Application to Boundary Problems for Elliptic Equations

Abstract

It is now common practice to study boundary value problems for an elliptic equation (say, in a domain of Euclidean space or, more generally, on a manifold with boundary) by transferring them to the boundary. For this purpose one makes use of the Poisson kernel for the Dirichlet problem, relative to the equation under study, and of the regularity results and estimates, now well established, in the latter problem. If appropriate results have been established for the problem on the boundary, it is then possible to reach the desired conclusions about the original problem, most often the regularity up to the boundary of its solutions, the finite dimensionality of its kernel and cokernel, etc. Of course, in an expository text, the author is still left with the task of establishing the classical properties of the Dirichlet problem.