DIRICHLET WEIGHT INTEGRAL ESTIMATION TO DIRICHLET PROBLEM SOLUTION FOR THE GENERAL SECOND ORDER ELLIPTIC EQUATIONS

Abstract

We consider the Dirichlet problem in a bounded domain $Q\subset R_n, \partial Q\in C^l$, for the second order linear elliptic equation $$-\sum\limits_{i ,j=1}^n (a_i, j(x)u_{x_i})_{x_j}+\sum\limits_{i=1}^n b_i(x) u_{x_i}+\sum\limits_{i=1}^n (c_i(x)u)_ {x_i}+d(x)u=$$ $$=f(x)- div F(x), x\in Q,$$ $$u|_{\partial Q}=u_0.$$

For the solution we prove boundedness of the Dirichlet integral with the weight $r( x)$ , i.e. the function $r(x)|\bigtriangledown u(x)|^2$ is integrable over $Q$ , where $r(x)$ is the distance from a point $x\in Q$ to the boundary $\partial Q$ .