Eigenvalue distributions of some degenerate elliptic operators: an approach via entropy numbers.
We consider a class of eigenvalue problems for polyharmonic operators, including Dirichlet and buckling-type eigenvalue problems. We prove an analyticity result for the dependence of the symmetric functions of the eigenvalues upon domain perturbations and compute Hadamard-type formulas for the Frechét differentials. We also consider isovolumetric domain perturbations and characterize the corresponding critical domains for the symmetric functions of the eigenvalues. Finally, we prove that balls are...
One may produce the qth harmonic of a string of length π by applying the 'correct touch' at the node during a simultaneous pluck or bow. This notion was made precise by a model of Bamberger, Rauch and Taylor. Their 'touch' is a damper of magnitude b concentrated at . The 'correct touch' is that b for which the modes, that do not vanish at , are maximally damped. We here examine the associated spectral problem. We find the spectrum to be periodic and determined by a polynomial of degree ....
Nous donnons des résultats analytiques sur les propriétés de régularité du laplacien hypoelliptique de Jean-Michel Bismut et plus généralement sur les opérateurs de type Fokker-Planck géométrique agissant sur le fibré cotangent d’une variété riemannienne compacte . En particulier, nous prouvons un résultat d’hypoellipticité maximale pour , et nous en déduisons des bornes sur la localisation de ses valeurs spectrales.
The Coupled Cluster (CC) method is a widely used and highly successful high precision method for the solution of the stationary electronic Schrödinger equation, with its practical convergence properties being similar to that of a corresponding Galerkin (CI) scheme. This behaviour has for the discrete CC method been analyzed with respect to the discrete Galerkin solution (the “full-CI-limit”) in [Schneider, 2009]. Recently, we globalized the CC formulation to the full continuous space, giving a root...
For a family of elliptic operators with rapidly oscillating periodic coefficients, we study the convergence rates for Dirichlet eigenvalues and bounds of the normal derivatives of Dirichlet eigenfunctions. The results rely on an estimate in for solutions with Dirichlet condition.
We survey recent results concerning estimates of the principal eigenvalue of the Dirichlet -Laplacian and the Navier -biharmonic operator on a ball of radius in and its asymptotics for approaching and . Let tend to . There is a critical radius of the ball such that the principal eigenvalue goes to for and to for . The critical radius is for any for the -Laplacian and in the case of the -biharmonic operator. When approaches , the principal eigenvalue of the Dirichlet...