On classes of solutions of elliptic linear partial differential equations

Abstract

Introduction. Among the theorems which deal with the functional properties of the solutions of elliptic linear partial differential equations, the most important ones are perhaps the following: (a) The solutions of equations with analytic coefficients are analytic. (b) The solutions of equations with indefinitely differentiable (i.d.) coefficients are i.d. In Part I of this paper we prove a result which connects the above mentioned theorems. Qualitatively we prove that the i.d. solutions of elliptic differential systems with i.d. coefficients have the same distance from analyticity as have the coefficients. More precisely, we define classes of i.d. functions and show, under certain assumptions, that if the coefficients belong to a certain class, then so do the solutions. In particular, we give a new proof to Theorem (a). In Part II we define another kind of classes of i.d. functions and prove, for some kinds of elliptic equations, results similar to that of Part I. All functions in this paper are real functions. I should like to express my gratitude to Dr. S. Agmon for his help and encouragement during the preparation of Part I of this paper.