Interior regularity of solutions of differential equations

Abstract

The simplest case of the results proved in this chapter is the fact that every u ∈ C² satisfying the Laplace equation

$${{\partial }^{2}}u/\partial {{x}^{2}}+{{\partial }^{2}}u/\partial {{y}^{2}}=0$$

is actually in C^∞ and can even be expanded in a convergent power series in x and y. The literature devoted to results of this kind is very extensive, so we shall only mention here a few papers which are particularly closely related to the results and methods of this chapter.