On hyperbolic boundary value problems
Linear electromagnetic waves or linear elastic waves are solutions to hyperbolic systems of partial differential equations (PDE). The propagation of waves in bounded regions leads to boundary value problems where the system of PDE gets augmented by a boundary condition. We review the classical theory of hyperbolic boundary value problems which is largely due to Friedrichs (1954), Hersh (1963), Sakamoto(1970) and Kreiss (1970). The main result is the well-posedness in the function space of square integrable functions.
Not all boundary conditions which are of practical interest fit the classical theory. The most inter-esting class of boundary conditions may be characterized as conservative. In this case, the well-posedness of the problem still holds; however, there is a loss of regularity along the boundary. This is a relevant scenario for Maxwell's equations with a perfect conductor as a boundary or for the elastic wave equations with traction forces along the boundary.
We close by classifying hyperbolic boundary problems as either strongly stable, strongly unstable or weakly regular of real type. This characterization is due to Benzoni-Gavage, Rousset, Serre, and Zumbrun (2002).