A Model Diffractive Boundary Value Problem on An Asymptotically Anti-deSitter Space.

Ha Pham
Stanford University

Room: UNIV 101

Feb 29, 2012 2:30 PM EST

On manifolds without boundaries, a theorem by Hörmander states that singularities of solution of PDE are propagated along bicharacteristics. In the presence of boundaries and tangency of bicharacteristics to the boundaries, the description becomes more complicated. The simplest scenario of tangency is at diffractive points where the order of tangency is simple. At such points, for smooth boundaries and incomplete metrics, results by Melrose, Sjöstrand, and Taylor give that singularities are propagated along diffractive rays and do not enter the 'shadow' region. We will discuss the diffractive problem (the existence of 'shadow region') on an asympotically anti-de Sitter space whose boundary is not totally geodesic. Our main result establishes conormality (to the boundary) for the forward fundamental solution, showing the presence of 'shadow region'. The approach adopted is a parametrix construction, motivated by that used for a conformally related problem studied by Friedlander.