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Comptes Rendus Mathématique
Volume 335, n° 1
pages 33-38 (2002)
Doi : S1631-073X(02)02389-0
Received : 25 Mars 2002 ;  accepted : 28 Mars 2002
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 a, Jean-Pierre Puel b
a Department of Mathematics, Iowa State University, 400 Carver Hall, Ames, IA 50011-2064, USA 
b Laboratoire de mathématiques appliquées, Université de Versailles St Quentin, 45, avenue des États Unis, 78035 Versailles cedex, France 

Note presented by Philippe G. Ciarlet

Abstract

We consider a general second order elliptic equation with right-hand side f+∑j=0N∂fj∂xjH−1(Ω) where f,fjL2(Ω) and Dirichlet boundary condition g H1/2(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 H1/2 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. To cite this article: O.Y. Imanuvilov, J.-P. Puel, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 33-38.

The full text of this article is available in PDF format.
Résumé

On considère une équation elliptique du second ordre générale avec second membre f+∑j=0N∂fj∂xjH−1(Ω), f,fjL2(Ω) et condition de Dirichlet g H1/2(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 H1/2 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. Pour citer cet article : O.Y. Imanuvilov, J.-P. Puel, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 33-38.

The full text of this article is available in PDF format.


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