Stabilization of the wave equation with Ventcel boundary condition

Buffe, Rémi

doi:10.1016/j.matpur.2016.11.001

Cited by 26 publications

(22 citation statements)

References 21 publications

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“…The starting point is boundary Carleman estimates for the operator −∆ − k 2 , k ≥ 1, for functions u satisfying the Robin boundary conditions (∂ ν − ik)u = 0 on ∂Ω. Such Carleman estimates are essentially well known and are discussed in Section 2, following the works by Fursikov-Imanuvilov [19], Lebeau-Robbiano [39], [40], Burq [10], and Buffe [8], see also [37] and [38]. Let us mention that the presence of the large parameter k in the boundary conditions makes the situation more complicated and in addition to 1/k, it becomes natural to introduce a second small parameter h such that 0 < h ≪ 1/k.…”

Section: Remarkmentioning

confidence: 99%

Stability estimates for partial data inverse problems for Schrödinger operators in the high frequency limit

Krupchyk

Uhlmann

2019

Journal de Mathématiques Pures et Appliquées

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We consider the partial data inverse boundary problem for the Schrödinger operator at a frequency k > 0 on a bounded domain in R n , n ≥ 3, with impedance boundary conditions. Assuming that the potential is known in a neighborhood of the boundary, we first show that the knowledge of the partial Robin-to-Dirichlet map at the fixed frequency k > 0 along an arbitrarily small portion of the boundary, determines the potential in a logarithmically stable way. We prove, as the principal result of this work, that the logarithmic stability can be improved to the one of Hölder type in the high frequency regime.Résumé. Nous considérons un problème inverse avec des données partielles pour l'opérateur de Schrödingerà la fréquence k > 0 sur un domaine borné dans R n , n ≥ 3, avec des conditions aux limites d'impédance. En supposant que le potential soit connu sur un voisinage du bord, nous montrons d'abord que la connaissance de l'opérateur Robin-Dirichet partielà la fréquence fixée k > 0 sur des sous-ensembles arbitrairement petits du bord, détermine le potentiel de manière logarithmiquement stable. Nous montrons, comme le résultat principal de ce travail, que la stabilité logarithmique peutêtre améliorée et remplacée par une estimation de stabilité de type Höldérienne dans le régime des hautes fréquences.

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Section: Remarkmentioning

confidence: 99%

Stability estimates for partial data inverse problems for Schrödinger operators in the high frequency limit

Krupchyk

Uhlmann

2019

Journal de Mathématiques Pures et Appliquées

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“…For the estimate of U 2 , we note that it satisfies a Ventcel boundary condition on J 1 and the Dirichlet boundary condition on J 0 . Hence, we use the following result, which is basically a consequence of a Carleman estimate obtained in [11]. However for the sake of completeness, we prove the next result in Appendix A.…”

Section: Estimates On the Velocitymentioning

confidence: 88%

“…Optimizing this inequality with respect to τ τ 4 (see, for instance, [11,Lemma 8.4]) allows us to conclude the proof of Theorem 5.3.…”

Section: Patching the Estimates Togethermentioning

confidence: 99%

“…The proof of Theorem 5.2 is mainly based on a Carleman estimate obtained in [11] that we recall here. We recall that Z, J 1 , J 0 are defined by (3.14) whereas Z and J 1 are defined by (5.2).…”

Section: A1 a Carleman Estimatementioning

confidence: 99%

“…The proof of the spectral inequality and thus of Theorem 1.1 is obtained at the end of Section 5. In the appendix, we show an interpolation estimate for the Ventcel boundary condition that is mainly a consequence of a Carleman estimate obtained in [11]. Notation 1.5.…”

Section: Introductionmentioning

confidence: 95%

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Controllability of a simplified fluid-structure interaction system

Buffe¹,

Takahashi²

2021

Preprint

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We are interested by the controllability of a fluid-structure interaction system where the fluid is viscous and incompressible and where the structure is elastic and located on a part of the boundary of the fluid's domain. In this article, we simplify this system by considering a linearization and by replacing the wave/plate equation for the structure by a heat equation. We show that the corresponding system coupling the Stokes equations with a heat equation at its boundary is null-controllable. The proof is based on Carleman estimates and interpolation inequalities. One of the Carleman estimates corresponds to the case of Ventcel boundary conditions. This work can be seen as a first step to handle the real system where the structure is modeled by the wave or the plate equation.

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Uniform decay rate estimates for the 2D wave equation posed in an inhomogeneous medium with exponential growth source term, locally distributed nonlinear dissipation, and dynamic Cauchy–Ventcel–type boundary conditions

Simion Antunes,

Cavalcanti,

Cavalcanti

2023

Mathematische Nachrichten

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We study the wellposedness and stabilization for a Cauchy–Ventcel problem in an inhomogeneous medium with dynamic boundary conditions subject to a exponential growth source term and a nonlinear damping distributed around a neighborhood ω of the boundary according to the geometric control condition. We, in particular, generalize substantially the work due to Almeida et al. (Commun. Contemp. Math. 23 (2021), no. 03, 1950072), in what concerns an exponential growth for source term instead of a polynomial one. We give a proof based on the truncation of a equivalent problem and passage to the limit in order to obtain in one shot, the energy identity as well as the observability inequality, which are the essential ingredients to obtain uniform decay rates of the energy. We show that the energy of the equivalent problem goes uniformly to zero, for all initial data of finite energy taken in bounded sets of finite energy phase space. One advantage of our proof is that the decay rate is independent of the nonlinearity.

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Stabilization of the wave equation with Ventcel boundary condition

Cited by 26 publications

References 21 publications

Stability estimates for partial data inverse problems for Schrödinger operators in the high frequency limit

Stability estimates for partial data inverse problems for Schrödinger operators in the high frequency limit

Controllability of a simplified fluid-structure interaction system

Uniform decay rate estimates for the 2D wave equation posed in an inhomogeneous medium with exponential growth source term, locally distributed nonlinear dissipation, and dynamic Cauchy–Ventcel–type boundary conditions

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