The Existence and Uniqueness of the Solution for a Kind of Second Order Elliptic Differential Equation of Variable Coefficient

Abstract

In this paper, we consider a kind of boundary value problem of a second order elliptic differential equation of variable coefficient. First, we give the variation inequality which is equal to this boundary value problem, and prove the existence and uniqueness of the solution of the variation inequality of this kind by using Green formula and variation method lemma. Then we can obtain the existence and uniqueness of the solution of the original boundary value problem. Finally, using regularization method, the variation inequality can be formulated as a differentiable variation equation since it includes an item which can't be differentiable. So we can solve the boundary value problem with variable coefficient by translating it to the corresponding variation form. These works provide wide methods and the theoretical basis for studying elliptic differential equations of variable coefficient.