Inverse boundary problems are a rapidly developing area of applied mathematics with applications throughout physics and the engineering sciences. However, the mathematical theory of inverse problems remains incomplete and needs further development to aid in the solution of many important practical problems.
Inverse Boundary Spectral Problems develop a rigorous theory for solving several types of inverse problems exactly. In it, the authors consider the following:
"Can the unknown coefficients of an elliptic partial differential equation be determined from the eigenvalues and the boundary values of the eigenfunctions?"
Along with this problem, many inverse problems for heat and wave equations are solved.
The authors approach inverse problems in a coordinate invariant way, that is, by applying ideas drawn from differential geometry. To solve them, they apply methods of Riemannian geometry, modern control theory, and the theory of localized wave packets, also known as Gaussian beams. The treatment includes the relevant background of each of these areas.
Although the theory of inverse boundary spectral problems has been in development for at least 10 years, until now the literature has been scattered throughout various journals. This self-contained monograph summarizes the relevant concepts and the techniques useful for dealing with them.
Table of Contents
ONE-DIMENSIONAL INVERSE PROBLEM
The Problem and the Main Result
Controllability and Projectors
BASIC GEOMETRICAL AND ANALYTICAL METHODS FOR INVERSE PROBLEMS
Basic Tools of Riemannian Geometry for Inverse Problems
Elliptic Operators on Manifolds and Gauge Transformation
Initial-Boundary Value Problem for Wave Equation
Carleman Estimates and Unique Continuation
GEL'FAND INVERSE BOUNDARY SPECTRAL PROBLEM FOR MANIFOLDS
Formulation of the Problem and the Main Result
Fourier Coefficients of Waves
Domains of Influence
Global Unique Continuation from the Boundary
Gaussian Beams from the Boundary
Domains of Influence and Gaussian Beams
Boundary Distance Functions
Reconstruction of the Riemannian Manifold
Reconstruction of the Potential
INVERSE PROBLEMS FOR WAVE AND OTHER TYPES OF EQUATIONS
Inverse Problems with Different Types of Data
Dynamical Inverse Problem for the Wave Equation
Continuation of Data
Inverse problems with Data Given on a Part of the Boundary
Inverse Problems for Operators in Rm
TABLE OF NOTATION
"[This book] contains a wealth of important methods and ideas, and the presentation is always very clear. … [A] very interesting and valuable contribution to the literature on inverse problems for partial differential equations."
- Zentralblatt MATH, Vol. 1037