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.