By Vladimir Maz'ya, Alexander Soloviev

The goal of this e-book is to provide a complete exposition of the idea of boundary indispensable equations for unmarried and double layer potentials on curves with external and inside cusps. the idea was once built via the authors over the last 20 years and the current quantity relies on their results.

The first 3 chapters are dedicated to harmonic potentials, and within the ultimate bankruptcy elastic potentials are handled. Theorems on solvability in numerous functionality areas and asymptotic representations for ideas close to the cusps are bought. Kernels and cokernels of the essential operators are explicitly defined. the strategy is predicated on a learn of auxiliary boundary worth difficulties that is of curiosity in itself.

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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. By D we denote the image of Ω+ under the mapping u + iv = (x + iy)1/2 . The boundary ∂ D near the origin is the graph of the function κ(u) such that 1 κ(u) = ± α± |u|2μ+1 1 + O u2 min{2μ,1} , u → ±0. 116) This asymptotic decomposition can be diﬀerentiated twice. Let θ denote a conformal mapping of R2+ onto D, normalized by θ(0) = 0.

3 (Privalov). Let f (ζ) = u(r, θ) + iv(r, ζ), ζ = reiθ , be holomorphic in the disk |ζ| < 1 and let u(r, θ) be continuous in |ζ| u(1, θ) satisﬁes the H¨ older condition |u(1, θ) − u(1, θ )| c|θ − θ |α with 0 < α < 1, then f satisﬁes the H¨ older condition |f (ζ) − f (ζ )| in the disk |ζ| 1. c|ζ − ζ |α 1. If Chapter 1. 7) with boundary data in N1,− p,β (Γ) for a domain Ω+ with outward peak belongs to the space Np,β (Γ). The functions in the space N1,− p,β (Γ) have a simple characterization under a conformal mapping of a domain with outward peak onto a strip.

147) Since Φ(ξ − t) − Φ(ξ + t) = β+ ξ 2μ−1 t + O(ξ 2μ tγ + ξ 2μ−1+γ t) for 0 < τ < ξ, it is clear that I2 (ξ) = c ξ 2μ + O ξ 2μ+γ . 148) Let us ﬁnd an asymptotic representation of I3 (ξ). We use the identity ξ2 t 1 ξ2 ξ 2m ξ 2m+2 . = − − 3 − · · · − 2m+1 − 2m+1 2 2 −t t t t t (ξ − t2 ) It suﬃces to consider the integral 1 2ξ tdt Φ(−t) 2 =− ξ − t2 1 m ξ k=0 2k 2ξ Φ(−t) dt + ξ 2(m+1) t2k+1 1 2ξ Φ(−t) dt .