DIFFERENTIAL GEOMETRY/PDE SEMINAR

Friday, February 11, 2005 Padelford C-401 2:30 pm (Note: special day, time and room!)

Geometric optics and the wave equation on manifolds with corners

András Vasy (MIT)

I will describe the propagation of smooth (C^{∞}) and Sobolev singularities for the wave equation on smooth manifolds with corners M equipped with a Riemannian metric g. That is, for X = M ×ℝ_{t}, P = D_{t}^{2} −Δ_{M}, and u ∈ H_{loc}^{1}(X) solving P u = 0 with homogeneous Dirichlet or Neumann boundary conditions, the appropriate wave front set WF_{b}(u) of u is a union of maximally extended generalized broken bicharacteristics. Since the latter follow the rules of geometric optics, i.e. those of classical dynamics, this result is a facet of the classical-quantum correspondence, namely that singularities of solutions of the wave equation follow geometric optics. This result is a smooth counterpart of Lebeau’s results for the propagation of analytic singularities on real analytic manifolds with appropriately stratified boundary.

I will indicate the key ideas of the proof, such as microlocalization with respect to the appropriate ps.d.o. algebra, Ψ_{b}(X), and gaining b-regularity (i.e. conormal regularity) relative to H_{loc}^{1}(X) via positive commutator estimates. Certain aspects of this problem are related to N-body scattering.

For more information about this seminar, visit the DG/PDE Seminar Web page (from the Math Department home page, www.math.washington.edu, follow the link Seminars, Colloquia, and Conferences).

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