Laplace Eigenfunctions and Damped Wave Equation on Product Manifolds

@article{Burq2015LaplaceEA,
  title={Laplace Eigenfunctions and Damped Wave Equation on Product Manifolds},
  author={Nicolas Burq and Claude Zuily},
  journal={Applied Mathematics Research Express},
  year={2015},
  volume={2015},
  pages={296-310}
}
  • N. Burq, C. Zuily
  • Published 18 March 2015
  • Mathematics
  • Applied Mathematics Research Express
- The purpose of this article is to study possible concentrations of eigenfunc-tions of Laplace operators (or more generally quasi-modes) on product manifolds. We show that the approach of the first author and Zworski [10, 11] applies (modulo rescalling) and deduce new stabilization results for weakly damped wave equations which extend to product manifolds previous results by Leautaud-Lerner [12] obtained for products of tori. 
View PDF on arXiv
9 Citations
Highly Influential Citations
1
Background Citations
3
Results Citations
1

9 Citations

Second Microlocalization and Stabilization of Damped Wave Equations on Tori
In this talk we present some recent results obtained in collaboration with P. Gerard (Stabilization of wave equations with rough dampings, 2016, in preparation) on the damped wave equation on two…
L^p-tauberian theorems and L^p-rates for energy decay
Stabilization Rates for the Damped Wave Equation with Hölder-Regular Damping
We study the decay rate of the energy of solutions to the damped wave equation in a setup where the geometric control condition is violated. We consider damping coefficients which are 0 on a strip…
$L^p$ concentration estimates for the Laplacian eigenfunctions near submanifolds
We study $L^p$ bounds on spectral projections for the Laplace operator on compact Riemannian manifolds, restricted to small frequency dependent neighborhoods of submanifolds. In particular, if…
Stabilization of wave equations on the torus with rough dampings
For the damped wave equation on a compact manifold with {\em continuous} dampings, the geometric control condition is necessary and sufficient for {uniform} stabilisation. In this article, on the two…
Optimal constants in nontrapping resolvent estimates and applications in numerical analysis
We study the resolvent for nontrapping obstacles on manifolds with Euclidean ends. It is well known that for such manifolds, the outgoing resolvent satisfies $\|\chi R(k) \chi\|_{L^2\to L^2}\leq…
Weyl Law Improvement for Products of Spheres
The classical Weyl Law says that if $N_M(\lambda)$ denotes the number of eigenvalues of the Laplace operator on a $d$-dimensional compact manifold $M$ without a boundary that are less than or equal…
Quelques problèmes de dynamique classique et quantique
Ce memoire d'habilitation traite de problemes a l'interface de la theorie des systemes dynamiques, de la theorie spectrale et de l'analyse microlocale avec en toile de fond des questions de physique…
Quantified Tauberian Theorems and Applications to Decay of Waves

References

SHOWING 1-10 OF 22 REFERENCES
Concentration of Laplace Eigenfunctions and Stabilization of Weakly Damped Wave Equation
In this article, we prove some universal bounds on the speed of concentration on small (frequency-dependent) neighbourhoods of sub-manifolds of L2-norms of quasi modes for Laplace operators on…
Sharp polynomial energy decay for locally undamped waves
In this note, we present the results of the article [LL14], and provide a complete proof in a simple case. We study the decay rate for the energy of solutions of a damped wave equation in a situation…
Semiclassical Lp Estimates
Abstract.The purpose of this paper is to use semiclassical analysis to unify and generalize Lp estimates on high energy eigenfunctions and spectral clusters. In our approach these estimates do not…
Optimal polynomial decay of functions and operator semigroups
We characterize the polynomial decay of orbits of Hilbert space C0-semigroups in resolvent terms. We also show that results of the same type for general Banach space semigroups and functions obtained…
Equation des Ondes Amorties
We study the large time behavior of the solutions of ∂ t 2 — Δ + 2a(x)∂ t on a compact Riemannian manifold M with boundary (a(x)≥ 0). We give a formula for the exponential decay rate in term of the…
Polynomial decay rate for the dissipative wave equation
Sharp polynomial decay rates for the damped wave equation on the torus
We address the decay rates of the energy for the damped wave equation when the damping coefficient $b$ does not satisfy the Geometric Control Condition (GCC). First, we give a link with the…
Energy decay for a locally undamped wave equation
We study the decay rate for the energy of solutions of a damped wave equation in a situation where the Geometric Control Condition is violated. We assume that the set of undamped trajectories is a…
Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary
For the observation or control of solutions of second-order hyperbolic equation in $\mathbb{R}_t \times \Omega $, Ralston’s construction of localized states [Comm. Pure Appl. Math., 22 (1969), pp. ...
Control in the presence of a black box
We apply the ``black box'' scattering theory to problems in control theory and in high energy eigenvalue scarring.
...
...