Stabilization of the damped plate equation under general boundary conditions

Le Rousseau, Jérôme; Zongo, Emmanuel Wend-Benedo

Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 1-65

We consider a damped plate equation on an open bounded subset of ${ℝ}^d$, 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 ${ℝ}^d$, 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.

Classification : 35B45, 35J30, 35J40, 74K20, 93D15
Keywords: Carleman estimates, stabilization, Lopatinskiĭ-Šapiro condition, resolvent estimate
Mots-clés : Inégalité de Carleman, stabilisation, condition de Lopatinskiĭ-Šapiro, estimée de résolvante

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