The simplest equation of mixed type having discontinuous coefficients is the Lavrent'ev-Bitsadze equation
(1) uyy + σ(y)uxx = 0 where σ(y) = +1 for y > 0, –1 for y < 0
The work of Bitsadze is concerned with finding a solution of equation (1) which is continuously differentiable in a suitably chosen domain and which assumes given boundary values on a suitably chosen part of the boundary (Tricomi and Frankl problem). In this paper, Bitsadze's work will be generalized.
In Section I, we shall consider the Tricomi problem for a linear second order differential equation whose hyperbolic part is self-adjoint. We shall show that uniqueness for this problem implies existence. Under suitable smoothness assumptions on the boundary of the domain, the coefficients in the equation and the data, the solutions will be a continuous function having Hölder continuous derivatives.
In Section II, we shall consider the Frankl problem for linear second order differential equations whose hyperbolic part is self-adjoint and whose elliptic part is of the form uxx + uyy + aux + buy + cy = 0. Here, we shall always take a segment of the real axis as the transition curve, and we shall assume that the Frankl curve is partly characteristic. We shall show that uniqueness for this problem implies existence and that the solution is a continuous function having Hölder continuous derivatives.
In Section III, we shall consider the Frankl problem for equation (1) with no restrictions on the Frankl line. Again, the real axis will be the transition curve. For certain domains we shall get a uniqueness theorem for this problem. We shall then indicate a possible procedure for getting "weak" solutions of this problem.