A counterexample to the “hot spots

Burdzy, Krzysztof; Werner, Wendelin

Displaying similar documents to “A counterexample to the “hot spots' conjecture.”

Equality in Wielandt’s eigenvalue inequality

Shmuel Friedland (2015)

Special Matrices

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In this paper we give necessary and sufficient conditions for the equality case in Wielandt’s eigenvalue inequality.

Dependence of a differential equation on the first eigenvalue of a suitable problem

Jan Bochenek (1980)

Annales Polonici Mathematici

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On Selberg's Eigenvalue Conjecture.

P. Sarnak, W. Luo, Z. Rudnick (1995)

Geometric and functional analysis

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A simple derivation of the eigenvalues of a tridiagonal matrix arising in biogeography

Qassem M. Al-Hassan, Mowaffaq Hajja (2015)

Applicationes Mathematicae

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In investigating a certain optimization problem in biogeography, Simon [IEEE Trans. Evolutionary Comput. 12 (2008), 702-713] encountered a certain specially structured tridiagonal matrix and made a conjecture regarding its eigenvalues. A few years later, the validity of the conjecture was established by Igelnik and Simon [Appl. Math. Comput. 218 (2011), 195-201]. In this paper, we give another proof of this conjecture that is much shorter, almost computation-free, and does not resort...

On some properties of eigenvalues and eigenfunctions of certain differential equations of the fourth order

Jan Bochenek (1971)

Annales Polonici Mathematici

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Monotonicity of the principal eigenvalue related to a non-isotropic vibrating string

Behrouz Emamizadeh, Amin Farjudian (2014)

Nonautonomous Dynamical Systems

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In this paper we consider a parametric eigenvalue problem related to a vibrating string which is constructed out of two different materials. Using elementary analysis we show that the corresponding principal eigenvalue is increasing with respect to the parameter. Using a rearrangement technique we recapture a part of our main result, in case the difference between the densities of the two materials is sufficiently small. Finally, a simple numerical algorithm will be presented which will...

An integro-differential inequality related to the smallest positive eigenvalue ofp(x)-Laplacian Dirichlet problem

Damian Wiśniewski, Mariusz Bodzioch (2016)

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

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We consider the eigenvalue problem for the p(x)-Laplace-Beltrami operator on the unit sphere. We prove same integro-differential inequalities related to the smallest positive eigenvalue of this problem.

A nonlinear eigenvalue problem with indefinite weights related to the Sobolev trace embedding.

Julián Fernández Bonder, Julio D. Rossi (2002)

Publicacions Matemàtiques

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In this paper we study the Sobolev trace embedding W(Ω) → L (∂Ω), where V is an indefinite weight. This embedding leads to a nonlinear eigenvalue problem where the eigenvalue appears at the (nonlinear) boundary condition. We prove that there exists a sequence of variational eigenvalues λ / +∞ and then show that the first eigenvalue is isolated, simple and monotone with respect to the weight. Then we prove a nonexistence result related to the first eigenvalue and we end...

Note on a paper of E. M. E. Zayed and S. F. M. Ibraham.

Eberhard, W., Freiling, G., Schneider, A. (1992)

International Journal of Mathematics and Mathematical Sciences

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Existence of multiple principal eigenvalues for some indefinite linear eigenvalue problems

J. Fleckinger, J. Hernández, F. Thélin (2004)

Bollettino dell'Unione Matematica Italiana

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We study the existence of principal eigenvalues for differential operators of second order which are not necessarily in divergence form. We obtain results concerning multiplicity of principal eigenvalues in both the variational and the general case. Our approach uses systematically the Krein-Rutman theorem and fixed point arguments for the spectral radius of some associated problems. We also use a variational characterization for both the self-adjoint and the general case.

On the nodal set of the second eigenfunction of the laplacian in symmetric domains in $R^{N}$

Lucio Damascelli (2000)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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We present a simple proof of the fact that if $Ω$ is a bounded domain in $R^{N}$ , $N \geq 2$ , which is convex and symmetric with respect to $k$ orthogonal directions, $1 \leq k \leq N$ , then the nodal sets of the eigenfunctions of the laplacian corresponding to the eigenvalues $λ_{2}, \dots, λ_{k + 1}$ must intersect the boundary. This result was proved by Payne in the case $N = 2$ for the second eigenfunction, and by other authors in the case of convex domains in the plane, again for the second eigenfunction.