The Boundary Differentiability Of A Class Of Degenerate Elliptic Equations

In this paper,we study a kind of degenerate elliptic equationprove the differentiability of the viscosity solution on the degenerate boundary.We work on the Dirichlet problem of the formwhere ? = {(x,y)|0? x ?a,|y|<b} is a rectangular area,?>0 is an arbitrary positive constant.We will prove that the solution of the equation is differentiable on the boundary when the functions f(x,y)and g(x,y)satisfy some well-posed conditions.We prove this mainly by the method of iteration.We first introduce a new metric which is related to the operator L? with the scaling of(1,1+?)and satisfy scaling structure d(rX,rY)= rd(X,Y),and then show that the graph of the solution can be bounded by two planes by choosing some barrier functions and using some basic knowledge of elliptic equation,finally,by the method of iteration,we prove that at each scale,the graph of the solution can always be bounded by two different planes and the two planes will close to each other as the scale tends to zero,which imply the differentiability of the solution.The main contents are organized as follows:In chapter 1,we introduce the background and research results of the degenerate equations.In chapter 2,we introduce some elementary knowledge,which is necessary in this paper.And give the main result of this thesis.In chapter 3,we prove the differentiability of the solution at the degenerate line by the means of iteration.In chapter 4,we prove the differentiability of the solution at the endpoint of the degenerate line.