By Vladimir Maz'ya, Alexander Soloviev
This ebook is a complete exposition of the speculation of boundary essential equations for unmarried and double layer potentials on curves with external and inside cusps. 3 chapters disguise harmonic potentials, and the ultimate bankruptcy treats elastic potentials.
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The aim of this ebook is to offer a finished exposition of the speculation of boundary crucial equations for unmarried and double layer potentials on curves with external and inside cusps. the speculation was once built through the authors over the past two decades and the current quantity is predicated on their effects.
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Extra resources for Boundary Integral Equations on Contours with Peaks (Operator Theory: Advances and Applications)
140) k=1 where |tk (ϕ)| c T Lp (R) + Q Lp (R2+ ) , k = −1, . . , n0 − 1, and |Rn0 (ξ)| c T Lp (R) + Q Lp (R2+ ) for small |ξ|. 141) k=1 where f # ∈ N1,− p,β (Γ), and bk (ϕ), k = 1, . . 140). Chapter 1. 141), the function g = κh + f is harmonic in Ω+ and can be represented as m ck (ϕ)Re z k−1/2 + g # (z), z ∈ Ω− , g(z) = k=1 with ck (ϕ) = ak (ϕ) + bk (ϕ). Moreover, m |c(k) | + g # N1,− p,β (Γ) k=1 c ϕ N1,+ p,β (Γ) and (g ◦ θ)(∞) = 0 by deﬁnition of g. Owing to (∂/∂s)g ∈ Lp,β+1 (Γ), one of the functions conjugate to −g is a harmonic extension of ϕ onto Ω+ with normal derivative in Lp,β+1 (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 .
Chapter 1. 124), we have δ δ |ϕ(x) − ϕ(x + h(x))|p x(β−μ)p dx c 0 1 xμ+1 . Similarly x+cxμ+1 p t |(dϕ/dt)(t)|dt xμ+1 xβ p dx x−cxμ+1 0 δ |(dϕ/dt)(t)|p x(β+1)p dx . 123). 127) < β + p1 < 1. 129) < 12 , and by (∂/∂ξ)H(ξ) = if 1 πξ 1 πξ 2 R < 1, satisﬁes (∂/∂ξ)H Lp,2β+1+1/p(R) c (d/dξ)Φ Lp,2β+1+1/p (R) . 130) Let n0 be the integer subject to the inequalities n0 − 1 2(μ − β − p−1 ) < n0 and let m be the largest integer satisfying 2m n0 . 3. Dirichlet and Neumann problems for a domain with peak 49 The function H(−) , deﬁned by H(−) (ξ) = 1 H(ξ) − H(−ξ) = 2 π R ξ−τ d (+) Φ (τ ) log dτ , dτ ξ can be written in the form ξ n0 π R Φ(+) (τ ) dτ − n τ 0 (ξ − τ ) n0 −1 k=k0 ξk π R Φ(+) (τ ) dτ .
Boundary Integral Equations on Contours with Peaks (Operator Theory: Advances and Applications) by Vladimir Maz'ya, Alexander Soloviev