The Scalar Case

Abstract

The aim of this chapter is to discuss some full (that is everywhere) regularity results for scalar-valued weak solutions to second order elliptic equations in divergence forms. Weak solutions only belong to some Sobolev space (consequently, neither are they necessarily continuous nor do derivatives a priori exist in the classical sense), and hence, their regularity needs to be investigated. We here prove local Holder regularity of weak solution and present two different (and classical) strategies of proof dating back to the late 1950s. First, we explain De Giorgi’s level set technique, in a unified approach that applies both to weak solutions of elliptic equations and to minimizers of variational integrals, via the study of Q-minimizers of suitable functionals. We then address, for the specific case of linear elliptic equations, an alternative proof of the everywhere regularity result of weak solutions via Moser’s iteration method.