A boundary value problem for cold plasma dynamics (Q1864542)

A boundary value problem is considered for a mixed elliptic-hyperbolic system \[ [x-\sigma(y)]u_{1x}+u_{2y}=f_1,\quad u_{1y}-u_{2x}=f_2. \] The condition of the form \[ u_1\frac{dx}{ds}+u_2\frac{dy}{ds}=0 \] is given on a part of the boundary (Guderley-Morawetz problem). This problem arises in models of wave propagation in cold plasma. It is proved that a weak solution exists in a weighted Hilbert space.