Acoustic and Electromagnetic Equations. Integral Representations for Harmonic Problems
This book investigates special solutions to two classical hyperbolic equations – the wave equations and the Maxwell equations on Minkowski space. In both cases, the author considers only solutions that are harmonic in the time variable t, and so can be written as a product of a function of space variables with function exp(ikt). The wave equation for such functions reduces to the Helmholtz equation, and similarly for solutions of the Maxwell equations. The second chapter is devoted to properties of the solutions of the Helmholtz equation, including preparatory sections describing classical spherical harmonics and interior and exterior problems for the Laplace equation in R3. Chapter 3 describes integral representations for Helmholtz equations and the corresponding integral equations, while Chapter 4 is devoted to basic facts on singular integral operators. The final chapter contains a treatment of solutions of the Maxwell equation. The choice of material is guided by the needs of applications in mathematical physics and engineering, and the book is suitable for graduate students.