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

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DIRICHLET WEIGHT INTEGRAL ESTIMATION TO DIRICHLET PROBLEM SOLUTION FOR THE GENERAL SECOND ORDER ELLIPTIC EQUATIONS V.Zh. Dumanian

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 .

DOI: 10.46991/PYSUA.2009.43.3.010 Physical and Mathematical Sciences, 43(3 (220) 10-21

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Keywords

Dirichlet problem, elliptic equation, Dirichlet's integral

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