Boundary value problems for a differential equation of the mixed type.

Boundary value problems for differential equations of the mixed type were first considered by tricomi in 1923. In his paper (1) he resolves the Dirichlet type problem for the equation yzxx + zyy = 0, (E) in a mixed domain by reducing it to a singular integral equation. Later Holmgren (2) and Gellerstedt (3), (4) generalized some of Tricomi's results to the equation ym a2z/ax2 + a2z/ay2 = 0 m being an odd positive integer. Due to the breathily of Holmgrens paper we have felt it necessary to obtain his results in detail (chapters 4 and 5). In Part I we completely resolve the Dirichlet type problem for the equation (E) in a mixed domain by reducing it to a singular integral equation. We solve this integral equation in a closed form by using the elegant theory developed by Gakhov and Chibrikova (5). In part II we consider a boundary value problem for (E) of the mixed type in a mixed domain. That is to say, we suppose that the value of the unknown solution is given on part of the boundary, and the value of a directional derivative of the solution is given on another part. Once again we obtain a singular integral equation, but in this case it cannot be solved in a closed form. We show the existence of a solution to the integral equation by reducing it to an equivalent fredholm equation, using the regularization method of Carleman-Vekua (see for example Gakhov (6)). I would like to thank my supervisor or professor T.V. Davies for his help and encouragement, and for suggesting the problems considered here.