This book is devoted to the study of the acoustic wave equation and of the Maxwell system, the two most common wave equations encountered in physics or in engineering. The main goal is to present a detailed analysis of their mathematical and physical properties. Wave equations are time dependent. However, use of the Fourier trans form reduces their study to that of harmonic systems: the harmonic Helmholtz equation, in the case of the acoustic equation, or the har monic Maxwell system. This book concentrates on the study of these harmonic problems, which are a first step toward the study of more general time-dependent problems. In each case, we give a mathematical setting that allows us to prove existence and uniqueness theorems. We have systematically chosen the use of variational formulations related to considerations of physical energy. We study the integral representations of the solutions. These representa tions yield several integral equations. We analyze their essential properties. We introduce variational formulations for these integral equations, which are the basis of most numerical approximations. Different parts of this book were taught for at least ten years by the author at the post-graduate level at Ecole Poly technique and the University of Paris 6, to students in applied mathematics. The actual presentation has been tested on them. I wish to thank them for their active and constructive participation, which has been extremely useful, and I apologize for forcing them to learn some geometry of surfaces.
From the reviews:
MATHEMATICAL REVIEWS
"A remarkable feature of this book is that this rather classical topic of applied mathematics is not approached in the framework of classical Holder spaces but is linked up with concepts of Sobolev spaces and variational methods. This feature gives this monograph a modern feel. On the other hand it has also a quite down to earth flavor, since it includes conrete calculations such as elaborate discussion of the case of the sphere as the domain boundary. That the book is based on lecture notes is still noticeable and makes it comfortable to read and suitable for self-study."
"The book investigates special solutions to two classical hyperbolic equations – the wave equations and the Maxwell equations on Minkowski space. ... The choice of material is guided by the needs of applications in mathematical physics and engineering, and the book is suitable for graduate students." (European Mathematical Society Newsletter, September, 2002)
"A remarkable feature of this book is that this rather classical topic of applied mathematics ... is linked up with concepts of Sobolev spaces and variational methods. This feature gives this monograph a modern feel. On the other hand it has also a quite down to earth flavour, since it includes concrete calculations ... . That the book is based on lecture notes is still noticeable and makes it comfortable to read and suitable for self-study." (Rainer Picard, Mathematical Reviews, Issue 2002 c)
"This book is based on lectures held by the author and is intended for graduate students in mathematics, physics, and engineering. It is self-contained and should be useful to anyone interested in problems of acoustic and electromagnetics." (Johannes Elschner, Zentralblatt MATH, Vol. 981, 2002)
"The book is a detailed description of the theory of classical boundary value problems ... . A very deep and detailed analysis for the case of a spherical obstacle is presented. This book stands out for an extremely careful and thorough discussion of necessary functional spaces ... and other auxiliary facts which makes it completely self-contained. Careful and complete proofs are given throughout. This is an indispensable reading for any person interested in boundary value problems for the time-harmonic wave equations." (V. V. Kravchenko, Zeitschrift für Analysis und ihre Anwendungen, Vol. 21 (1), 2002)
