Natural Science, Mathematics, 2025
DIRICHLET WEIGHT INTEGRAL ESTIMATION TO DIRICHLET PROBLEM SOLUTION FOR THE GENERAL SECOND ORDER ELLIPTIC EQUATIONS
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Submitted: 2025-02-27; Published: 2025-02-27
© 2025 by author(s) and The Gufo Inc.
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(CC BY-NC 4.0).
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 .