Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates. Part II: L 2(ω)-estimates


We consider a general non-conservative Schrödinger equation defined on an open bounded domain Ω in ℝ n, with C 2-boundary Γ = ∂Ω= Γ 0∪Γ 1, Γ 0∩Γ 1 = θ, subject to (Dirichlet and, as a main focus, to) Neumann boundary conditions on the entire boundary Γ. Here, Γ 0 and Γ 1 are the unobserved (or uncontrolled) and observed (or controlled) parts of the boundary, respectively, both being relatively open in Γ. The Schrödinger equation includes energy-level (H 1(Ω)-level) terms, which accordingly may be viewed as unbounded perturbations. The first goal of the paper is to provide Carleman-type inequalities at the H 1-level, which do not contain lowerorder terms; this is a distinguishing feature over most of the literature. This goal is accomplished in a few steps: the paper obtains first pointwise Carleman estimates for C 2-solutions; and next, it turns these pointwise estimates into integral-type Carleman estimates with no lower-order terms, originally for H 2-solutions, and ultimately for H 1-solutions. The passage from H 2- to H 1-solutions is readily accomplished in the case of Dirichlet B.C., but it requires a delicate regularization argument in the case of Neumann B.C. This is so since finite energy solutions are known to have L 2-normal traces in the case of Dirichlet B.C., but by contrast do not produce H 1-traces in the case of Neumann B.C. From Carleman-type inequalities with no lower-order terms, one then obtains the sought-after benefits. These consist of deducing, in one shot, as a part of the same flow of arguments, two important implications: (i) global uniqueness results for H 1-solutions satisfying over-determined boundary conditions, and - above all - (ii) continuous observability (or stabilization) inequalities with an explicit constant. The more demanding purely Neumann boundary conditions requires the same geometrical conditions on the triple {Ω,Γ 0,Γ 1} that arise in the corresponding problems for second-order hyperbolic equations. The most general result, with weakest geometrical conditions, is, in fact, deferred to Section 9. Sections 1 through 8 provide the main body of our treatment with one vector field under a preliminary working geometrical condition, which is then removed in Section 9, by use of two suitable vector fields. The second and final goal of this paper is to shift the Carleman estimates (Hence, the continuous observability/stabilization inequalities) by one unit downward to the lower L 2(Ω)-level. This is accomplished in Section 10 by means of pseudo-differential analysis, and accordingly, it contains lower-order terms. Applications of these L 2(Ω)-Carleman estimate includes a new uniform stabilization of the conservative Schrödinger equation in the state space L 2(Ω), by an attractive boundary feedback.

Publication Title

Journal of Inverse and Ill-Posed Problems