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

Abstract

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: H 1 ( Ω ) -estimates, J. Inverse Ill-Posed Problems, 12 (1) (2004) 1–81]–of the following result: The only solution { u , p } of the Oseen problem { ( − ν 0 Δ ) u + L e ( u ) + ∇ p = λ u in Ω ; (a) div u ≡ 0 in Ω ; (b) u ≡ 0 in ω . (c) defined on a bounded Ω ⊂ R d , d = 2 , 3 with interior condition u ≡ 0 in an arbitrary sufficiently smooth subdomain ω ⊂ Ω , is the trivial solution u ≡ 0 , p ≡ const in Ω . The method of proof applies equally well to the corresponding Riemannian setting [where, in particular, Δ g is now the Laplace–Bertrami operator], by relying on the pointwise Carleman estimate for Δ g 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 H 1 ( Ω ) -level (with X. Xu), AMS Contemp. Math. 426 (2007), 339–404].