Geometric optics expansions for quarter-space boundary value problems III: glancing modes and multiple self-interaction

This article aims to continue the study of geometric optics expansions for hyperbolic boundary value problems in the quarter-space initiated in [2]. The motivations are linked to the range of effective applicability of the theorem establishing the existence of the geometric optics expansions. Compared to [2], we ameliorate the range of applicability by adding two distinct features. The first one is that now we can consider glancing modes in the expansions by using the results of [17]. The second one, which is proper to quarter-space problems, is that we can now consider rather “complicated” self-interaction phenomena. It is a first step in the study of geometric optics expansions in bounded domains. A direct consequence of the first point of amelioration is that no condition on glancing modes is required to intialize the construction of the geometric optics expansion. It seems to indicate that the expected condition characterizing the strong well-posedness of corner problems, established in [14], can be relaxed to the hyperbolic component of the stable subspace only.