General Properties of Solutions of Linear Elliptic Partial Differential Equations

Abstract

In these lectures I shall be concerned mostly with the behavior of the ‘general’ solution of a linear differential equation in the small. The properties of solutions, on which information will be obtained, include analyticity, existence of derivatives of a certain order, nature of isolated singularities, and unique continuation properties. The methods used will be elementary. The main trick will be to replace the differential equation by integral relations connecting the values of a solution in points a finite distance apart. The situation is similar to that in the theory of complex functions, where the expression of f(z) in terms of an integral by Cauchy’ s formula makes it easy to derive statements on the functional behaviour of f(z). Most of the results obtained here will be restricted to the case of linear equations with analytic coefficients. The order of the equation and the number of independent variables will be kept arbitrary.