The kernel of the neumann operator for a strictly diffractive analytic problem

We consider p a partial differential operator of order 2 and Rⁿ= ω⁺ ∪ ?ω ∪ ω^? a partition of Rⁿ , such that (p, ω⁺) admits a strictly diffractive point (in the sense of Friedlander and Melrose). We compute the trace and the trace of the normal derivative on ?ω of the solution u of the diffraction problem pu= 0 in ω⁺ u satisfying a mixed boundary condition on ?ω, ?ω analytic. That is done using the construction by Lebeau of a Gevrey 3 parametrix in the neighborhood of the strictly diffractive point.

This result generalizes, for a mixed boundary condition, the Gevrey 3 propogation result of Lebeau. We use this result to compute the leading term in the shadow region of the diffracted wave outside a strictly convex analytical obstacle with a mixed boundary condition and a given incoming wave.