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Boundary integral equations on contours with peaks by Vladimir Maz'ya, Alexander Soloviev

By Vladimir Maz'ya, Alexander Soloviev

The goal of this publication is to provide a finished exposition of the idea of boundary imperative equations for unmarried and double layer potentials on curves with external and inside cusps. the speculation was once built via the authors over the last 20 years and the current quantity is predicated on their results.

The first 3 chapters are dedicated to harmonic potentials, and within the ultimate bankruptcy elastic potentials are taken care of. Theorems on solvability in a variety of functionality areas and asymptotic representations for ideas close to the cusps are received. Kernels and cokernels of the indispensable operators are explicitly defined. the tactic relies on a examine of auxiliary boundary worth difficulties that's of curiosity in itself.

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It is clear that I6 is represented in the form m c(k) xk + (R2 σ)(z), k=0 where m |c(k) | + R2 σ k=0 Lp,β−μ (Γ∩{|q|<δ/2}) c σ Lp,β+1 (Γ) . Finally, m (πσ − T σ)(z) = ±π σ(z+ ) − σ(z− ) + c(k) xk + (Rσ)(z), z ∈ Γ+ , k=0 where m |c(k) | + Rσ k=0 with Lp,β−μ (Γ∩{|q|<δ/2}) c σ Lp,β+1 (Γ) , 4 (Rσ)(z) = Ik (z) + (R1 σ)(z) + (R2 σ)(z) . 14. Thus we obtain the boundedness of (πI − T ) : L1p,β+1 (Γ) −→ M1,+ p,β (Γ). 18. 19. Let Ω+ have an outward peak, and let 0 < β + p−1 < min{μ, 1}. Then the operator L1p,β+1 (Γ) is continuous.

Since, in addition, (∂G1 /∂τ )(ξ, 0) = O(ξ −2 ) as |ξ| → ∞ , the normal derivative (∂g1 /∂n) of g1 = G1 ◦ γ belongs to Lp,β+1 (Γ) and satisfies ∂g1 /∂n Lp,β+1 (Γ) c ϕ1 L1p,β+1 (Γ) . Let λ be a conformal mapping of the unit disk D onto Ω+ and let G2 be the harmonic extension of ϕ2 ◦ λ on D. From the representation 1 ∂G2 (ζ) = ∂n 2π π −π s−t d ϕ2 (λ(eit )) cot dt, ζ = eit , dt 2 Chapter 1. Lp -theory of Boundary Integral Equations 42 it follows that g2 = G2 ◦ λ−1 satisfies ∂g2 /∂n Lp,β+1 (Γ) c1 ∂G2 /∂n Lp (∂D) c2 d(ϕ2 ◦ λ)/dt Lp (∂D) c3 ϕ2 L1p,β+1 (Γ) .

102) ∂u/∂n In particular, ∂u/∂n Proof. We start with the case when ϕ ∈ N1,− p,β (Γ) vanishes on Γ ∩ {|q| < δ/2}. Let θ be a conformal mapping of the unit disk D onto Ω+ . We introduce the harmonic extension F of the continuous function ϕ◦ θ onto D. The normal derivative ∂F/∂n on ∂D has the form ∂F 1 (ζ) = ∂n 2π π −π s−t d ϕ(θ(eit )) cot dt, ζ = eis . dt 2 Hence u = F ◦ θ−1 satisfies ∂u/∂n Lp,γ+1 (Γ) c1 ∂F/∂n Lp (∂D) c2 (ϕ ◦ θ) Lp (∂D) c3 ϕ N1,− p,γ (Γ) for every real γ. Since (∂u/∂n)ds = 0, Γ ∂u/∂n is represented in the form ∂v/∂s, where the function v satisfies v Lp,β−μ (Γ) + v Lp,β+1 (Γ) c ϕ N1,− p,β (Γ) .

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