[Stabilisation de l’équation des plaques amorties sous des conditions au bord générales]
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.
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.
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Keywords: Carleman estimates, stabilization, Lopatinskiĭ-Šapiro condition, resolvent estimate
Mot 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 -
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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, Tome 10 (2023), pp. 1-65. doi : 10.5802/jep.213. https://jep.centre-mersenne.org/articles/10.5802/jep.213/
[1] - “Piecewise multiplier method and nonlinear integral inequalities for Petrowsky equation with nonlinear dissipation”, J. Evol. Equ. 6 (2006) no. 1, p. 95-112 | DOI | MR | Zbl
[2] - “Sharp energy estimates for nonlinearly locally damped PDEs via observability for the associated undamped system”, J. Funct. Anal. 260 (2011) no. 8, p. 2424-2450 | DOI | MR | Zbl
[3] - “Nonlinear damped partial differential equations and their uniform discretizations”, J. Funct. Anal. 273 (2017) no. 1, p. 352-403 | DOI | MR | Zbl
[4] - “A sharp geometric condition for the boundary exponential stabilizability of a square plate by moment feedbacks only”, in Control of coupled partial differential equations, Internat. Ser. Numer. Math., vol. 155, Birkhäuser, Basel, 2007, p. 1-11 | DOI | MR | Zbl
[5] - “Exact controllability of the superlinear heat equation”, Appl. Math. Optim. 42 (2000) no. 1, p. 73-89 | DOI | MR | Zbl
[6] - “Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary”, SIAM J. Control Optim. 30 (1992) no. 5, p. 1024-1065 | DOI | MR | Zbl
[7] - “Non-uniform stability for bounded semi-groups on Banach spaces”, J. Evol. Equ. 8 (2008) no. 4, p. 765-780 | DOI | MR | Zbl
[8] - “Carleman estimates for elliptic operators with complex coefficients. Part I: Boundary value problems”, J. Math. Pures Appl. (9) 104 (2015) no. 4, p. 657-728 | DOI | MR | Zbl
[9] - Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011 | DOI
[10] - “Global uniqueness of a class of multidimensional inverse problems”, Soviet Math. Dokl. 24 (1981), p. 244-247 | Zbl
[11] - “Uniqueness in the Cauchy problem for partial differential equations”, Amer. J. Math. 80 (1958), p. 16-36 | DOI | MR | Zbl
[12] - “Sur un problème d’unicité pour les systèmes d’équations aux dérivées partielles à deux variables indépendantes.”, Ark. Mat. Astron. Fys. 26 (1939) no. 17, p. 1-9 | MR | Zbl
[13] - “A structurally damped plate equation with Dirichlet-Neumann boundary conditions”, J. Differential Equations 259 (2015) no. 4, p. 1323-1353 | DOI | MR | Zbl
[14] - “Sharp Carleman estimates and unique continuation”, Duke Math. J. 129 (2005) no. 3, p. 503-550 | DOI | MR | Zbl
[15] - “Limiting Carleman weights and anisotropic inverse problems”, Invent. Math. 178 (2009) no. 1, p. 119-171 | DOI | MR | Zbl
[16] - “Null and approximate controllability for weakly blowing up semilinear heat equations”, Ann. Inst. H. Poincaré C Anal. Non Linéaire 17 (2000) no. 5, p. 583-616 | DOI | MR | Zbl
[17] - Controllability of evolution equations, Lect. Notes Series, vol. 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996
[18] - “On the uniqueness of the Cauchy problem”, Math. Scand. 6 (1958), p. 213-225 | DOI | Zbl
[19] - Linear partial differential operators, Grundlehren Math. Wiss., vol. 116, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963 | DOI
[20] - The analysis of linear partial differential operators. III, Classics in Math., Springer, Berlin, 2007 | DOI
[21] - “An inverse problem for the dynamical Lamé system with two sets of boundary data”, Comm. Pure Appl. Math. 56 (2003) no. 9, p. 1366-1382 | DOI | Zbl
[22] - Inverse problems for partial differential equations, Applied Math. Sciences, vol. 127, Springer, Cham, 2017 | DOI
[23] - “Contrôle interne exact des vibrations d’une plaque rectangulaire”, Portugal. Math. 47 (1990) no. 4, p. 423-429 | Zbl
[24] - “Unique continuation and absence of positive eigenvalues for Schrödinger operators”, Ann. of Math. (2) 121 (1985) no. 3, p. 463-494, With an appendix by E. M. Stein | DOI | Zbl
[25] - “Pseudo-périodicité et séries de Fourier lacunaires”, Ann. Sci. École Norm. Sup. 79 (1962), p. 93-150 | DOI | Zbl
[26] - “The Calderón problem with partial data”, Ann. of Math. (2) 165 (2007) no. 2, p. 567-591 | DOI | Zbl
[27] - “Carleman estimates and unique continuation for second-order elliptic equations with nonsmooth coefficients”, Comm. Pure Appl. Math. 54 (2001) no. 3, p. 339-360 | DOI | MR | Zbl
[28] - “Sharp counterexamples in unique continuation for second order elliptic equations”, J. reine angew. Math. 542 (2002), p. 133-146 | DOI | MR | Zbl
[29] - “Dispersive estimates for principally normal pseudodifferential operators”, Comm. Pure Appl. Math. 58 (2005) no. 2, p. 217-284 | DOI | MR | Zbl
[30] - “Uniqueness in inverse hyperbolic problems—Carleman estimate for boundary value problems”, J. Math. Kyoto Univ. 40 (2000) no. 3, p. 451-473 | DOI | MR | Zbl
[31] - Elliptic Carleman estimates and applications to stabilization and controllability. Vol. I. Dirichlet boundary conditions on Euclidean space, Progress in Nonlinear Differential Equations and their Applications, vol. 97, Birkhäuser/Springer, Cham, 2022 | DOI
[32] - Elliptic Carleman estimates and applications to stabilization and controllability. Vol. II. General boundary conditions on Riemannian manifolds, Progress in Nonlinear Differential Equations and their Applications, vol. 98, Birkhäuser/Springer, Cham, 2022 | DOI
[33] - “Spectral inequality and resolvent estimate for the bi-Laplace operator”, J. Eur. Math. Soc. (JEMS) 22 (2020) no. 4, p. 1003-1094 | DOI | MR | Zbl
[34] - “Contrôle de l’équation de Schrödinger”, J. Math. Pures Appl. (9) 71 (1992) no. 3, p. 267-291 | Zbl
[35] - “Équation des ondes amorties”, in Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), Math. Phys. Stud., vol. 19, Kluwer Acad. Publ., Dordrecht, 1996, p. 73-109 | DOI | Zbl
[36] - “Contrôle exact de l’équation de la chaleur”, Comm. Partial Differential Equations 20 (1995) no. 1-2, p. 335-356 | DOI
[37] - “Stabilisation de l’équation des ondes par le bord”, Duke Math. J. 86 (1997) no. 3, p. 465-491 | DOI | Zbl
[38] - “Internal stabilization of the plate equation in a square: the continuous and the semi-discretized problems”, J. Math. Pures Appl. (9) 85 (2006) no. 1, p. 17-37 | DOI | MR | Zbl
[39] - “Exponential decay of solutions to hyperbolic equations in bounded domains”, Indiana Univ. Math. J. 24 (1974), p. 79-86 | DOI | MR
[40] - “Oscillatory integrals and unique continuation for second order elliptic differential equations”, J. Amer. Math. Soc. 2 (1989) no. 3, p. 491-515 | DOI | MR | Zbl
[41] - “Well-posedness and stability of a hinged plate equation with a localized nonlinear structural damping”, Nonlinear Anal. 71 (2009) no. 12, p. e2288-e2297 | DOI | MR | Zbl
[42] - “Well-posedness and stabilization of an Euler-Bernoulli equation with a localized nonlinear dissipation involving the -Laplacian”, Discrete Contin. Dynam. Systems 32 (2012) no. 6, p. 2315-2337 | DOI | MR | Zbl
[43] - Uniqueness and non-uniqueness in the Cauchy problem, Progress in Math., vol. 33, Birkhäuser, Cham, 1983 | DOI
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