Regularity theory for the local and nonlocal equations

Abstract

This paper is a book which is written based on the contents of reference [2] and [3]. In this paper, we study fractional Laplacian and property of viscosity solutions. In chapter 1, we learn that the fractional Laplacian can be derived from the extension problem. It is the key point of chapter 1 that we can apply the theory of the local equations to the non-local equation fractional Laplacian. In chapter 2, we define the viscosity solution of fully nonlinear elliptic equations and study the properties of it. The ABP-estimate gives us a information of a lower bound of the viscosity supersolutions. From this fact, we proves Harnack's inequality which is the goal of this chapter.