Boundary-value problems for light and water waves near a caustic

A quasilinear system of elliptic-hyperbolic partial differential equations arising from the uniformly asymptotic approximation of solutions to the Helmholtz equation is discussed. These equations produce a model for high-frequency waves which is stable on both sides of a smooth, convex caustic. We prove the existence of strong solutions to a class of inhomogeneous boundary-value problems for an arbitrarily small lower-order perturbation of such a system in its hodograph linearization. The boundary is allowed to extend into both the elliptic and hyperbolic regions of the equations. This extends work by Kravtsov and Ludwig, who independently developed the asymptotic approximation in the 1960s, and also recent work by Magnanini and Talenti, who showed the existence of singular solutions for the case in which the boundary is restricted to the elliptic region of the equations. We also give conditions on the boundary sufficient to exclude the existence of classical solutions to the Dirichlet problem, and discuss the extension of these results to water waves. MSC2000: 35M10, 78A05, 76B15