 By Denisenko V. V.

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Extra resources for A Boundary Value Problem for an Elliptic Equation with Asymmetric Coefficients in a Nonschlicht Domain

Example text

Let j = 1, 2 and i = 1, . . , k. We have: ρ2 V (x)e Pk h=1 P Uh P ψij = 64π 2 Ω + G(ξl , ξh ) + l=h 1 8π k l=1 as ρ → 0. 4 deduce that, ρ2 V (x)e Pk h=1 P Uh P ψij ≤ Cρ2 |P ψij | = O(ρ). 3 we get ρ2 V (x)e Pk h=1 P Uh  1 log V (ξi ) + O(ρ). 13) P ψij B(ξl ,ε) =ρ P 2 B(ξl ,ε) V (x)e8π(H(x,ξl )+ h=l G(x,ξh )) (τl2 ρ2 + |x − ξl |2 )2 8π P = 8π B(0, τερ ) V (τl ρy + ξl )e8π(H(τl ρy+ξl ,ξl )+ τl2 (1 + |y|2 )2 ∂G (x, ξi ) + O(ρ2 ) ∂(ξi )j h=l G(τl ρy+ξl ,ξh )) × l ∂G (τl ρy + ξl , ξi ) + O(ρ2 ) × ∂(ξi )j ∂G = 64π 2 (ξl , ξi ) + O(ρ).

K. We have: ρ2 V (x)e Pk h=1 P Uh P ψij = 64π 2 Ω + G(ξl , ξh ) + l=h 1 8π k l=1 as ρ → 0. 4 deduce that, ρ2 V (x)e Pk h=1 P Uh P ψij ≤ Cρ2 |P ψij | = O(ρ). 3 we get ρ2 V (x)e Pk h=1 P Uh  1 log V (ξi ) + O(ρ). 13) P ψij B(ξl ,ε) =ρ P 2 B(ξl ,ε) V (x)e8π(H(x,ξl )+ h=l G(x,ξh )) (τl2 ρ2 + |x − ξl |2 )2 8π P = 8π B(0, τερ ) V (τl ρy + ξl )e8π(H(τl ρy+ξl ,ξl )+ τl2 (1 + |y|2 )2 ∂G (x, ξi ) + O(ρ2 ) ∂(ξi )j h=l G(τl ρy+ξl ,ξh )) × l ∂G (τl ρy + ξl , ξi ) + O(ρ2 ) × ∂(ξi )j ∂G = 64π 2 (ξl , ξi ) + O(ρ).

S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473-483. H. Weston, On the asymptotic solution of a partial differential equation with exponential nonlinearity, SIAM J. Math. Anal. 9 (1978), 1030-1053.