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

Authors

  • V. Zh. Dumanian Chair of Numerical Analysis and Mathematical Modeling, YSU, Armenia

DOI:

https://doi.org/10.46991/PYSU:A/2009.43.3.010

Keywords:

Dirichlet problem, elliptic equation, Dirichlet's integral

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 $\partial Q$ .

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Published

2009-10-04

How to Cite

Dumanian, V. Z. (2009). DIRICHLET WEIGHT INTEGRAL ESTIMATION TO DIRICHLET PROBLEM SOLUTION FOR THE GENERAL SECOND ORDER ELLIPTIC EQUATIONS. Proceedings of the YSU A: Physical and Mathematical Sciences, 43(3 (220), 10–21. https://doi.org/10.46991/PYSU:A/2009.43.3.010

Issue

Section

Mathematics