In constructing solutions of
steady-state wave propagation problems, a common procedure is to assume that
the frequency has a small imaginary part and, with an eventual solution
in hand, to let this imaginary part go to zero — the principle of limiting
absorption. There are three basic problems involved here. The first is to
establish the principle of limiting absorption itself, i.e., to show in a rigorous
manner that a steady-state solution can actually be obtained in this fashion.
The second problem is to find a class of functions in which the solution
so constructed is unique (a “radiation condition”). While in problems in
exterior domains or with bounded perturbations of the coefficients uniqueness
classes are essentially dictated by the asymptotic behavior at large spatial
distances of the free-space Green functions, in the problems with infinite
boundary considered below it is not immediately clear how to specify uniqueness
classes. For example, must the asymptotic behavior at large distances of
eventual surface-wave components of the solution be included in the conditions
designating such classes? Finally, since, strictly speaking, steady-state solutions are
physically meaningless (they fail to have finite energy), a third problem is to
determine in what sense they are approximations for large times to the original
time-dependent solutions—the principle of limiting amplitude. In this paper we
study these questions for the steady-state versions of the dissipative problems
for Maxwell’s equations considered previously. While these problems are
particular examples, the results obtained do provide a guide and generalize
to other problems. For example, although the equations of elasticity are
much more difficult to deal with, the steady-state Rayleigh surface wave of
elasticity theory has basically the same properties as the surface wave in the
selfadjoint Leontovich case considered below. As far as we know, questions of
uniqueness for steady-state wave propagation problems in domains with
infinite boundary have not been considered previously, although Eidus has
recently established the principle of limiting absorption for some such domains.
Unfortunately, the abstract approach used there apparently does not provide enough
information to make it possible to define uniqueness classes for the solutions so
constructed.
