34 research outputs found
Estimates of the gaps between consecutive eigenvalues of Laplacian
By the calculation of the gap of the consecutive eigenvalues of
with standard metric, using the Weyl's asymptotic formula, we know the order of
the upper bound of this gap is We conjecture that this order
is also right for general Dirichlet problem of the Laplace operator, which is
optimal if this conjecture holds, obviously. In this paper, using new method,
we solve this conjecture in the Euclidean space case intrinsically. We think
our method is valid for the case of general Riemannian manifolds and give some
examples directly.Comment: 15 page
A second eigenvalue bound for the Dirichlet Schroedinger operator
Let be the th eigenvalue of the Schr\"odinger
operator with Dirichlet boundary conditions on a bounded domain and with the positive potential . Following the spirit of the
Payne-P\'olya-Weinberger conjecture and under some convexity assumptions on the
spherically rearranged potential , we prove that . Here denotes the ball, centered at the
origin, that satisfies the condition .
Further we prove under the same convexity assumptions on a spherically
symmetric potential , that decreases
when the radius of the ball increases.
We conclude with several results about the first two eigenvalues of the
Laplace operator with respect to a measure of Gaussian or inverted Gaussian
density
Universal Inequalities for Eigenvalues of the Buckling Problem of Arbitrary Order
We investigate the eigenvalues of the buckling problem of arbitrary order on
compact domains in Euclidean spaces and spheres. We obtain universal bounds for
the th eigenvalue in terms of the lower eigenvalues independently of the
particular geometry of the domain.Comment: 24 page
"Universal" inequalities for the eigenvalues of the biharmonic operator
In this paper, we establish universal inequalities for eigenvalues of the
clamped plate problem on compact submanifolds of Euclidean spaces, of spheres
and of real, complex and quaternionic projective spaces. We also prove similar
results for the biharmonic operator on domains of Riemannian manifolds
admitting spherical eigenmaps (this includes the compact homogeneous Riemannian
spaces) and nally on domains of the hyperbolic space.Comment: Soumis (15 pages
A lower bound to the spectral threshold in curved tubes
We consider the Laplacian in curved tubes of arbitrary cross-section rotating
together with the Frenet frame along curves in Euclidean spaces of arbitrary
dimension, subject to Dirichlet boundary conditions on the cylindrical surface
and Neumann conditions at the ends of the tube. We prove that the spectral
threshold of the Laplacian is estimated from below by the lowest eigenvalue of
the Dirichlet Laplacian in a torus determined by the geometry of the tube.Comment: LaTeX, 13 pages; to appear in R. Soc. Lond. Proc. Ser. A Math. Phys.
Eng. Sc