We prove trace inequalities of type $\left|\right|u^{\text{'}}{\left|\right|}_{L^2}^2+{\sum }_{j\in \mathbb{Z}}{k}_j|u\left(a_j\right){|}^2{\ge \lambda \left|\right|u\left|\right|}_{L^2}^2$ where $u\in H^1\left(ℝ\right)$, under suitable hypotheses on the sequences ${a_j}_{j\in \mathbb{Z}}$ and ${k_j}_{j\in \mathbb{Z}}$, with the first sequence increasing and the second bounded.

We consider the variational problem         inf{αλ1(Ω) + βλ2(Ω) + (1 − α − β)λ3(Ω) | Ω open in ℝn, |Ω| ≤ 1}, for α, β ∈ [0, 1], α + β ≤ 1, where λk(Ω) is the kth eigenvalue of the Dirichlet Laplacian acting in L2(Ω) and |Ω| is the Lebesgue measure of Ω. We investigate for which values of α, β every minimiser is connected.

Soit $M$ une variété hyperbolique compacte de dimension 3, de diamètre $d$ et de volume $\le V$. Si on note ${\mu }_i\left(M\right)$ la $i$-ième valeur propre du laplacien de Hodge-de Rham agissant sur les 1-formes coexactes de $M$, on montre que ${\mu }_1\left(M\right)\ge \frac{c}{d^3{e}^{2kd}}$ et ${\mu }_{k+1}\left(M\right)\ge \frac{c}{d^2}$, où $c\>0$ est une constante ne dépendant que de $V$, et $k$ est le nombre de composantes connexes de la partie mince de $M$. En outre, on montre que pour toute 3-variété hyperbolique $M_{\infty }$ de volume fini avec cusps, il existe une suite $M_i$ de remplissages compacts de $M_{\infty }$, de diamètre $d_i\to +\infty$ telle que et ${\mu }_1\left(M_i\right)\ge \frac{c}{d_i^2}$.

We state a superconvergence result for the lowest order Raviart-Thomas approximation of eigenvalue problems. It is known that a similar superconvergence result holds for the mixed approximation of Laplace problem; here we introduce a new proof, since the one given for the source problem cannot be generalized in a straightforward way to the eigenvalue problem. Numerical experiments confirm the superconvergence property and suggest that it also holds for the lowest order Brezzi-Douglas-Marini...

We prove two explicit bounds for the multiplicities of Steklov eigenvalues ${\sigma }_k$ on compact surfaces with boundary. One of the bounds depends only on the genus of a surface and the index $k$ of an eigenvalue, while the other depends as well on the number of boundary components. We also show that on any given Riemannian surface with smooth boundary the multiplicities of Steklov eigenvalues ${\sigma }_k$ are uniformly bounded in $k$.

The paper deals with error estimates and lower bound approximations of the Steklov eigenvalue problems on convex or concave domains by nonconforming finite element methods. We consider four types of nonconforming finite elements: Crouzeix-Raviart, $Q_1^{\mathrm{rot}}$, $E{Q}_1^{\mathrm{rot}}$ and enriched Crouzeix-Raviart. We first derive error estimates for the nonconforming finite element approximations of the Steklov eigenvalue problem and then give the analysis of lower bound approximations. Some numerical results are presented to...

Many problems in quantum chemistry deal with the computation of fundamental or excited states of molecules and lead to the resolution of eigenvalue problems. One of the major difficulties in these computations lies in the very large dimension of the systems to be solved. Indeed these eigenfunctions depend on $3n$ variables where $n$ stands for the number of particles (electrons and/or nucleari) in the molecule. In order to diminish the size of the systems to be solved, the chemists have proposed many...

Many problems in quantum chemistry deal with the computation of fundamental or excited states of molecules and lead to the resolution of eigenvalue problems. One of the major difficulties in these computations lies in the very large dimension of the systems to be solved. Indeed these eigenfunctions depend on 3n variables where n stands for the number of particles (electrons and/or nucleari) in the molecule. In order to diminish the size of the systems to be solved, the chemists have proposed...