By Vladimir Maz'ya, Alexander Soloviev
This e-book is a entire exposition of the idea of boundary indispensable 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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Extra resources for Boundary Integral Equations on Contours with Peaks (Operator Theory: Advances and Applications)
Since (∂Z/∂n)ds = 0 , ∂G the function ∂Z/∂n can be represented in the form ∂X/∂s, where X satisﬁes X c Lp,1−α (Γ) + Φ + − Φ− X Lp,1−α (Γ) Lp,1−α (R) Φ± + Lp,−α (R) + ± Φ± Lp,1−α (R) . We set F = Z + χ(Φ(−) + Φ(+) ) . 105), the function u = F ◦ ω −1 is the required harmonic extension of ϕ onto Ω+ . Since 1 ∈ N1,− p,β (Γ), the next proposition follows from the Cauchy-Riemann conditions. 5. Let ψ belong to Np,β (Γ), where 0 < β + p−1 < 1 . 8) in Ω+ with boundary data ψ has a solution v in N1,− p,β (Γ) satisfying v N1,− (Γ) c ψ Np,β (Γ) .
Therefore, ∂ T σ(z) = − ∂s σ (q) Γ ∂ |z| dsq log ∂nz |z − q| and it suﬃces to verify the continuity of the operator T∗ : Lp,β+1 (Γ) → Lp,β+1 (Γ) given by ∂ |z| dsq . σ (q) log T∗ σ(z) = ∂nz |z − q| Γ+ ∪Γ− We represent T∗ σ(z) in the form σ (q) + Γc− (x) Γc+ (x) ∂ |z| dsq + I(z), log ∂nz |z − q| where the last term admits the estimate |I(z)| x c x2 (|σ+ (u)| + |σ− (u)|)udu 0 δ c + x (|σ+ (u)| + |σ− (u)|)du , z ∈ Γ+ ∪ Γ− . 22) that I Lp,β+1 (Γ+ ∪Γ− ) c σ Lp,β+1 (Γ) . In the sequel we assume that z ∈ Γ+ .
It is clear that I1 N1,− p,β (Γ) = I1 L1p,β+1 (Γ) c σ Since 1 ∂ log ∂n |q| c 1 , |q| L1p,β+1 (Γ) . 83) Chapter 1. Lp -theory of Boundary Integral Equations 24 we have δ |I2 (z)| |σ(τ )| dτ, z ∈ Γ+ ∪ Γ− . 22), we obtain I2 N1,− p,β (Γ) = I2 c σ L1p,β+1 (Γ) L1p,β+1 (Γ) . 84) Now we estimate I3 . We have Re k z q ∂ ∂nq = Re z k ∂ ∂ Im q −k − Im z k Re q −k , ∂nq ∂nq where q = u + iκ± (u) ∈ Γ± . Since the inequalities |Im z k | and |(∂/∂nq )q −k | c xk+μ c u−k−1 hold, it follows that ∂ Re q −k | ∂nq |Im z k c xk+μ .
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