By Vladimir Maz'ya, Alexander Soloviev

This booklet is a entire exposition of the speculation of boundary critical equations for unmarried and double layer potentials on curves with external and inside cusps. 3 chapters hide harmonic potentials, and the ultimate bankruptcy treats elastic potentials.

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**Sample text**

22) imply the required estimate in Lp,β−μ (Γ+ ∪Γ− ) for the left-hand side. 80). We obtain for q ∈ Γc+ (x): q − z− = q − z+ 1+ z+ − z− u−x 1− 1− u−x q − z+ z+ − z− (u − x) + (z+ − z− ) Since Re (z+ − z− ) = 0, we have z+ − z− (u − x) + (z+ − z− ) 1. Taking into account that 1− u−x κ+ (u) − κ+ (x) u − x =i = O(xμ ) q − z+ u−x q − z+ and z+ /z− = 1 + O(xμ ), we deduce 1 − q/z− 1 − q/z+ (1 + c xμ ) 1 + c xμ+1 |u − x| . Chapter 1. Lp -theory of Boundary Integral Equations 20 Hence, 1 − q/z− dsq 1 − q/z+ σ(q)log Γc+ (x) (1+ε)x (1+ε)x |σ+ (u)|du + c μ cx (1−ε)x |σ+ (u)|log 1 + cxμ+1 |u − x| du = J1 + J2 .

Chapter 1. Lp -theory of Boundary Integral Equations 38 We introduce the bounded harmonic function Φ(+) on the strip Π taking the same value (Φ+ (τ ) + Φ− (τ ))/2 at the points (τ, 1) and (τ, −1) on ∂Π. The Fourier transform of Φ(+) (τ, ν) with respect to τ is given by c Φ+ (ξ) + Φ− (ξ) cosh (νξ) (cosh ξ)−1 , where Φ± denotes the Fourier transform of Φ± . Therefore, the Fourier transform of (∂/∂n)Φ(+) (τ, ±1) is equal to c i Φ+ (ξ) + Φ− (ξ) tanh ξ . Hence ∂Φ(+) (τ, ±1) = c ∂n R d π (Φ+ (t) + Φ− (t)) sinh (τ − t) dt 2 −1 dt .

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 .