Elliptic Boundary Value Problems

Abstract

This chapter is devoted to general boundary value problems for second-order elliptic differential operators. We begin in Sect. 6.1 with a summary of the basic facts about existence, uniqueness and regularity of solutions of the Dirichlet problem in the framework of Hölder spaces. In Sect 6.2, using the calculus of pseudo-differential operators, we prove existence, uniqueness and regularity theorems for the Dirichlet problem in the framework of Sobolev spaces. In Sect. 6.3 we formulate general boundary value problems, and show that these problems can be reduced to the study of pseudo-differential operators on the boundary. The virtue of this reduction is that there is no difficulty in taking adjoints after restricting the attention to the boundary, whereas boundary value problems in general do not have adjoints. This allows us to discuss the existence theory more easily.