Concentration of Laplace Eigenfunctions and Stabilization of Weakly Damped Wave Equation

Abstract

Abstract. -In this article, we prove some universal bounds on the speed of concentration on small (frequency-dependent) neighborhoods of submanifolds of L 2 -norms of quasi modes for Laplace operators on compact manifolds. We deduce new results on the rate of decay of weakly damped wave equations.Résumé. -On démontre dans cet article des bornes universelles sur la vitesse de concentration dans de petits voisinages (dépendant de la fréquence) de sous variétés pour les normes L 2 de quasimodes du Laplacian sur une variété compacte. On en déduit de nouveaux résultats sur la dćroissance deséquations des ondes faiblement amorties.

Non-uniform Stability of Damped Contraction Semigroups

“…Note that this is worse than the rate obtained under additional assumptions in [36,17]. On the other hand, it is stated in [12,Rem. 1.5] that in the general setting the rate t −1/κ cannot be improved.…”

“…Consider the damped wave equation (6.1) on the square Ω = (0, 1) 2 but with periodic rather than Dirichlet boundary conditions. These boundary conditions allow us to view Ω as the 2-torus T 2 and hence to apply the results of [12]. Note that the setting is not exactly the same as in Section 2.…”

“…This allows us to apply our general machinery and obtain resolvent estimates under the assumption that the damping coefficient b(•) satisfies a certain type of lower bound away from a submanifold Σ of T 2 . A typical example would be for Σ to be a circle of the form Σ = { (ξ 1 , ξ 2 ) ∈ Ω | ξ 1 ∈ (0, 1) } for some fixed ξ 2 ∈ (0, 1), but the results in [12] also apply in a much more general setting than this. The following result is a simple extension of [12,Cor.…”

“…A typical example would be for Σ to be a circle of the form Σ = { (ξ 1 , ξ 2 ) ∈ Ω | ξ 1 ∈ (0, 1) } for some fixed ξ 2 ∈ (0, 1), but the results in [12] also apply in a much more general setting than this. The following result is a simple extension of [12,Cor. 1.3] in our special case.…”

A note on constructing families of sharp examples for 𝐿^{𝑝} growth of eigenfunctions and quasimodes

“…This feature only persists for an O(h) region so when β 1 we cannot expect to get sharp examples for low p from Observation 3. However Burq and Zuily [2] have, in the case k ≤ n − 3, obtained…”