Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems

Oleg Yu. Imanuvilov; Jean-Pierre Puel

doi:10.1016/S1631-073X(02)02389-0

Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems
[Estimations de Carleman globales pour des solutions faibles de problèmes elliptiques avec condition de Dirichlet non homogène]

Oleg Yu. Imanuvilov ¹ ; Jean-Pierre Puel ²

¹ Department of Mathematics, Iowa State University, 400 Carver Hall, Ames, IA 50011-2064, USA
² Laboratoire de mathématiques appliquées, Université de Versailles St Quentin, 45, avenue des États Unis, 78035 Versailles cedex, France

Comptes Rendus. Mathématique, Volume 335 (2002) no. 1, pp. 33-38.

Résumé
Abstract

On considère une équation elliptique du second ordre générale avec second membre $f + \sum_{j = 0}^{N} \frac{\partial f_{j}}{\partial x_{j}} \in H^{- 1} (Ω)$ , $f, f_{j} \in L^{2} (Ω)$ et condition de Dirichlet g∈H^1/2(Γ). On montre une estimation de Carleman globale pour la solution y de cette équation en termes de normes L² à poids de f et f_j et de la norme H^1/2 de g. Cette estimation dépend de deux paramètres réels s et λ qui sont supposés assez grands et est optimale en ce qui concerne les exposants de ces paramètres. Ceci nous permet d'obtenir, par exemple, des estimations fines sur la pression dans les équations de Navier–Stokes linéarisées et se révèle fort utile dans l'étude des problèmes de contrôlabilité.

We consider a general second order elliptic equation with right-hand side $f + \sum_{j = 0}^{N} \frac{\partial f_{j}}{\partial x_{j}} \in H^{- 1} (Ω)$ where $f, f_{j} \in L^{2} (Ω)$ and Dirichlet boundary condition g∈H^1/2(Γ). We prove a global Carleman estimate for the solution y of this equation in terms of the weighted L² norms of f and f_j and the H^1/2 norm of g. This estimate depends on two real parameters s and λ which are supposed to be large enough and is sharp with respect to the exponents of these parameters. This allows us to obtain, for example, sharper estimates on the pressure term in the linearized Navier–Stokes equations and it turns out to be very useful in the context of controllability problems.

Reçu le : 2002-03-25
Accepté le : 2002-03-28
Publié le : 2002-01-01

DOI : 10.1016/S1631-073X(02)02389-0

Affiliations des auteurs :

Oleg Yu. Imanuvilov ¹ ; Jean-Pierre Puel ²

¹ Department of Mathematics, Iowa State University, 400 Carver Hall, Ames, IA 50011-2064, USA
² Laboratoire de mathématiques appliquées, Université de Versailles St Quentin, 45, avenue des États Unis, 78035 Versailles cedex, France

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     title = {Global {Carleman} estimates for weak solutions of elliptic nonhomogeneous {Dirichlet} problems},
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     doi = {10.1016/S1631-073X(02)02389-0},
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Oleg Yu. Imanuvilov; Jean-Pierre Puel. Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems. Comptes Rendus. Mathématique, Volume 335 (2002) no. 1, pp. 33-38. doi : 10.1016/S1631-073X(02)02389-0. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02389-0/

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