Singularities in Boundary Value Problems: Proceedings of the NATO Advanced Study Institute held at Maratea, Italy, September 22 – October 3, 1980

The 1980 Maratea NATO Advanced Study Institute (= ASI) followed the lines of the 1976 Liege NATO ASI. Indeed, the interest of boundary problems for linear evolution partial differential equations and systems is more and more acute because of the outstanding position of those problems in the mathematical description of the physical world, namely through sciences such as fluid dynamics, elastodynamics, electro­ dynamics, electromagnetism, plasma physics and so on. In those problems the question of the propagation of singularities of the solution has boomed these last years. Placed in its definitive mathematical frame in 1970 by L. Hormander, this branch -of the theory recorded a tremendous impetus in the last decade and is now eagerly studied by the most prominent research workers in the field of partial differential equations. It describes the wave phenomena connected with the solution of boundary problems with very general boundaries, by replacing the (generailly impossible) computation of a precise solution by a convenient asymptotic approximation. For instance, it allows the description of progressive waves in a medium with obstacles of various shapes, meeting classical phenomena as reflexion, refraction, transmission, and even more complicated ones, called supersonic waves, head waves, creeping waves, •••••• The !'tudy of singularities uses involved new mathematical concepts (such as distributions, wave front sets, asymptotic developments, pseudo-differential operators, Fourier integral operators, microfunctions, ••• ) but emerges as the most sensible application to physical problems. A complete exposition of the present state of this theory seemed to be still lacking.

Table of Contents:
Sur le Comportement Semi Classique du Spectre et de l’Amplitude de Diffusion d’un Hamiltonien Quantique.- General Initial-Boundary Problems for Second Order Hyperbolic Equations.- Note on a Singular Initial-Boundary Value Problem.- Pseudo-Differential Operators of Principal Type.- Mixed Problems for the Wave Equation.- Microlocal Analysis of Boundary Value Problems with Applications to Diffraction.- Transformation Methods for Boundary Value Problems.- Propagation of Singularities and the Scattering Matrix.- Propagation at the Boundary of Analytic Singularities.- Lower Bounds at Infinity for Solutions of Differential Equations with Constant Coefficients in Unbounded Domains.- Analytic Singularities of Solutions of Boundary Value Problems.- Diffraction Effects in the Scattering of Waves.- Singularities of Elementary Solutions of Hyperbolic Equations with Constant Coefficients.- The Mixed Problem for Hyperbolic Systems.