We consider a general second order elliptic equation with right-hand side where 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 fj 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.
On considère une équation elliptique du second ordre générale avec second membre , 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 fj 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.
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Oleg Yu. Imanuvilov 1; Jean-Pierre Puel 2
@article{CRMATH_2002__335_1_33_0, 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}, publisher = {Elsevier}, volume = {335}, number = {1}, year = {2002}, doi = {10.1016/S1631-073X(02)02389-0}, language = {en}, }
TY - JOUR AU - Oleg Yu. Imanuvilov AU - Jean-Pierre Puel TI - Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems JO - Comptes Rendus. Mathématique PY - 2002 SP - 33 EP - 38 VL - 335 IS - 1 PB - Elsevier DO - 10.1016/S1631-073X(02)02389-0 LA - en ID - CRMATH_2002__335_1_33_0 ER -
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/
[1] Prolongement unique des solutions de l'équation de Stokes, Comm. Partial Differential Equations, Volume 21 (1996), pp. 573-596
[2] Controllability of Evolution Equations, Lecture Notes Series, 34, Seoul National University, 1996
[3] Linear Partial Differential Operators, Academic Press–Springer-Verlag, New York, Berlin, 1963
[4] On exact controllability for the Navier–Stokes equations, ESAIM: Control Optim. Calc. Var., Volume 3 (1998), pp. 97-131 www.emath.fr/cocv/
[5] 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] Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971
[9] Pseudodifferential Operators and Nonlinear PDE, Birkhäuser, Berlin, 1991
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