Eigenfunction expansions for the equations of crystal optics in a half-space

Abstract

Most wave propagation problems of classical physics can be formulated as first order symmetric hyperbolic systems of partial differential equations. The symbol of the spatial part for these systems is seldom elliptic and very often there are several nonzero propagation speeds which can coincide in various directions. These facts greatly complicate the study of such systems. Further, the solution of the mixed initial-boundary value problem is also complicated by the possible coupling of modes at the boundary and, depending on the particular choice of boundary condition, the unitary group delivering solutions to the Cauchy problem is represented as a superposition of both plane waves and surface waves. It was observed in [6] that the existence of these surface waves that propagate with a speed k(p’) which vanish for nonzero p’ implies noncoercivity of the boundary condition. A detailed study of obstacle scattering for such systems requires that the problems mentioned above be dealt with; and the group solving the Cauchy problem be obtained in a form which exhibits the coupling of modes, surface waves, etc. Thus far, only the half space problems for the isotropic Maxwell system in iR: [6] and the equations of elasticity in IR’, [ 71 have been considered within a unified framework. Maxwell’s equations in uzcuo present a 6 X 6 example of an isotropic system with one nonzero propagation speed of multiplicity two and the symbol has constant rank four. The constant, maximal, energy-preserving boundary conditions for Maxwell’s equations in IR: [5] are described by two one-parameter families Bi (“classical”) and 8: (“strange”). In [6], the unitary groups U:(t) corresponding to Bt are represented as a superposition of plane waves and all such conditions are shown to be coercive, whereas the group U:(t), corresponding to Bt, all admit surface waves and are shown to be noncoercive. The isotropic elastic system in R’, [7] is a 5 x 5 system with two distinct propagation modes, S (shear) and P (pressure), and the symbol has constant rank four. The classical boundary conditions are shown to couple the modes and admit surface waves, while other boundary conditions do not couple the 600 OC22-247)</80/08060~23$02.00/0