Chapter 1 Linear Elliptic Equations In this chapter, we review briefly three basic topics in the theory of lin- ear elliptic equations: the maximum principle, Krylov-Safonov's Harnack inequality, and the Schauder theory. In Section 1.1, we review Hopf's maximum principle. The maximum principle is an important method to study elliptic differential equations of the second order. In this section, we review the weak maximum principle and the strong maximum principle and derive several forms of a priori estimates of solutions. In Section 1.2, we review Krylov-Safonov's Harnack inequality. The Harnack inequality is an important result in the theory of elliptic differential equations of the second order and plays a fundamental role in the study of nonlinear elliptic differential equations. In Section 1.3, we review the Schauder theory for uniformly elliptic linear equations. Three main topics are a priori estimates in Holder norms, the regularity of arbitrary solutions, and the solvability of the Dirichlet problem. Among these topics, a priori estimates are the most fundamental and form the basis for the existence and the regularity of solutions. We will review both the interior Schauder theory and the global Schauder theory. These three sections play different roles in the rest of the book. In the study of quasilinear elliptic equations in Part 1, the maximum principle will be used to derive estimates of derivatives up to the first order, the Harnack inequality will be used to derive estimates of the Holder semi- norms of derivatives of the first order, and the Schauder theory will be used 由扫描全能王扫描创建由 扫描全能王 扫描创建