  Energy inequalities are derived for an elliptic-hyperbolic operator arising
in plasma physics. These inequalities imply the existence of distribution and
weak solutions to various closed boundary-value problems. An existence theorem
is proven for a related class of Keldysh equations, and the failure of expected
methods for obtaining uniqueness is discussed. The proofs use ideas recently
introduced by Lupo, Morawetz, and Payne for a generalized Tricomi operator. The
existence of strong solutions under open boundary conditions is also proven.
