Abstract
We give a rigorous justification of geometric optics for a class of Kreiss well-posed semilinear boundary problems where both resonant interactions and glancing modes are present. Errors are o8(818)8 in L827 as the wavelength tends to zero. We emphasize the features that distinguish boundary problems from hyperbolic problems in free space. These include:
(818)8 the apparent failure of coherence and symmetry hypotheses alone to guarantee existence of the exact solution on a fixed domain independent of the wavelength,
(828)8 inconsistent transport equations for glancing modes connected with the presence of a glancing boundary layer,
(838)8 the need to use (generalized) eigenvectors associated to nonreal eigenvalues in constructing approximate solutions and the related presence of an elliptic boundary layer,
(848)8 and the appearance of unbounded families of projection operators (associated to eigenvalues of high multiplicity) in the profile equations.