[Stabilisation de l’équation des plaques amorties sous des conditions au bord générales]
We consider a damped plate equation on an open bounded subset of , 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 , 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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Keywords: Carleman estimates, stabilization, Lopatinskiĭ-Šapiro condition, resolvent estimate
Mots-clés : Inégalité de Carleman, stabilisation, condition de Lopatinskiĭ-Šapiro, estimée de résolvante
Jérôme Le Rousseau 1 ; Emmanuel Wend-Benedo Zongo 2
CC-BY 4.0
@article{JEP_2023__10__1_0,
author = {J\'er\^ome Le Rousseau and Emmanuel Wend-Benedo Zongo},
title = {Stabilization of the damped plate equation under general boundary conditions},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {1--65},
publisher = {\'Ecole polytechnique},
volume = {10},
year = {2023},
doi = {10.5802/jep.213},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.213/}
}
TY - JOUR AU - Jérôme Le Rousseau AU - Emmanuel Wend-Benedo Zongo TI - Stabilization of the damped plate equation under general boundary conditions JO - Journal de l’École polytechnique — Mathématiques PY - 2023 SP - 1 EP - 65 VL - 10 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.213/ DO - 10.5802/jep.213 LA - en ID - JEP_2023__10__1_0 ER -
%0 Journal Article %A Jérôme Le Rousseau %A Emmanuel Wend-Benedo Zongo %T Stabilization of the damped plate equation under general boundary conditions %J Journal de l’École polytechnique — Mathématiques %D 2023 %P 1-65 %V 10 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.213/ %R 10.5802/jep.213 %G en %F JEP_2023__10__1_0
Jérôme Le Rousseau; Emmanuel Wend-Benedo Zongo. Stabilization of the damped plate equation under general boundary conditions. Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 1-65. doi : 10.5802/jep.213. https://jep.centre-mersenne.org/articles/10.5802/jep.213/
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