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

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.

Abstract
Résumé

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.

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.

Received: 2002-03-25
Accepted: 2002-03-28
Published online: 2002-01-01

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

Author's affiliations:

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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     author = {Oleg Yu. Imanuvilov and Jean-Pierre Puel},
     title = {Global {Carleman} estimates for weak solutions of elliptic nonhomogeneous {Dirichlet} problems},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {33--38},
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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/

References
Cited by

[1] C. Fabre; G. Lebeau Prolongement unique des solutions de l'équation de Stokes, Comm. Partial Differential Equations, Volume 21 (1996), pp. 573-596

[2] A. Fursikov; O. Imanuvilov Controllability of Evolution Equations, Lecture Notes Series, 34, Seoul National University, 1996

[3] L. Hörmander Linear Partial Differential Operators, Academic Press–Springer-Verlag, New York, Berlin, 1963

[4] O. Imanuvilov On exact controllability for the Navier–Stokes equations, ESAIM: Control Optim. Calc. Var., Volume 3 (1998), pp. 97-131 www.emath.fr/cocv/

[5] O. Imanuvilov Remarks on exact controllability for Navier–Stokes equations, ESAIM: Control Optim. Calc. Var., Volume 6 (2001), pp. 39-72 www.emath.fr/cocv/

[6] O. Imanuvilov, J.-P. Puel, Global Carleman estimates for weak elliptic nonhomogeneous Dirichlet problem, to appear

[7] O. Imanuvilov, M. Yamamoto, Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations, UTMS 98-46

[8] J.-L. Lions Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971

[9] M. Taylor Pseudodifferential Operators and Nonlinear PDE, Birkhäuser, Berlin, 1991

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