic Eigenvalue Problems

We study the asymptotic behavior of the nonlinear parabolic flows $u_{t}=F(D^2 u^m)$ when $t\ra \infty$ for $m\geq 1$, and the geometric properties for solutions of the following elliptic nonlinear eigenvalue problems: F(D^2 \vp) &+ μ\vp^{p}=0, \quad \vp>0\quad\text{in $Ω$} \vp&=0\quad\text{on $\pΩ$} posed in a (strictly) convex and smooth domain $Ω\subset\re^n$ for $0< p \leq 1,$ where $F(\cdot)$ is uniformly elliptic, positively homogeneous of order one and concave. We establish that $\log (\vp)$ is concave in the case $p=1$ and that the function $\vp^{\frac{1-p}{2}}$ is concave for $0<p<1.$

Uniform stability of the ball with respect to the first Dirichlet and Neumann $\infty-$eigenvalues

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…

抽象希尔伯特空间设定下变双线性形式特征值的稳定性及其应用

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。

Eigenvalue, global bifurcation and positive solutions for a class of fully nonlinear problems

In this paper, we shall study global bifurcation phenomenon for the following Kirchhoff type problem \begin{equation} \left\{ \begin{array}{l} -\left(a+b\int_Ω\vert \nabla u\vert^2\,dx\right)Δu=λu+h(x,u,λ)\,\,\text{in}\,\, Ω,\\ u=0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{on}\,\,Ω. \end{array} \right.\nonumber \end{equation} Under some natural hypotheses on $h$, we show that $\left(aλ_1,0\right)$ is a bifurcation point of the above problem. As applications of the above result, we shall determine the interval of $λ$, in which there exist positive solutions for the above problem wit…

On the existence and stability of modified Maxwell Steklov eigenvalues

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…

Global branches of solutions for a class of non uniformly fully nonlinear elliptic equations

We consider singular perturbations of eigenvalue problems. We prove that to these problems correspond simple eigenvalues and we study their asymptotic behavior. As a result, we prove global bifurcation results for non uniformly and fully nonlinear elliptic equations. These equations are defined on smooth enough bounded domains and the solutions belonging in these branches are smooth and positive.

微极剪切流的稳定性与特征值界

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。

Upper bounds for the eigenvalues of Hessian equations

We prove some upper bounds for the Dirichlet eigenvalues of a class of fully nonlinear elliptic equations, namely the Hessian equations

The Dirichlet problem for singular fully nonlinear operators

In this paper we prove existence of (viscosity) solutions of Dirichlet problems concerning fully nonlinear elliptic operator, which are either degenerate or singular when the gradient of the solution is zero. For this class of operators it is possible to extend the concept of eigenvalue, this paper concerns the cases when the inf of the principal eigenvalues is positive i.e. when both the maximum and the minimum principle holds.

Stability of isoperimetric inequalities for Laplace eigenvalues on surfaces

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…

The eigenvalue problem for the Monge-Ampère operator on general bounded convex domains

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…

Spectral stability under removal of small capacity sets and applications to Aharonov-Bohm operators

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.

通过第二特征值对 Dirichlet 拉普拉斯算子全谱的定量稳定性控制

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。

Laplacian算子第一特征值的极值区域的稳定性

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。

Spectral stability of nonlinear waves in KdV-type evolution equations

This paper concerns spectral stability of nonlinear waves in KdV-type evolution equations. The relevant eigenvalue problem is defined by the composition of an unbounded self-adjoint operator with a finite number of negative eigenvalues and an unbounded non-invertible symplectic operator $\partial_x$. The instability index theorem is proven under a generic assumption on the self-adjoint operator both in the case of solitary waves and periodic waves. This result is reviewed in the context of other recent results on spectral stability of nonlinear waves in KdV-type evolution equations.

第一Robin-Neumann特征值的稳定性结果：一种双扰动方法

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。

Conformal Spectral Stability Estimates for the Neumann Laplacian

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.

An eigenvalue problem for fully nonlinear elliptic equations with gradient constraints

We consider the problem of finding $λ\in \mathbb{R}$ and a function $u:\mathbb{R}^n\rightarrow\mathbb{R}$ that satisfy the PDE $$ \max\left\{λ+ F(D^2u) -f(x),H(Du)\right\}=0, \quad x\in \mathbb{R}^n. $$ Here $F$ is elliptic, positively homogeneous and superadditive, $f$ is convex and superlinear, and $H$ is typically assumed to be convex. Examples of this type of PDE arise in the theory of singular ergodic control. We show that there is a unique $λ^*$ for which the above equation has a solution $u$ with appropriate growth as $|x|\rightarrow \infty$. Moreover, associated to $λ^*$ is a convex so…

Orbital Stability of Periodic Traveling-Wave Solutions for a Dispersive Equation

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…

Stability estimates for the lowest eigenvalue of a Schrödinger operator

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.

Linear and orbital stability analysis for solitary-wave solutions of variable-coefficient scalar field equations

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 …

Spectral stability of small amplitude solitary waves of the Dirac equation with the Soler-type nonlinearity

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…

Stability of fully discrete variational schemes for elastodynamics with a polyconvex stored energy

In this article we develop a fully discrete variational scheme that approximates the equations of three dimensional elastodynamics with polyconvex stored energy. The fully discrete scheme is based on a time-discrete variational scheme developed by S.~Demoulini, D.~M.~A.~Stuart and A.~E.~Tzavaras (2001). We show that the fully discrete scheme is unconditionally stable. The proof of stability is based on a relative entropy estimation for the fully discrete approximates.

Nonlinear stability of time-periodic viscous shocks

In order to understand the nonlinear stability of many types of time-periodic travelling waves on unbounded domains, one must overcome two main difficulties: the presence of embedded neutral eigenvalues and the time-dependence of the associated linear operator. This problem is studied in the context of time-periodic Lax shocks in systems of viscous conservation laws. Using spatial dynamics and a decomposition into separate Floquet eigenmodes, it is shown that the linear evolution for the time-dependent operator can be represented using a contour integral similar to that of the standard time-in…

On stability and instability of standing waves for 2d-nonlinear Schrödinger equations with point interaction

We study existence and stability properties of ground-state standing waves for two-dimensional nonlinear Schrödinger equation with a point interaction and a focusing power nonlinearity. The Schrödinger operator with a point interaction describes a one-parameter family of self-adjoint realizations of the Laplacian with delta-like perturbation. The perturbed Laplace operator always has a unique simple negative eigenvalue. We prove that if the frequency of the standing wave is close to the negative eigenvalue, it is stable. Moreover, if the frequency is sufficiently large, we have the stability i…

Sharp bounds for the first eigenvalue and the torsional rigidity related to some anisotropic operators

We prove a sharp upper bound for the first Dirichlet eigenvalue of a class of nonlinear elliptic operators which includes the p-Laplace and the pseudo-p-Laplace operators. Moreover, we prove a stability result by means of a suitable isoperimetric deficity. Finally, we give a sharp lower bound for the anisotropic p-torsional rigidity.

穿孔球中完全非线性方程的主特征值与特征函数

Principal eigenvalues and eigenfunctions for fully nonlinear equations in punctured balls

本文研究含奇异势的穿孔球上完全非线性一致椭圆方程的主特征值问题，分析不同γ参数下解的存在性与性质，并给出Pucci算子下的显式特征值，推广了拉普拉斯算子的Hardy–Sobolev常数。

Uniform stability of the Dirichlet spectrum for rough outer perturbations

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.

Hermitian和正常矩阵的特征值稳定性

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。

Pinching of the first eigenvalue for second order operators on hypersurfaces of the Euclidean space

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.

包含 Logistic 增长的一维极小 Keller--Segel 模型中尖峰的存在性、稳定性与慢动力学

Existence, Stability and Slow Dynamics of Spikes in a 1D Minimal Keller--Segel Model with Logistic Growth

本文基于趋化剂小扩散率奇异极限，分析了一维极小 Keller--Segel 模型中多尖峰图案的存在性、线性稳定性与慢动力学，揭示了由细胞扩散率和弛豫时间引发的失稳机制，并推导了控制其慢动力学的微分代数系统。

Stability estimates for resolvents, eigenvalues and eigenfunctions of elliptic operators on variable domains

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…

Eigenvalue bounds and spectral stability of Lamé operators with complex potentials

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.

Nonlinear eigenvalue methods for linear pointwise stability of nonlinear waves

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…

Stochastic stability of Pollicott-Ruelle resonances

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.

Spectral stability of the critical front in the extended Fisher-KPP equation

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…

Nonlinear stability of shock-fronted travelling waves under nonlocal regularization

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…

Stability for the multi-dimensional Borg-Levinson theorem with partial spectral data

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.

Numerical Computations for Bifurcations and Spectral Stability of Solitary Waves in Coupled Nonlinear Schrödinger Equations

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…

Stability of ground state eigenvalues of non-local Schrödinger operators with respect to potentials and applications

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…