Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems

We consider a general second order elliptic equation with right-hand side f+∑j=0N∂fj∂xj∈H−1(Ω) where f,fj∈L2(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 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. To cite this article: O.Y. Imanuvilov, J.-P. Puel, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 33–38.