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

Claudia Anedda; Fabrizio Cuccu

Displaying similar documents to “Symmetry breaking in the minimization of the first eigenvalue for the composite clamped punctured disk”

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.

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.

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...

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

Mohamed Mourad Lhannafi Ait Yahia, Hamid Haddadou (2021)

Applications of Mathematics

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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...

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

Victor Ivrii (1998-1999)

Séminaire Équations aux dérivées partielles

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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.

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 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.

On the spectrum and eigenfunctions of the operator $(V f) (x) = \int_{0}^{x^{α}} f (t) d t$

I. Yu. Domanov (2007)

Banach Center Publications

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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...

About a Pólya-Schiffer inequality

Bodo Dittmar, Maren Hantke (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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For simply connected planar domains with the maximal conformal radius 1 it was proven in 1954 by G. Pólya and M. Schiffer that for the eigenvalues $λ$ of the fixed membrane for any $n$ the following inequality holds $\sum_{k = 1}^{n} \frac{1}{λ_{k}} \geq \sum_{k = 1}^{n} \frac{1}{λ_{k}^{(σ)}},$ where $λ_{k}^{(σ)}$ are the eigenvalues of the unit disk. The aim of the paper is to give a sharper version of this inequality and for the sum of all reciprocals to derive formulas which allow in some cases to calculate exactly this sum.

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 $k$ .

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.

Asymptotic properties of a $ϕ$ -Laplacian and Rayleigh quotient

Waldo Arriagada, Jorge Huentutripay (2020)

Commentationes Mathematicae Universitatis Carolinae

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In this paper we consider the $ϕ$ -Laplacian problem with Dirichlet boundary condition, $- div (ϕ (| \nabla u |) \frac{\nabla u}{| \nabla u |}) {= λ g (\cdot) ϕ (u) in Ω, λ \in ℝ and u |}_{\partial Ω} = 0 .$ The term $ϕ$ is a real odd and increasing homeomorphism, $g$ is a nonnegative function in $L^{\infty} (Ω)$ and $Ω \subseteq ℝ^{N}$ is a bounded domain. In these notes an analysis of the asymptotic behavior of sequences of eigenvalues of the differential equation is provided. We assume conditions which guarantee the existence of stationary solutions of the system. Under these rather stringent hypotheses we prove that any extremal is both...

Monotonicity of first eigenvalues along the Yamabe flow

Liangdi Zhang (2021)

Czechoslovak Mathematical Journal

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We construct some nondecreasing quantities associated to the first eigenvalue of $- Δ_{φ} + c R$ $(c \geq \frac{1}{2} (n - 2) / (n - 1))$ along the Yamabe flow, where $Δ_{φ}$ is the Witten-Laplacian operator with a $C^{2}$ function $φ$ . We also prove a monotonic result on the first eigenvalue of $- Δ_{φ} + \frac{1}{4} (n / (n - 1)) R$ along the Yamabe flow. Moreover, we establish some nondecreasing quantities for the first eigenvalue of $- Δ_{φ} + c R^{a}$ with $a \in (0, 1)$ along the Yamabe flow.