Spectral inequality and resolvent estimate for the bi-Laplace operator
type: Item
instance of: scholarly article
title: Spectral inequality and resolvent estimate for the bi-Laplace operator
zbMATH Open document ID: 1439.35351
DOI: 10.4171/JEMS/939
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2020-05-07T00:00:00Z
full work available at URL: https://arxiv.org/abs/1509.02098
review text: On a compact Riemannian manifold with boundary, the authors prove a spectral inequality for the bi-Laplace operator in the case of so-called ``clamped'' boundary conditions, that is, homogeneous Dirichlet and Neumann conditions simultaneously. They also prove a resolvent estimate for the generator of the damped plate semigroup associated with these boundary conditions. The spectral inequality allows one to observe finite sums of eigenfunctions for this fourth-order elliptic operator, from an arbitrary open subset of the manifold. Moreover, the constant that appears in the inequality grows as \(\exp(C \mu^{1/4})\) where \(\mu\) is the largest eigenvalue associated with the eigenfunctions appearing in the sum. This type of inequality is known for the Laplace operator. As an application, they obtain a null-controllability result for a higher-order parabolic equation. The resolvent estimate provides the spectral behavior of the plate semigroup generator on the imaginary axis. This type of estimate is known in the case of the damped wave semigroup. As an application, they deduce a stabilization result for the damped plate equation, with a log-type decay. The proofs of both the spectral inequality and the resolvent estimate are based on the derivation of different types of Carleman estimates for an elliptic operator related to the bi-Laplace operator, in the interior and at some boundaries. One of these estimates exhibits a loss of one full derivative. Its proof requires the introduction of an appropriate semi-classical calculus and a delicate microlocal argument.
Mathematics Subject Classification ID: 35P15
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35J30
Mathematics Subject Classification ID: 35L35
Mathematics Subject Classification ID: 35S15
Mathematics Subject Classification ID: 93B05
Mathematics Subject Classification ID: 93D15
zbMATH DE Number: 7198102
zbMATH Keywords: high-order operators
zbMATH Keywords: interpolation inequality
zbMATH Keywords: controllability
zbMATH Keywords: stabilization
zbMATH Keywords: Carleman estimate
zbMATH Keywords: semi-classical calculus
author: Jérôme Le Rousseau
author: L. Robbiano
reviewed by: Kaïs Ammari
MaRDI profile type: Publication
arXiv ID: 1509.02098
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Spectral inequality and resolvent estimate for the bi-Laplace operator (OpenAlex ID)
OpenAlex ID: W2996665200