caa a22 4500 378883283 CHVBK 20180305123437.0 cr unu---uuuuu 161128e20040201xx s 000 0 eng 10.1515/156939404773972761 doi (NATIONALLICENCE)gruyter-10.1515/156939404773972761 Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates. Part I: H 1(Ω)-estimates [Elektronische Daten] [I. Lasiecka, R. Triggiani, X. Zhang] We consider a general non-conservative Schrödinger equation defined on an open bounded domain Ω in , with C 2-boundary subject to (Dirichlet and, as a main focus, to) Neumann boundary conditions on the entire boundary Γ. Here, Γ0 and Γ1 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 lower-order 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,Γ0,Γ1} 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. Copyright 2004, Walter de Gruyter Lasiecka I. University of Virginia, Department of Mathematics, P. O. Box 400137, Charlottesville, VA 22904-4137. E-mails: I.L., il2v@virginia.edu; R.T., rt7u@virginia.edu. aut Triggiani R. University of Virginia, Department of Mathematics, P. O. Box 400137, Charlottesville, VA 22904-4137. E-mails: I.L., il2v@virginia.edu; R.T., rt7u@virginia.edu. aut Zhang X. School of Mathematics, Sichuan University, Chengdu 610064, China; and Departamento de Mathematicas, Facultad de Ciencias, Universidad Autonoma de Madrid, 28049, Madrid, Spain. aut Journal of Inverse and Ill-posed Problems Walter de Gruyter 12/1(2004-02-01), 43-123 0928-0219 12:1<43 2004 12 jiip https://doi.org/10.1515/156939404773972761 text/html Onlinezugriff via DOI 1 research article jats NATIONALLICENCE 856 40 https://doi.org/10.1515/156939404773972761 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Lasiecka I. University of Virginia, Department of Mathematics, P. O. Box 400137, Charlottesville, VA 22904-4137. E-mails: I.L., il2v@virginia.edu; R.T., rt7u@virginia.edu aut NATIONALLICENCE 700 1- Triggiani R. University of Virginia, Department of Mathematics, P. O. Box 400137, Charlottesville, VA 22904-4137. E-mails: I.L., il2v@virginia.edu; R.T., rt7u@virginia.edu aut NATIONALLICENCE 700 1- Zhang X. School of Mathematics, Sichuan University, Chengdu 610064, China; and Departamento de Mathematicas, Facultad de Ciencias, Universidad Autonoma de Madrid, 28049, Madrid, Spain aut NATIONALLICENCE 773 0- Journal of Inverse and Ill-posed Problems Walter de Gruyter 12/1(2004-02-01), 43-123 0928-0219 12:1<43 2004 12 jiip CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence SWISSBIB 378883283 BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-gruyter