2009; Physical and Mathematical Sciences, 43(3 (220): 10–21
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DIRICHLET WEIGHT INTEGRAL ESTIMATION TO DIRICHLET PROBLEM SOLUTION FOR THE GENERAL SECOND ORDER ELLIPTIC EQUATIONS

V.Zh. Dumanian*

Received: 2025-02-27 · Published: 2009-10-04

© 2009 The Author(s) and the original publisher.

CC BY 4.0 This work is licensed under a Creative Commons Attribution 4.0 International License.
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Original title
DIRICHLET WEIGHT INTEGRAL ESTIMATION TO DIRICHLET PROBLEM SOLUTION FOR THE GENERAL SECOND ORDER ELLIPTIC EQUATIONS
Author
V.Zh. Dumanian
Published
2009-10-04
Licence
Creative Commons Attribution 4.0 International

Abstract

We consider the Dirichlet problem in a bounded domain Q⊂Rn,∂Q∈Cl, for the second order linear elliptic equation−∑i,j=1n(ai,j(x)uxi)xj+∑i=1nbi(x)uxi+∑i=1n(ci(x)u)xi+d(x)u==f(x)−divF(x),x∈Q,u|∂Q=u0. For the solution we prove boundedness of the Dirichlet integral with the weight r(x) , i.e. the function r(x)|▽u(x)|2 is integrable over Q , where r(x) is the distance from a point x∈Q to the boundary ∂Q .
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