Interior Regularity of the Dirichlet Problem for the Tricomi Equation

For the Tricomi equation with Dirichlet boundary conditions, we study the relationship between singularites at the boundary and singularities in the interior of a bounded planar region with smooth non-characteristic boundary. Necessary and sufficient conditions for interior smoothness are stated in terms of microlocal regularity at the boundary and are proven via known microlocal propagation of singularities results along the generalized bicharacteristic flow. In particular, a trapped gliding ray phenomenon at parabolic boundary points is demonstrated under a sharp geometric hypothesis, which provides a microlocal explanation for the possibility of having only isolated singularities at the boundary, which is a question left open in the work of Morawetz