Application Of Fractal Pseudo-differential Operators In Differential Equations

Based on the basic knowledge of fractal ordinary differential equations and fractal partial differential equations,this paper studies the role of two pseudo-differential operators on a particular fractal set.The main content of the research involves two aspects:one is to study the boundary value problem for Laplacian on a domain ? consisting of the leve-3 Sierpinski gasket.The Dirichlet boundary value problem on the left half domain of the leve-3 Sierpinski gasket and the exact expressions for the Green function on the level-3 Sierpinski gasket are discussed in-depth solving methods and processes;The other is to study the Schrodinger operator with fractal potential like H?=id-?+?tr?,where ? stands for the Dirichlet-Laplacian operator in R2,tr? is closely connected with the trace operator tr? and ? is a nonistropic fractal set.In this paper,by introducing anisotropic Sobolev spaces and regularized nonisotropic fractal sets,we try to further extend the theoretical results of the original Wely estimate of pseudo-differential operator on isotropic and anisotropic fractal sets onto a nonisotropic fractal set and obtain a fine estimate of its operator's negative spectrum.