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

We present joint work with Daniela Lupo and Cathleen Morawetz on the question of existence and uniqueness of solutions to the Dirichlet problem for mixed type equations. While it is well known that the presence of hyperbolicity renders such a problem overdetermined for solutions with classical regularity, we show well-posedness for solutions belonging to suitably weighted Sobolev spaces. This follows from global energy estimates which are obtained by exploiting integral variants of Friedrichs’ multiplier method. Attention is paid to the problem of obtaining results with minimal restrictions on the boundary geometry and the form of the type change function in preparation for the construction of stream functions in the hodograph plane for transonic flows about profiles.