On Elliptic And Parabolic Equations With Inverse-square Potentials

This thesis is mainly composed of three parts.In Chapter 2,we prove Holder continuity of weak solutions of the uniformly elliptic and parabolic equations with critical or supercritical 0-order term coefficients which are beyond De Giorgi-Nash-Moser's Theory.We also prove,in some special cases,weak solutions are even differentiable.This is a joint work with Prof.Qi S.Zhang.In Chapter 3,we derive Schauder estimates to the solutions of the following uni-formly elliptic equation with an inverse-square potential and nonhomogeneous term which lead to the existence and sharp regularity results of the classical solutions.More precisely,we prove that u ?Cn+2,? provided f ? Cn,?,aij?Cn,? and A>?(2 + n +?)(d + n + ?).In Chapter 4,two main results are presented.First,the precise heat kernel and Green function of the operator-(?-1/r2)under the axially symmetric condition,and some weighted LP estimates of the Green function are given.This will serve as a tool for the study of axially symmetric Navier-Stokes equations.As an application,we prove the regularity of solutions to axially symmetric Navier-Stokes equations under a critical(or a subcritical)assumption on the angular component ?? of the vorticity.Second,we prove the regularity of solutions to axially symmetric Navier-Stokes equations under a log supercritical assumption on the horizontally radial component ur and vertical component uz,accompanied by a log subcritical assumption on the horizontally angular component u? of the velocity.This is a part of the ongoing paper with Dr.Xinghong Pan.