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Propagation of singularities for the wave equation on conic manifolds. (English) Zbl 1088.58011

From the introduction: Solutions to the wave equation (for the Friedrichs extension of the Laplacian) associated to a conic metric on a compact manifold with boundary exhibit a diffractive, or “ringing”, effect when singularities strike the boundary. The main results of this paper describe the relationship between the strength of the singularities incident on the boundary and the strength of the diffracted singularities. We first show that if no singularities arrive at the boundary at a time \(\overline t\) then the solution is smooth near the boundary at that time, in the sense that it is locally in the intersection of the domains of all powers of the Laplacian. We then show that if there are singularities incident on the boundary at time \(\overline t\) and in addition the solution satisfies an appropriate nonfocusing condition with respect to the boundary, then the strongest singularities leaving the boundary at that time are on the geometric continuations of those incoming bicharacteristics which carry singularities, whereas on the diffracted, i.e., not geometrically continued, rays the singularities are weaker. If the incident wave satisfies a conormality condition, then the singularity on the diffracted front is shown to be conormal. Applying this analysis to the forward fundamental solution gives an extension of results of Cheeger and Taylor from the product-conic to the general conic case.

MSC:

58J47 Propagation of singularities; initial value problems on manifolds
35L05 Wave equation
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