Appl Math Optim 46:331–375 (2002)
2002 Springer-Verlag New York Inc.
Carleman Estimates with No Lower-Order Terms for General
Riemann Wave Equations. Global Uniqueness and
Observability in One Shot
and P. F. Yao
Department of Mathematics, University of Virginia,
Charlottesville, VA 22904, USA
Institute of Systems Science, Academy of Mathematics and Systems Science,
Chinese Academy of Sciences, Beijing 100080, People’s Republic of China
Abstract. This paper considers a fully general (Riemann) wave equation on a
ﬁnite-dimensional Riemannian manifold, with energy level (H
essentially minimal smoothness assumptions on the variable (in time and space)
coefﬁcients. The paper provides Carleman-type inequalities: ﬁrst pointwise, for C
solutions, then in integral form for H
(Q)-solutions. The aim of the present ap-
proach 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 ar-
guments. Constants in the estimates are, therefore, generally explicit. The paper
emphasizes the more challenging pure Neumann B.C. case. The paper is a general-
ization 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 ap-
proach is Riemann geometric, but different from—indeed, more ﬂexible than—the
one in [LTY1].
Key Words. Riemann wave, Carleman estimates, Observability.
AMS Classiﬁcation. 35, 49, 53, 93.
The research by R. Triggiani was partially supported by the National Science Foundation under Grant
DMS-9804056 and the Army Research Ofﬁce under Grant DAA4-96-1-0059. The research by P. F. Yao was
performed while he was visiting the Mathematics Department, University of Virginia, Charlottesville, VA
22904, USA, and he was partially supported by NSF Grant DMS-9804056, and by the National Science
Foundation of China.