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. 

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References

SHOWING 1-10 OF 26 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

Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds

We give estimates for the $L^p$ norm ($2\leq p \leq +\infty$) of the restriction to a curve of the eigenfunctions of the Laplace Beltrami operator on a Riemannian surface. If the curve is a geodesic,

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 L p ESTIMATES

The purpose of this paper is to use semiclassical analysis to unify and generalize L estimates on high energy eigenfunctions and spectral clusters. In our approach these estimates do not depend on

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