Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems, Volume 144
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.
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Acoustic and Electromagnetic Equations: Integral Representations ..., Volume 144
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admits a unique belongs bilinear form boundary condition bounded coefficients coercive compact compute continuous curl H curlp curlr defined Dirichlet problem divp divr double layer potential eigenvalue equivalent estimate expression exterior Dirichlet problem exterior problem follows Fredholm alternative fundamental solution given Green formula Helmholtz equation Hilbert space identity inequality integral equations integral operators integral representation interior and exterior introduce kernel Laplace equation Laplace-Beltrami operator Legendre Lemma Maxwell equations Maxwell system Moreover multipoles Neumann problem norm normal derivative obtain orthogonal perfect conductor plane wave polynomials properties prove radiation condition recursion formula regularity right-hand side satisfies scattering amplitude shows single layer potential Sobolev spaces space H spherical harmonics Stokes formula surface takes the form tangent plane tends to infinity tends to zero trace theorem unique solution unit sphere vanishes variable variational formulation yields