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

V. Zh. Dumanyan

Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 3 (2009), pp. 10-21

Résumé

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 $\partial Q$.

Détail

Keywords: Dirichlet problem, Dirichlet's integral.
Mots-clés : elliptic equation