By Vladimir Maz'ya, Alexander Soloviev
This booklet is a complete exposition of the idea of boundary fundamental equations for unmarried and double layer potentials on curves with external and inside cusps. 3 chapters conceal harmonic potentials, and the ultimate bankruptcy treats elastic potentials.
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Additional resources for Boundary Integral Equations on Contours with Peaks (Operator Theory: Advances and Applications)
Then the function d(ζ) = d0 (ζ) + γ(ζ) if 2μ ∈ N , if 2μ ∈ / N, d0 (ζ) satisﬁes θ ◦ d (ξ) = ξ + O |ξ|2μ+1 . 121) is a conformal mapping of a neighborhood of ζ = 0 in the peak in Ω+ which admits the representation x := Re θ0 (ξ) = ξ 2 + O |ξ|2μ+2 R2+ onto a neighborhood of as ξ → ±0 . 122) The inverse mapping θ0−1 satisﬁes ξ := Re θ0−1 (z) = ±x1/2 + O xμ+1/2 on Γ± . By diminishing δ in the deﬁnition of Γ± , we can assume that θ0 is deﬁned on Γ+ ∪ Γ − . Let n0 be an integer subject to the inequalities n0 − 1 2(μ − β − p−1 ) < n0 and let m be the largest integer satisfying 2m m = [μ − β − 1/p].
8) in Ω+ . 3 Auxiliary boundary value problems for a domain with peak The concluding principal proposition in this section concerns a representation of the harmonic conjugate for a harmonic extension of a function in N1,+ p,β (Γ). Before passing to this result, we state an auxiliary assertion which is proved at the end of the section. 8. 114) Chapter 1. Lp -theory of Boundary Integral Equations 44 near the peak and Λ(0) = 0. 115) [2μ]−1 c(k) ξ k + c(m) ξ m log |ξ| + c(±) |ξ|2μ + O ξ 2μ+γ for [2μ] = 2μ k=0 with γ < γ.
100) 1 1 1 1 + Re m+1 Im z m+1 Re . Re z m+1 Im q m+1 q−z q q−z Since Im z k Re q −k−1 = O xμ u−1 and Re z k Im q −k−1 = xk Im q −k−1 + O xμ u−1 , we arrive at m zk Im k=0 q k+1 m = xk Im q −k−1 + O xμ u−1 . 100) does not exceed c xm uμ−m−2 . 100) Re q −m−1 Re z m+1 Im (q − z)−1 c (xm+1 uμ−m−2 + xμ u−1 ) . 100) satisﬁes Re q −m−1 Im z m+1 Re (q − z)−1 c xμ u−1 . Thus we have, for q ∈ Γr+ (x) ∪ Γr− (x), ∂ 1 = log ∂nq |q − z| m xk −Re k=0 where |I(q, z)| 1 q k+1 cos(nq , 0x) + Im 1 q k+1 cos(nq , 0y) + I(q, z), c (xm+1 uμ−m−2 + xμ u−1 ).