Partial Differential Equations with more than Two Independent Variables in the Complex Domain

摘要

In this paper we shall be primarily concerned with linear partial differential equations of the second order whose coefficients are analytic. We shall be interested in properties of the solutions of such equations which can be studied by means of an analytic continuation into the domain of complex values of the independent variables. The case of two independent variables has been thoroughly investigated from this point of view, and our contribution here will therefore be to problems involving three or more variables. The significance of such a study lies in the hope it offers of extending to several variables some of the theories for partial differential equations in two variables which depend on conformal mapping or on the introduction of characteristic coordinates.