Carleman Estimates with No Lower-Order Terms for General Riemann Wave Equations. Global Uniqueness and Observability in One Shot

Journal Article · Thu Dec 19 04:00:00 EST 2002 · Applied Mathematics and Optimization

DOI:https://doi.org/10.1007/S00245-002-0751-5· OSTI ID:21067481

Triggiani, R ^[1]; Yao, P F ^[2]

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1. Department of Mathematics, University of Virginia, Charlottesville, VA 22904 (United States)
2. Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080 (China)

This paper considers a fully general (Riemann) wave equation on a finite-dimensional Riemannian manifold, with energy level (H{sup 1} x L{sub 2}) -terms, under essentially minimal smoothness assumptions on the variable (in time and space) coefficients. The paper provides Carleman-type inequalities: first pointwise, for C{sup 2} -solutions, then in integral form for H{sup 1,1}(Q) -solutions. The aim of the present approach is to provide Carleman inequalities which do not contain lower-order terms, a distinguishing feature over most of the literature. Accordingly, global uniqueness results for overdetermined problems as well as Continuous Observability/ Uniform Stabilization inequalities follow in one shot, as a part of the same stream of arguments. Constants in the estimates are, therefore, generally explicit. The paper emphasizes the more challenging pure Neumann B.C. case. The paper is a generalization from the Euclidean to the Riemannian setting of [LTZ] in the more difficult case of purely Neumann B.C., and of [KK1] in the case of Dirichlet B.C. The approach is Riemann geometric, but different from-indeed, more flexible than-the one in [LTY1].

• OSTI ID: 21067481
• Journal Information: Applied Mathematics and Optimization, Journal Name: Applied Mathematics and Optimization Journal Issue: 2-3 Vol. 46; ISSN 0095-4616
• Country of Publication: United States
• Language: English

Related Subjects

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