Bellassoued M., Yamamoto M. Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems

Springer Japan KK, 2017. — 267 p. — (Springer Monographs in Mathematics) — ISBN: 978-4-431-56598-7In this book, focusing on hyperbolic systems, we give self-contained descriptions of • derivations of Carleman estimates; • methods for application of Carleman estimates to stability of inverse problems. Confining ourselves to equations of hyperbolic type, we survey previous and recent results concerning the applicability of Carleman estimates.
We do not intend to pursue any general treatment of the Carleman estimates themselves; rather by arguing in a direct manner, we mainly aim to demonstrate the applicability of Carleman estimates to inverse problems. In many places, we choose direct arguments based on basic calculus, rather than more general sophisticated methods. Because inverse problems are strongly connected with the respective partial differential equations under consideration and, for example, we have to specify unknown coefficients more concretely, and the direct method is more relevant for inverse problems. Moreover, we do not intend the current book to be encyclopedic in any sense, and the references are limited.Basics of Carleman Estimates
Basic Tools of Riemannian Geometry
Well-Posedness and Regularity for the Wave Equation with Variable Coefficients
Carleman Estimate for the Wave Equation on a Riemannian Manifold
Inverse Problems for Wave Equations on a Riemannian Manifold
Realization of the Convexity of the Weight Function
Carleman Estimates for Some Thermoelasticity
Inverse Heat Source Problem for the Thermoelasticity System
Inverse Problem for a Hyperbolic Equation with a Finite Set of Boundary Data
Supplementary Research Problems