Unique continuation from an arbitrary interior subdomain of the variable-coefficient Oseen equation

This paper presents a proof–based on a pointwise Carleman estimate for the Laplacian Δ from [I. Lasiecka, R. Triggiani, Z. Zhang, Nonconservative wave equations with unobserved Neumann B.C.: Global uniqueness and observability, AMS Contemp. Math., 268 (2000) 227–326] or [I. Lasiecka, R. Triggiani, Z. Zhang, Global uniqueness, observability and stabilization of non-conservative Schrödinger equations via pointwise Carleman estimates. Part I: H1(Ω)H1(Ω)-estimates, J. Inverse Ill-Posed Problems, 12 (1) (2004) 1–81]–of the following result: The only solution {u,p}{u,p} of the Oseen problem {(−ν0Δ)u+Le(u)+∇p=λuin Ω;(a)divu≡0in Ω;(b)u≡0in ω.(c) defined on a bounded Ω⊂RdΩ⊂Rd, d=2,3d=2,3 with interior condition u≡0u≡0 in an arbitrary sufficiently smooth subdomain ω⊂Ωω⊂Ω, is the trivial solution u≡0u≡0, p≡p≡ const in ΩΩ. The method of proof applies equally well to the corresponding Riemannian setting [where, in particular, Δg is now the Laplace–Bertrami operator], by relying on the pointwise Carleman estimate for Δg in this case from [R. Triggiani, P.F. Yao, Carleman estimates with no lower order terms for general Riemannian wave equations. Global uniqueness and observability in one shot, Appl. Math. Optim. 46 (2–3) (2002), 331–375] or [R. Triggiani, X. Xu, Pointwise Carleman estimates, global uniqueness, observability and stabilization for non-conservative Schrödinger equations on Riemannian manifolds at the H1(Ω)H1(Ω)-level (with X. Xu), AMS Contemp. Math. 426 (2007), 339–404].