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Journal Articles Journal de l'École polytechnique — Mathématiques Year : 2023

STABILIZATION OF THE DAMPED PLATE EQUATION UNDER GENERAL BOUNDARY CONDITIONS

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Abstract

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.
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Dates and versions

hal-03868483 , version 1 (23-11-2022)

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Cite

Jérôme Le Rousseau, Emmanuel Wend-Benedo Zongo. STABILIZATION OF THE DAMPED PLATE EQUATION UNDER GENERAL BOUNDARY CONDITIONS. Journal de l'École polytechnique — Mathématiques, 2023, 10, pp.1-65. ⟨10.5802/jep.213⟩. ⟨hal-03868483⟩
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