Boundary Estimation Of Fractional Laplace Equations

In this paper,we mainly study the boundary regularity for solutions of fractional Dirichlet boundary value problems.We use the A-B-P estimate and iterative method to prove the C^? and C^1,? boundary regularity by constructing some non-local barrier functions.In the first chapter,we introduce the background and development of fractional partial differential equations,and summarize the research status of internal regularity and boundary regularity estimation of fractional partial differential equations.At the end of this chapter,we expound the research content and work arrangement of this paper.The second chapter mainly gives some preliminaries to prove the boundary regularity of fractional partial differential equations.We first introduce fractional Laplace operators and Pucci type extreme value operators,and then give the definitions of regularity for domain's boundary and regularity for functions.As the end of this chapter,we summarize some known results for fractional Laplace equations,such as the definition of viscosity solutions,the A-B-P estimate,and the internal Harnack inequality.In Chapter 3,we consider the following Dirichlet problems:(?)Constructing a class of suitable barrier functions to control viscosity solutions,we prove the C^?(0<?<1)boundary estimate by iterative methods and A-B-P estimate under regularity conditions that the boundary is Lipschitz,f(x)?L^?(?)and g(x)?C^?(Rn\?).This conclusion can be generali-zed to more general fractional Laplace equations.The Chapter 4 explores the boundary regularity of the above equations.By the A-B-P estimate and Harnack inequality,we prove that solutions of our equations are C^1,?(0<?<1)near the boundary when the domain's boundary is C^1,?,g(x)?C^1,?(Rn\?)and f(x)?C^?(?).