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Spectrum of the Laplacian in narrow tubular neighbourhoods of hypersurfaces with combined Dirichlet and Neumann boundary conditions

David Krejčiřík (2014)

Mathematica Bohemica

We consider the Laplacian in a domain squeezed between two parallel hypersurfaces in Euclidean spaces of any dimension, subject to Dirichlet boundary conditions on one of the hypersurfaces and Neumann boundary conditions on the other. We derive two-term asymptotics for eigenvalues in the limit when the distance between the hypersurfaces tends to zero. The asymptotics are uniform and local in the sense that the coefficients depend only on the extremal points where the ratio of the area of the Neumann...

Spectrum of the weighted Laplace operator in unbounded domains

Alexey Filinovskiy (2011)

Mathematica Bohemica

We investigate the spectral properties of the differential operator - r s Δ , s ≥ 0 with the Dirichlet boundary condition in unbounded domains whose boundaries satisfy some geometrical condition. Considering this operator as a self-adjoint operator in the space with the norm ∥ u ∥ L 2 , s ( Ω ) 2 = ∫ Ω r - s | u | 2 d x , we study the structure of the spectrum with respect to the parameter s . Further we give an estimate of the rate of condensation of discrete spectra when it changes to continuous.

Stability and semiclassics in self-generated fields

László Erdős, Soren Fournais, Jan Philip Solovej (2013)

Journal of the European Mathematical Society

We consider non-interacting particles subject to a fixed external potential V and a self-generated magnetic field B . The total energy includes the field energy β ∫ B 2 and we minimize over all particle states and magnetic fields. In the case of spin-1/2 particles this minimization leads to the coupled Maxwell-Pauli system. The parameter β tunes the coupling strength between the field and the particles and it effectively determines the strength of the field. We investigate the stability and the semiclassical...

Stabilization of Schrödinger equation in exterior domains

Lassaad Aloui, Moez Khenissi (2007)

ESAIM: Control, Optimisation and Calculus of Variations

We prove uniform local energy estimates of solutions to the damped Schrödinger equation in exterior domains under the hypothesis of the Exterior Geometric Control. These estimates are derived from the resolvent properties.

Strong diamagnetism for general domains and application

Soeren Fournais, Bernard Helffer (2007)

Annales de l’institut Fourier

We consider the Neumann Laplacian with constant magnetic field on a regular domain in ℝ 2 . Let B be the strength of the magnetic field and let λ 1 ( B ) be the first eigenvalue of this Laplacian. It is proved that B ↦ λ 1 ( B ) is monotone increasing for large B . Together with previous results of the authors, this implies the coincidence of all the “third” critical fields for strongly type 2 superconductors.

Superconvergence of a stabilized approximation for the Stokes eigenvalue problem by projection method

Pengzhan Huang (2014)

Applications of Mathematics

This paper presents a superconvergence result based on projection method for stabilized finite element approximation of the Stokes eigenvalue problem. The projection method is a postprocessing procedure that constructs a new approximation by using the least squares method. The paper complements the work of Li et al. (2012), which establishes the superconvergence result of the Stokes equations by the stabilized finite element method. Moreover, numerical tests confirm the theoretical analysis.

Sur le spectre semi-classique d’un système intégrable de dimension 1 autour d’une singularité hyperbolique

Olivier Lablée (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

Dans cette article on décrit le spectre semi-classique d’un opérateur de Schrödinger sur ℝ avec un potentiel type double puits. La description qu’on donne est celle du spectre autour du maximum local du potentiel. Dans la classification des singularités de l’application moment d’un système intégrable, le double puits représente le cas des singularités non-dégénérées de type hyperbolique.

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

Claudia Anedda, Fabrizio Cuccu (2015)

Applicationes Mathematicae

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 domain.

The effect of reduced integration in the Steklov eigenvalue problem

Maria G. Armentano (2004)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this paper we analyze the effect of introducing a numerical integration in the piecewise linear finite element approximation of the Steklov eigenvalue problem. We obtain optimal order error estimates for the eigenfunctions when this numerical integration is used and we prove that, for singular eigenfunctions, the eigenvalues obtained using this reduced integration are better approximations than those obtained using exact integration when the mesh size is small enough.

The effect of reduced integration in the Steklov eigenvalue problem

María G. Armentano (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we analyze the effect of introducing a numerical integration in the piecewise linear finite element approximation of the Steklov eigenvalue problem. We obtain optimal order error estimates for the eigenfunctions when this numerical integration is used and we prove that, for singular eigenfunctions, the eigenvalues obtained using this reduced integration are better approximations than those obtained using exact integration when the mesh size is small enough.

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