Linear Second Order Ordinary Differential Equations with Nonsmooth Coefficients

This paper deals with a Dirichlet boundary value problem for a linear second order ordinary differential operator, whose coefficients belong to certainL^p-spaces. Its solution is to be understood in the sense of Sobolev, so that the Fredholm alternative holds. The main purpose of this paper is, in case of unique solvability, to introduce a Green's function by means of which the solution can be given explicitly by integrals. We give the precise definition of the Green's function via Riesz' Representation Theorem and establish some of its basic properties. As a preliminary tool the Cauchy initial value problem is considered.