Schauder estimates for linear elliptic PDEs

Abstract

Partial differential equations (PDEs) are fundamental tools in mathematics. They serve as the backbone for modeling a vast array of physical phenomena and have many applications in a very wide range of mathematical subjects. In this project, we will focus on the study of second-order linear elliptic partial differential equations, though we will not address them in their most general form for simplicity. Our primary focus will be on regularity. However, before diving into the mathematical aspects, Chapter 2 will provide motivation by presenting examples from physics and probability where the equations we are interested in appear. For this chapter, we will primarily follow the work in [3] (Chapter 12) and [6] (Chapter 1). In Chapter 3, we will explore the key properties of harmonic functions, primarily following the approach in [4]. These properties will be fundamental for proving the project’s main theorems. In Chapter 4, we will introduce the concept of Hölder continuity and establish important results concerning Hölder spaces, drawing from [4] and [6]. We will see how Hölder continuity is particularly well-suited for the study of partial differential equations. In Chapter 5, following [6], we will establish interior Schauder estimates for equations in both divergence form (providing one proof) and non-divergence form (offering two different proofs). Following that, in Chapter 6, we will prove global Schauder estimates for non-divergence form equations and state the corresponding results for divergence form equations. We will also examine the critical role that boundary regularity plays in this context. Finally, in Chapter 7, we will explore how Schauder estimates, in conjunction with the Continuity Method, can be utilized to prove the regularity and existence of solutions for linear elliptic PDEs.