Elliptic boundary problems

Abstract

To solve a boundary problem means to find a solution of a given differential equation in an open set (or manifold with boundary) Ω which on the boundary of Ω satisfies some other given differential equations, called the boundary conditions. In the previous chapters we have only discussed the Cauchy boundary problem. Very little is known yet concerning boundary problems for general partial differential operators. We shall therefore restrict ourselves to studying elliptic differential operators and elliptic boundary conditions, that is, boundary conditions which ensure smoothness of the solutions also at the boundary. By repeating some arguments used in section 4.1 we are led to a formal definition of elliptic boundary problems in section 10.1. In close analogy with section 3.1 we then give in sections 10.2 and 10.3 an existence theory for elliptic boundary problems with constant coefficients in a half space in R _n . A local existence theory for elliptic boundary problems with variable coefficients is then developed in section 10.4 by means of the perturbation argument used in Chapter VII. The passage from local to global results is made in section 10.5, where we also give a number of examples. A brief discussions of elliptic boundary problems for systems is given in section 10.6.