Stabilization of the damped plate equation under general boundary conditions
Journal de l’École polytechnique — Mathématiques, Volume 10 (2023), pp. 1-65.

We consider a damped plate equation on an open bounded subset of d , or a smooth manifold, with boundary, along with general boundary operators fulfilling the Lopatinskiĭ-Šapiro condition. The damping term acts on an internal region without imposing any geometrical condition. We derive a resolvent estimate for the generator of the damped plate 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.

Nous considérons une équation des plaques amorties sur un ouvert borné régulier de d , ou sur une variété lisse et compacte à bord, avec des opérateurs au bord généraux qui satisfont la condition de Lopatinskiĭ-Šapiro. Le terme d’amortissement agit sur une région interne et aucune condition géométrique n’est imposée. Nous démontrons une estimée de résolvante pour le générateur du semi-groupe associé qui implique une décroissance logarithmique de l’énergie de la solution de l’équation des plaques. Cette estimée de résolvante est conséquence d’une inégalité de Carleman obtenue pour le bi-laplacien muni d’un paramètre spectral et sous les conditions au bord considérées. L’obtention de cette inégalité passe tout d’abord par des estimations microlocales, puis locales et enfin une estimation globale.

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Accepted:
Published online:
DOI: 10.5802/jep.213
Classification: 35B45,  35J30,  35J40,  74K20,  93D15
Keywords: Carleman estimates, stabilization, Lopatinskiĭ-Šapiro condition, resolvent estimate
Jérôme Le Rousseau 1; Emmanuel Wend-Benedo Zongo 2

1 Laboratoire Analyse, Géométrie et Applications, Université Sorbonne Paris Nord, CNRS, Université Paris 8 Villetaneuse, France
2 Dipartimento di Matematica, Università degli Studi di Milano Via C. Saldini, 50, 20133 Milan, Italy & Laboratoire Analyse, Géométrie et Applications, Université Sorbonne Paris Nord, CNRS, Université Paris 8 Villetaneuse, France Current address: Laboratoire de Mathématiques d’Orsay, Université Paris-Saclay, CNRS 91405 Orsay Cedex, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Jérôme Le Rousseau; Emmanuel Wend-Benedo Zongo. Stabilization of the damped plate equation under general boundary conditions. Journal de l’École polytechnique — Mathématiques, Volume 10 (2023), pp. 1-65. doi : 10.5802/jep.213. https://jep.centre-mersenne.org/articles/10.5802/jep.213/

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