Gradient Estimation Of Second Order Divergence Elliptic Equations

In this paper,we consider a divergence elliptic equation of second order(?)j(aij(x)ui)=(?)jfj(x),χ ∈ B1(0),where B1(0)is the unit ball in the Euclidaen space Rn.This paper discuss the gradient estimates for linear divergence elliptic equation of second order.We assume that coefficient of aij satisfies uniform elliptic conditions,under the condition of that both the coefficient and the right item are Dini continu-ous,it is proved that the gradient of the weak solution also satisfy Dini continuous.Using the W1,2 eatimate,the local L∞ estimate,the Caccioppoli inequality of weak solution and iterative method,the gradient estimates for solution of equation are proved.Moreover,when the coefficient and the right item are Holder continuous,the result also contains Holder continuous of the gradient for solution.The main contents are organized as follows:In chapter 1,we state the background of the regularity of partial differential equations and research results of Schauder estimates,what we explore and its de-velopment are put forward.In chapter 2,we state basic concepts,fundamental lemmas and inferences.In chapter 3,using the W1,2 eatimate,the local L∞ estimate,the Caccioppoli inequality of weak solution and iterative method prove the gradient estimates for linear divergence elliptic equation of second order.