By Vladimir G. Maz’ya, Alexander A. Soloviev (auth.), Tatyana Shaposhnikova (eds.)
The function of this e-book is to offer a finished exposition of the idea of boundary fundamental equations for unmarried and double layer potentials on curves with external and inside cusps. the idea used to be constructed by way of 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 a variety of functionality areas and asymptotic representations for strategies close to the cusps are acquired. Kernels and cokernels of the indispensable operators are explicitly defined. the strategy is predicated on a learn of auxiliary boundary price difficulties that's of curiosity in itself.
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Additional info for Boundary Integral Equations on Contours with Peaks
We notice that there exists a function of the form ⎧ [2μ] ⎪ ⎪ ⎪ ζ + ⎪ a(k) ζ k if 2μ ∈ N, ⎪ ⎨ k=2 d0 (ζ) = [2μ]+1 ⎪ ⎪ ⎪ ⎪ ⎪ a(k) ζ k if 2μ ∈ N, ζ + ⎩ k=2 Chapter 1. Lp -theory of Boundary Integral Equations 46 deﬁned on R2+ and satisfying (θ ◦ d0 )(ζ) = ζ + γ(ζ) . Here γ(ζ) is a holomorphic function on a neighborhood of ζ = 0 in R2+ taking real values near ξ = 0 in R and having the representation γ(ξ) = b[2μ]+1 ξ [2μ]+1 log |ξ| + O |ξ|2μ+1 if 2μ ∈ N, O |ξ| if 2μ ∈ / N, 2μ+1 where ξ = Re ζ. 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 .
Proof. First, we estimate the norm of (πI − T )σ(z+ ) − (πI − T )σ(z− ) in Lp,β−μ (Γ+ ∪ Γ− ). To this end we represent (πI − T )σ(z) in the form π[σ(z+ ) + σ(z− )] − σ(q) Γr+ (x)∪Γr− (x) N −1 − σ(q) k=1 Γr+ (x)∪Γr− (x) − σ(q) Γc∓ (x) − 1 ∂ Re k ∂nq ∂ 1 log dsq ∂nq |q| z q k dsq − σ(q) Γ\(Γ+ ∪Γ− ) ∂ 1 dsq + πσ(z∓ ) − log ∂nq |z − q| σ(q) ∂ 1 dsq log ∂nq |z − q| σ(q) |q| ∂ − log ∂nq |z − q| σ(q) Γc± (x) ∂ 1 dsq log ∂nq |z − q| ∂ 1 dsq log ∂nq |z − q| Γ+ (x)∪Γ− (x) − Γr+ (x)∪Γr− (x) N −1 k=1 1 Re k z q 8 k dsq = Ik (z), k=1 where z = x + iκ± (x) ∈ Γ± .
We represent the function Im (z/q)m+1 (q − z)−1 in the form Re m+1 z q Im 1 q−z m 1 + Im q (Re z)m+1 + Im z m+1 Re k=1 Since |Im (q − z)−1 | zk q m+1 1 . q−z (Re z)m−k Re c uμ−1 on Γr+ (x) ∪ Γr− (x), we obtain We have m+1 z q Re Im m Im z Re k=1 zk q m+1 and Im 1 q−z c xm+1 uμ−m−2 . 94) c uμ−m−1 . 95) that Im (z/q)m+1 q−z c uμ−m−2 xm+1 on Γr+ (x) ∪ Γr− (x). Therefore, δ |I8 (z)| (|σ+ (u)| + |σ− (u)|) uμ−m−2 du . m+1 cx x Now set m = [μ]. 22), the integral on the right-hand side belongs to Lp,β−μ (0, δ) and the norm in Lp,β−μ (Γ+ ∪ Γ− ) of I8 is estimated by c σ L .
Boundary Integral Equations on Contours with Peaks by Vladimir G. Maz’ya, Alexander A. Soloviev (auth.), Tatyana Shaposhnikova (eds.)