Linear Partial Differential Equations with Analytic Coefficients

Abstract

In the theory of partial differential equations solutions are commonly determined, which satisfy additional conditions, e.g. assume given Cauchy data or boundary values on a certain surface. Of paramount importance from the point of view of both theory and applications is the question to what extent the solution is determined by the data, and how far the data can be described arbitrarily. It is well known that for different types of equations different sets of data appear to be appropriate. In this paper the question of unique determination of a solution by Cauchy data and the arbitrariness of the data will be treated for the most general linear equation with analytic coefficients. The theorems derived will be valid for any type or order of the equation and for any number of independent variables. Simple conditions on the equation and initial surface will be given, which are necessary if a solution of the Cauchy problem for arbitrary data is to exist. The impossibility to solve the Cauchy problem for certain types of equations will be recognized as tied up with the functional character of the solutions, which belong to a class of functions with the property of “unique continuation” similar to the analytic functions. Before describing the results of this paper we shall review some of the known properties of solutions and introduce the descriptive terminology that will be used here.