Weak well-posedness for the Dirichlet problem for equations of mixed elliptic-hyperbolic type

Equations of mixed elliptic-hyperbolic type with a homogeneous Dirichlet condition imposed on the entire boundary will be discussed. Such closed problems are typically overdetermined in spaces of classical solutions in contrast to the well-posedness for classical solutions that can result from opening the boundary by prescribing the boundary condition only on a proper subset of the boundary. Closed problems arise, for example, in models of transonic fluid flow about a given profile, but very little is known on the well-posedness in spaces of weak solutions. We present recent progress, obtained in collaboration with D. Lupo and C. S. Morawetz, on the well-posedness in weighted Sobolev spaces as well as the beginnings of a regularity theory.