Eigenvalue problems with indefinite weight

Andrzej Szulkin; Michel Willem

Displaying similar documents to “Eigenvalue problems with indefinite weight”

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...

On the uniqueness and simplicity of the principal eigenvalue

Marcello Lucia (2005)

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

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Given an open set $Ω$ of $R^{N}$ $(N > 2)$ , bounded or unbounded, and a function $w \in L^{\frac{N}{2}} (Ω)$ with $w^{+} \neq 0$ but allowed to change sign, we give a short proof that the positive principal eigenvalue of the problem $- △ u = λ w (x) u, u \in D_{0}^{1, 2} (Ω)$ is unique and simple. We apply this result to study unbounded Palais-Smale sequences as well as to give a priori estimates on the set of critical points of functionals of the type $I (u) = \frac{1}{2} \int_{Ω} {|\nabla u|}^{2} d x - \int_{Ω} G (x, u) d x, u \in D_{0}^{1, 2} (Ω),$ when $G$ has a subquadratic growth at infinity.

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.

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.

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.

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

Jan Bochenek (1980)

Annales Polonici Mathematici

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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.

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...