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

Carleman Estimates with No Lower-Order Terms for General Riemann Wave Equations. Global... Abstract. This paper considers a fully general (Riemann) wave equation on a finite-dimensional Riemannian manifold, with energy level (H 1 × L 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 2 -solutions, then in integral form for H 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]. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

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

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References (29)

Publisher
Springer Journals
Copyright
Copyright © Inc. by 2002 Springer-Verlag New York
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation
ISSN
0095-4616
eISSN
1432-0606
DOI
10.1007/s00245-002-0751-5
Publisher site
See Article on Publisher Site

Abstract

Abstract. This paper considers a fully general (Riemann) wave equation on a finite-dimensional Riemannian manifold, with energy level (H 1 × L 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 2 -solutions, then in integral form for H 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].

Journal

Applied Mathematics and Optimization – Springer Journals

Published: Dec 19, 2002

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