Il est bien connu que les fréquences propres associées à un d'Alembertien amorti sont confinées dans une bande parallèle à l'axe réel. Nous rappelons l'asymptotique de Weyl pour la distribution des parties réelles des fréquences propres, nous montrons que «presque toutes» les fréquences propres appartiennent à une bande déterminée par la limite de Birkhoff du coefficient d'amortissement. Nous montrons aussi que certaines moyennes des parties imaginaires convergent vers la moyenne du coefficient...

This article studies the asymptotic behavior of the number $N\left(\lambda \right)$ of the negative eigenvalues $\<-\lambda$ as $\lambda \to +0$ of the two dimensional Pauli operators with electric potential $V\left(x\right)$ decaying at $\infty$ and with nonconstant magnetic field $b\left(x\right)$, which is assumed to be bounded or to decay at $\infty$. In particular, it is shown that $N\left(\lambda \right)=(1/2\pi ){\int }_{V\left(x\right)\>\lambda }b\left(x\right)dx(1+o\left(1\right))$, when $V\left(x\right)$ decays faster than $b\left(x\right)$ under some additional conditions.

For Schrödinger operator $P$ on Riemannian manifolds with conical end, we study the contribution of zero energy resonant states to the singularity of the resolvent of $P$ near zero. Long-time expansion of the Schrödinger group $U\left(t\right)={\mathrm{e}}^{-itP}$ is obtained under a non-trapping condition at high energies.

In this paper we consider the $\varphi$-Laplacian problem with Dirichlet boundary condition, $$-\mathrm{div}\left(\varphi \left(\right|\nabla u\left|\right)\frac{\nabla u}{\left|\nabla u\right|}\right){=\lambda g(·)\varphi \left(u\right)\text{in}\Omega ,\lambda \in ℝ\text{and}u|}_{\partial \Omega }=0.$$ The term $\varphi$ is a real odd and increasing homeomorphism, $g$ is a nonnegative function in $L^{\infty }\left(\Omega \right)$ and $\Omega \subseteq {ℝ}^N$ 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 a minimizer...

2000 Mathematics Subject Classification: 35J05, 35C15, 44P05In this paper, we consider the variations of eigenvalues and eigenfunctions for the Laplace operator with homogeneous Dirichlet boundary conditions under deformation of the underlying domain of definition. We derive recursive formulas for the Taylor coefficients of the eigenvalues as functions of the shape-perturbation parameter and we establish the existence of a set of eigenfunctions that is jointly holomorphic in the spatial and boundary-variation ...

In the recent years, many results have been established on positive solutions for boundary value problems of the form $-div\left(\right|\nabla u\left(x\right){|}^{p-2}\nabla u\left(x\right))=\lambda f\left(u\left(x\right)\right)$ in Ω, u(x)=0 on ∂Ω, where λ > 0, Ω is a bounded smooth domain and f(s) ≥ 0 for s ≥ 0. In this paper, a priori estimates of positive radial solutions are presented when N > p > 1, Ω is an N-ball or an annulus and f ∈ C¹(0,∞) ∪ C⁰([0,∞)) with f(0) < 0 (non-positone).