8 1.Linear Elliptic Equations to solve the linearized equations.In the study of fully nonlinear elliptic equtions n Part2,the maximum principle will be used to derive estimates of derivatives up to the second order,the Harnack inequality will be used to derive estimates of the Holder semi-norms of derivatives of the second order and the Schauder theory will be used to solve the linearized equations. It is not our intention to present a complete review of the linear theory. Notably missing from this short review are the W2.P-theory for linear equa- tions of the nondivergence form and theHtheory and the de Gorgi-Moser theory for linear equations of the divergence form.Refer to Chapters 2-9 of (59]for a complete account of the linear theory. 1.1.The Maximum Principle The maximun principle is an important method to study elliptic differential equtions of the second order.In,we review the weak maximum principle and the strong maximum principle and derive several forms of a priori estimates of solutions.Refer to Chapter 3 of [59]for details. Throughout this section,we let be a bounded domain in Rn and let eeaicen7waenaenaa融y-od (1.1.1) Lu aijdiju+bidiu+cu in s, for any uC2().The operator L is always assumed to be strictly elliptic in;namely,for any x∈nandξ∈Rm, (1.1.2) a4(x)5≥A|2, for some positive constant.For later reference,Lis callednformlyelliptic if,for any x∈nandξ∈R", (1.1.3) A52≤a(x)5i≤A512, for some positive constants A and A,which are usually called the ellipticity constants. 1.1.1.The Weak Maximum Principle.In this subsection, we review the weak maximum principle and its corollaries.We first introduce subso lutions and supersolutions. Definition 1.1.1.For some f C(),a C2()-function u is called a sub solution (or supersolution)of Lw f if Lu f (or Lu f)in S. If aij =6ij,bi=c=0,and f=0,subsolutions (or supersolutions)are subharmonic (or superharmonic) Now we prove the weak maximum principle for subsolutions.Recall that u+is the nonnegative part of u,defined by ut=maxto,u. 由扫描全能王扫描创建由 扫描全能王 扫描创建