Displaying similar documents to “On the nodal set of the second eigenfunction of the laplacian in symmetric domains in R N ”

Eigenvalue problems with indefinite weight

Andrzej Szulkin, Michel Willem (1999)

Studia Mathematica

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We consider the linear eigenvalue problem -Δu = λV(x)u, u ∈ D 0 1 , 2 ( Ω ) , and its nonlinear generalization - Δ p u = λ V ( x ) | u | p - 2 u , u ∈ D 0 1 , p ( Ω ) . The set Ω need not be bounded, in particular, Ω = ℝ N is admitted. The weight function V may change sign and may have singular points. We show that there exists a sequence of eigenvalues λ n → ∞ .

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 ∈ L N 2 ⁢ Ω with w + ≠ 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 ∈ 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 = 1 2 ⁢ ∫ Ω ∇ ⁡ u 2 ⁢ d ⁢ x - ∫ Ω G ⁢ x , u ⁢ d ⁢ x , u ∈ 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.

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...

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.

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 ∫ 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...