83 research outputs found
Lower bounds for the first eigenvalue of the magnetic Laplacian
We consider a Riemannian cylinder endowed with a closed potential 1-form A
and study the magnetic Laplacian with magnetic Neumann boundary conditions
associated with those data. We establish a sharp lower bound for the first
eigenvalue and show that the equality characterizes the situation where the
metric is a product. We then look at the case of a planar domain bounded by two
closed curves and obtain an explicit lower bound in terms of the geometry of
the domain. We finally discuss sharpness of this last estimate.Comment: Replaces in part arXiv:1611.0193
Two properties of volume growth entropy in Hilbert geometry
The aim of this paper is to provide two examples in Hilbert geometry which
show that volume growth entropy is not always a limit on the one hand, and that
it may vanish for a non-polygonal domain in the plane on the other hand
The spectral gap of graphs and Steklov eigenvalues on surfaces
Using expander graphs, we construct a sequence of smooth compact surfaces
with boundary of perimeter N, and with the first non-zero Steklov eigenvalue
uniformly bounded away from zero. This answers a question which was raised in
[9]. The genus grows linearly with N, this is the optimal growth rate.Comment: 9 pages, 1 figur
Eigenvalues estimate for the Neumann problem on bounded domains
In this note, we investigate upper bounds of the Neumann eigenvalue problem
for the Laplacian of a bounded domain (with smooth boundary) in a given
complete (not compact a priori) Riemannian manifold with Ricci bounded below .
For this, we use test functions for the Rayleigh quotient subordinated to a
family of open sets constructed in a general metric way, interesting for
itself. As application, we get upper bounds for the Neumann spectrum which is
clearly in agreement with the Weyl law and which is analogous to Buser's upper
bounds of the spectrum of a closed Riemannian manifold with lower bound on the
Ricci curvature.Comment: 9 pages, submitted december 200
Hilbert domains quasi-isometric to normed vector spaces
We prove that a Hilbert domain which is quasi-isometric to a normed vector
space is actually a convex polytope
Spectrum of the Laplacian with weights
Given a compact Riemannian manifold (M, g) and two positive functions
and , we are interested in the eigenvalues of the Dirichlet energy
functional weighted by , with respect to the L 2 inner product weighted
by . Under some regularity conditions on and , these
eigenvalues are those of the operator ^{-1} div(u) with
Neumann conditions on the boundary if M = . We investigate
the effect of the weights on eigenvalues and discuss the existence of lower and
upper bounds under the condition that the total mass is preserved
Extremal Eigenvalues of the Laplacian on Euclidean domains and closed surfaces
We investigate properties of the sequences of extremal values that could be
achieved by the eigenvalues of the Laplacian on Euclidean domains of unit
volume, under Dirichlet and Neumann boundary conditions, respectively. In a
second part, we study sequences of extremal eigenvalues of the Laplace-Beltrami
operator on closed surfaces of unit area
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