Abstract

This chapter is broadly divided into two parts. In the first, a brief but self-contained review of the interior regularity theory of elliptic PDEs is presented, including relevant preliminaries on function spaces. In the second, as an application of the tools introduced in the first part, a detailed study of the Einstein condition on Riemannian manifolds with metrics of Holder regularity is undertaken, introducing important techniques such as the use of harmonic coordinates and giving some consideration to the smoothness of the differentiable structure of the underlying manifold.

Introduction

Partial differential equations (PDEs) play a fundamental role in many areas of pure and applied sciences. Particularly known is their ubiquitous presence in the description of physical systems, but it is less obvious to students how their application to geometry is not only useful but often decisive. As an introduction to the world of PDEs in geometry, the aim of this chapter is two-fold. First, to give a brief (and rather condensed) overview of the minimum number of basic preliminaries that allow us to write down precise and complete statements of several important theorems from the regularity theory of elliptic PDEs, which is one of the most needed aspects for applications. Second, to present a detailed treatment of some useful techniques required to study geometric equations involving the Ricci tensor and, in particular, to put into use the definitions and tools introduced in the first part to study the smoothness of weak solutions to the Einstein condition on Riemannian manifolds with metrics of Hölder regularity C1,α.