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 Proceedings of the YSU, Physical and Mathematial Scineces: Year: Volume: Issue: Page: Find

 Proceedings of the YSU, Physical and Mathematial Scineces, 2009, Issue 3, Pages 10–21 (Mi uzeru230)

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.

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Accepted:31.03.2009
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Citation: V. Zh. Dumanyan, “Dirichlet weight integral estimation to Dirichlet problem solution for the general second order elliptic equations”, Proceedings of the YSU, Physical and Mathematial Scineces, 2009, no. 3, 10–21

Citation in format AMSBIB
\Bibitem{Dum09} \by V.~Zh.~Dumanyan \paper Dirichlet weight integral estimation to Dirichlet problem solution for the general second order elliptic equations \jour Proceedings of the YSU, Physical and Mathematial Scineces \yr 2009 \issue 3 \pages 10--21 \mathnet{http://mi.mathnet.ru/uzeru230}