Spectrum of the Laplacian with weights
Author(s)
El Soufi, Ahmad
Date issued
March 4, 2019
In
Annals of Global Analysis and Geometry
Vol
2
No
55
From page
149
To page
180
Reviewed by peer
1
Subjects
eigenvalue Laplacian density Cheeger inequality upper bounds
Abstract
Given a compact Riemannian manifold $(M,g)$ and two positive functions $\rho$ and $\sigma$, we are interested in the eigenvalues of the Dirichlet energy functional weighted by $\sigma$, with respect to the $L^2$ inner product weighted by $\rho$. Under some regularity conditions on $\rho$ and $\sigma$, these eigenvalues are those of the operator
$-\rho^{-1} \mbox{div}(\sigma \nabla u)$
with Neumann conditions on the boundary if $\partial M\ne \emptyset$.
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.
$-\rho^{-1} \mbox{div}(\sigma \nabla u)$
with Neumann conditions on the boundary if $\partial M\ne \emptyset$.
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.
Project(s)
Publication type
journal article
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