A boundary value problem for an equation of mixed type

Abstract

where K(y) is a continuous nondecreasing function with K(O) =0. Particular cases of (1.1) have been the subject of several investigations. In particular Darboux [4](1), Tricomi [8], and Cibrario [3] have investigated the case K(y) =y. Gellerstedt [6] considered the case K(y) =y' . Within the last few years interest in equation (1.1) has been stimulated by the problems of transsonic flow. The equation of Chaplygin for a twodimensional gas flow, when transformed to the hodograph plane, is of the form (1.1) with K(y) positive for subsonic speeds and negative for supersonic speeds. In this connection Frankl [5] investigated initial value problems for an equation of mixed type similar to (1.1). For (1.1) Bers announced an existence and uniqueness theorem for the Dirichlet problem in which the domain lies in the elliptic portion of the plane but has part of its boundary on the parabolic line [1J. He solved the Cauchy problem for a hyperbolic domain with the data prescribed along the parabolic line [2]. Recently Germain and Bader [7 ] have announced results similar to those of Bers and of the author. The similarities are discussed later in this section. We consider the boundary value problem for a hyperbolic-parabolic domain in which the data are prescribed along one characteristic and the parabolic line. We prove the following