Asymptotic lower bounds for eigenvalues of the Steklov eigenvalue problem with variable coefficients

Yu Zhang; Hai Bi; Yidu Yang

Displaying similar documents to “Asymptotic lower bounds for eigenvalues of the Steklov eigenvalue problem with variable coefficients”

A lower bound sequence for the minimum eigenvalue of Hadamard product of an $M$ -matrix and its inverse

Wenlong Zeng, Jianzhou Liu (2022)

Czechoslovak Mathematical Journal

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We propose a lower bound sequence for the minimum eigenvalue of Hadamard product of an $M$ -matrix and its inverse, in terms of an $S$ -type eigenvalues inclusion set and inequality scaling techniques. In addition, it is proved that the lower bound sequence converges. Several numerical experiments are given to demonstrate that the lower bound sequence is sharper than some existing ones in most cases.

On graphs whose second least eigenvalue is at least $- 1$

M. Petrović (1990)

Matematički Vesnik

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Behaviour of the first eigenvalue of the p-Laplacian in a domain with a hole

M. Sango (2001)

Colloquium Mathematicae

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We investigate the behaviour of a sequence $λ_{s}$ , s = 1,2,..., of eigenvalues of the Dirichlet problem for the p-Laplacian in the domains $Ω_{s}$ , s = 1,2,..., obtained by removing from a given domain Ω a set $E_{s}$ whose diameter vanishes when s → ∞. We estimate the deviation of $λ_{s}$ from the eigenvalue of the limit problem. For the derivation of our results we construct an appropriate asymptotic expansion for the sequence of solutions of the original eigenvalue problem.

A general homogenization result of spectral problem for linearized elasticity in perforated domains

Mohamed Mourad Lhannafi Ait Yahia, Hamid Haddadou (2021)

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The goal of this paper is to establish a general homogenization result for linearized elasticity of an eigenvalue problem defined over perforated domains, beyond the periodic setting, within the framework of the $H^{0}$ -convergence theory. Our main homogenization result states that the knowledge of the fourth-order tensor $A^{0}$ , the $H^{0}$ -limit of $A^{ε}$ , is sufficient to determine the homogenized eigenvalue problem and preserve the structure of the spectrum. This theorem is proved essentially by using...

Symmetry breaking in the minimization of the first eigenvalue for the composite clamped punctured disk

Claudia Anedda, Fabrizio Cuccu (2015)

Applicationes Mathematicae

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Let D₀=x∈ ℝ²: 0<|x|<1 be the unit punctured disk. We consider the first eigenvalue λ₁(ρ ) of the problem Δ² u =λ ρ u in D₀ with Dirichlet boundary condition, where ρ is an arbitrary function that takes only two given values 0 < α < β and is subject to the constraint $\int_{D ₀} ρ d x = α γ + β (| D ₀ | - γ)$ for a fixed 0 < γ < |D₀|. We will be concerned with the minimization problem ρ ↦ λ₁(ρ). We show that, under suitable conditions on α, β and γ, the minimizer does not inherit the radial symmetry of the...

Eigenvalue asymptotics for Neumann Laplacian in domains with ultra-thin cusps

Victor Ivrii (1998-1999)

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Asymptotics with sharp remainder estimates are recovered for number $N (τ)$ of eigenvalues of the generalized Maxwell problem and for related Laplacians which are similar to Neumann Laplacian. We consider domains with ultra-thin cusps (with $exp (- | x |^{m + 1}$ ) width ; $m > 0$ ) and recover eigenvalue asymptotics with sharp remainder estimates.

On the existence of five nontrivial solutions for resonant problems with p-Laplacian

Leszek Gasiński, Nikolaos S. Papageorgiou (2010)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In this paper we study a nonlinear Dirichlet elliptic differential equation driven by the p-Laplacian and with a nonsmooth potential. The hypotheses on the nonsmooth potential allow resonance with respect to the principal eigenvalue λ₁ > 0 of $(- Δ ₚ, W ₀^{1, p} (Z))$ . We prove the existence of five nontrivial smooth solutions, two positive, two negative and the fifth nodal.

Some properties complementary to Brualdi-Li matrices

Chuanlong Wang, Xuerong Yong (2015)

Czechoslovak Mathematical Journal

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In this paper we derive new properties complementary to an $2 n \times 2 n$ Brualdi-Li tournament matrix $B_{2 n}$ . We show that $B_{2 n}$ has exactly one positive real eigenvalue and one negative real eigenvalue and, as a by-product, reprove that every Brualdi-Li matrix has distinct eigenvalues. We then bound the partial sums of the real parts and the imaginary parts of its eigenvalues. The inverse of $B_{2 n}$ is also determined. Related results obtained in previous articles are proven to be corollaries.

On the principal eigencurve of the p-Laplacian related to the Sobolev trace embedding

Abdelouahed El Khalil, Mohammed Ouanan (2005)

Applicationes Mathematicae

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We prove that for any λ ∈ ℝ, there is an increasing sequence of eigenvalues μₙ(λ) for the nonlinear boundary value problem ⎧ $Δ ₚ u = {| u |}^{p - 2} u$ in Ω, ⎨ ⎩ ${| \nabla u |}^{p - 2} \partial u / \partial ν = {λ ϱ (x) | u |}^{p - 2} u + μ {| u |}^{p - 2} u$ on crtial ∂Ω and we show that the first one μ₁(λ) is simple and isolated; we also prove some results about variations of the density ϱ and the continuity with respect to the parameter λ.

Sublinear eigenvalue problems on compact Riemannian manifolds with applications in Emden-Fowler equations

Alexandru Kristály, Vicenţiu Rădulescu (2009)

Studia Mathematica

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Let (M,g) be a compact Riemannian manifold without boundary, with dim M ≥ 3, and f: ℝ → ℝ a continuous function which is sublinear at infinity. By various variational approaches, existence of multiple solutions of the eigenvalue problem $- Δ_{g} ω + α (σ) ω = K ̃ (λ, σ) f (ω)$ , σ ∈ M, ω ∈ H₁²(M), is established for certain eigenvalues λ > 0, depending on further properties of f and on explicit forms of the function K̃. Here, $Δ_{g}$ stands for the Laplace-Beltrami operator on (M,g), and α, K̃ are smooth positive functions. These...

The eigenvalues of symmetric Sturm-Liouville problem and inverse potential problem, based on special matrix and product formula

Chein-Shan Liu, Botong Li (2024)

Applications of Mathematics

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The Sturm-Liouville eigenvalue problem is symmetric if the coefficients are even functions and the boundary conditions are symmetric. The eigenfunction is expressed in terms of orthonormal bases, which are constructed in a linear space of trial functions by using the Gram-Schmidt orthonormalization technique. Then an $n$ -dimensional matrix eigenvalue problem is derived with a special matrix $𝐀 : = [a_{i j}]$ , that is, $a_{i j} = 0$ if $i + j$ is odd.Based on the product formula, an integration method with a fictitious...

New method for computation of discrete spectrum of radical Schrödinger operator

Ivan Úlehla, Miloslav Havlíček (1980)

Aplikace matematiky

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A new method for computation of eigenvalues of the radial Schrödinger operator $- d^{2} / d x^{2} + v (x), x \geq 0$ is presented. The potential $v (x)$ is assumed to behave as $x^{- 2 + ϵ}$ if $x \to 0_{+}$ and as $x^{- 2 - ϵ}$ if $x \to + \infty, ϵ \geq 0$ . The Schrödinger equation is transformed to a non-linear differential equation of the first order for a function $z (x, ℵ)$ . It is shown that the eigenvalues are the discontinuity points of the function $z (\infty, ℵ)$ . Moreover, it is shown how to obtain an arbitrarily accurate approximation of eigenvalues. The method seems to be much more economical in...

Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces

Mikhail Karpukhin, Gerasim Kokarev, Iosif Polterovich (2014)

Annales de l’institut Fourier

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We prove two explicit bounds for the multiplicities of Steklov eigenvalues $σ_{k}$ on compact surfaces with boundary. One of the bounds depends only on the genus of a surface and the index $k$ of an eigenvalue, while the other depends as well on the number of boundary components. We also show that on any given Riemannian surface with smooth boundary the multiplicities of Steklov eigenvalues $σ_{k}$ are uniformly bounded in $k$ .