STABILIZATION OF THE DAMPED PLATE EQUATION UNDER GENERAL BOUNDARY CONDITIONS

We consider a damped plate equation on a smooth open bounded subset of R^d , or a smooth compact manifold with boundary, along with general boundary operators fulfilling the Lopatinskiȋ-Šapiro condition. The damping term acts on a internal region without imposing a geometrical condition. We derive a...

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Bibliographic Details
Published inJournal de l'École polytechnique. Mathématiques Vol. 10; pp. 1 - 65
Main Authors Le Rousseau, Jérôme, Zongo, Emmanuel Wend-Benedo
Format Journal Article
LanguageEnglish
Published École polytechnique 01.01.2023
Subjects
Mathematics
Lopatinskiȋ-Šapiro condition
35J30
74K20
35J40
stabilization
93D15
Carleman estimates
resolvent estimate AMS 2020 subject classification: 35B45
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ISSN2429-7100
2270-518X
DOI10.5802/jep.213

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Summary:We consider a damped plate equation on a smooth open bounded subset of R^d , or a smooth compact manifold with boundary, along with general boundary operators fulfilling the Lopatinskiȋ-Šapiro condition. The damping term acts on a internal region without imposing a geometrical condition. We derive a resolvent estimate for the generator of associated semigroup that yields a logarithmic decay of the energy of the solution to the plate equation. The resolvent estimate is a consequence of a Carleman inequality obtained for the bi-Laplace operator involving a spectral parameter under the considered boundary conditions. The derivation goes first through microlocal estimates, then local estimates, and finally a global estimate.
ISSN:2429-7100
2270-518X
DOI:10.5802/jep.213