De Giorgi-Nas-Moser Theory

Abstract

This chapter is interested in Sobolev spaces and their applications in solving elliptic partial differential equations (PDEs), focusing on regularity results for weak solutions of these equations. Key points: Introduction to Sobolev spaces: Definitions and fundamental properties of Sobolev spaces, essential for the study of weak PDE solutions. Weak solutions of elliptic PDEs: Definition of weak solutions for elliptic PDEs of the form div(A(x)∇u(x)) = 0. Moser Iteration: Presentation of Moser iteration, a technique for proving regularity results for weak solutions of elliptic PDEs. The chapter provides a solid foundation for understanding the use of Sobolev spaces in the study of weak solutions of elliptic PDEs. The following sections will delve deeper into the regularity aspects using Moser's iteration and Harnack's inequality