On Extending Solutions to Wave Equations across Glancing Boundaries

(for example, the wave operator acting in the exterior of a smooth obstacle). After extending the coefficients of P smoothly across <9M, we can view P as an operator on some open extension M of M. The problem is easily solved in the two cases where no null bicharacteristics tangent to dT*M are present. When dM is everywhere elliptic with respect to P , classical theory implies that the desired ü can be found if and only if u\dM 6 C°°{dM). When dM is everywhere hyperbolic, nothing has to be assumed about U\QM € D'(dM), for the extension ü can be produced simply by solving the Cauchy problem in a neighborhood of dM with Cauchy data given by u. Here we are interested in the two cases where null bicharacteristics tangent to dT*M to first order are present, the diffractive and gliding cases. An example given in [8] shows that if the boundary is diffractive, even when U\QM is smooth, it may happen that no extension as a solution (in fact, no extension u such that p £ WF Pu where p € dT*M is a point of null bicharacteristic tangency) exists. Our main result (Theorem 2) implies that, in contrast to the diffractive case, near gliding points extensions as microlocal solutions always exist when U\SM is smooth. We construct such an extension after showing that, near a gliding point a £ WFU\QM, any distribution u satisfying Pu G C°°(M) has the series expansion given in Theorem 1. The proof of Theorem 2 makes essential use of the recent unified treatment of the diffractive and gliding parametrices [5], in which the eikonal and transport equations are solved on both sides of the boundary. Full proofs will appear in [9]. We proceed to recall some terminology.