Mathematics

Dirichlet weight integral estimation to Dirichlet problem solution for the general second order elliptic equations

V. Zh. Dumanyan

Chair of Numerical Analysis and Mathematical Modeling YSU, Armenia

Abstract: We consider the Dirichlet problem in a bounded domain $Q\subset R_n$ $\partial Q\in C^1$, for the second order linear elliptic equation
$$-\sum_{i,j=1}^n(a_{ij}(x)U_{x_i})_{x_j}+\sum_{i=1}^nb_i(x)u_{x_i}-\sum_{i=1^n}c_i(x)u)_{x_i}+d(x)u=f(x)-divF(x), \ x\in Q, \ u|_{\partial Q}=u_0.$$
For the solution we prove boundedness of the Dirichlet integral with the weight $r(x)$, i.e. the function $r(x)| \nabla u(x)|^2$ is integrable over $Q$ , where $r(x) $ is the distance from a point $x\in Q$ to the boundary $\partial Q$.

Keywords: Dirichlet problem, elliptic equation, Dirichlet's integral.

Received: 27.02.2009
Accepted: 31.03.2009

Language: English