SECTION 4 - Second-Order Elliptic Differential Equations

Abstract

This chapter discusses the second-order elliptic differential equations. For elliptic partial equations, the Cauchy or mixed problems are not the natural ones to consider. While on one hand, the initial conditions have to be analytic to assure the existence of solutions, on the other hand, even such strong conditions do not assure the continuous dependence of the solution on the initial data. It turns out that the correctly posed problems for elliptic equations are the boundary value problems. The simplest elliptic equation of the second order is the Laplace equation whose solutions are called harmonic functions. The chapter presents the fundamental boundary value problems for elliptic operators and describes the general properties of elliptic operators. It also illustrates the application of the Green's function in the solution of Dirichlet's problem for a half-plane. The physical interpretation of the Green's function is the potential at M because of a positive unit charge placed at Mi. The surface S is assumed to be a grounded conductor.