Singularities in Boundary Value Problems

This volume gathers the material from the 1980 NATO Advanced Study Institute held in Maratea, which continued the program begun at the 1976 institute in Liege. Its focus is linear evolution partial differential equations and systems with boundary conditions, a area that has grown in importance for describing the physical world in fields such as fluid dynamics, elastodynamics, electrodynamics, electromagnetism and plasma physics. Research on how singularities of solutions propagate has expanded rapidly, especially since Hormander gave it its present mathematical foundation in 1970.

The theory explains wave behavior in settings with irregular boundaries by replacing an exact solution, which is usually impossible to compute, with a suitable asymptotic approximation. It accounts for progressive waves moving past obstacles of many shapes, including familiar effects such as reflection, refraction and transmission, as well as more intricate phenomena known as supersonic waves, head waves and creeping waves. Its analysis relies on advanced tools, among them distributions, wave front sets, asymptotic expansions, pseudo-differential operators, Fourier integral operators and microfunctions.

The text is aimed at mathematicians and researchers working on partial differential equations and their physical applications. It was prepared to fill a gap, since a full and up-to-date account of this theory had not yet been published.