By Vladimir G. Maz’ya, Alexander A. Soloviev (auth.), Tatyana Shaposhnikova (eds.)

The goal of this publication is to provide a accomplished exposition of the idea of boundary necessary equations for unmarried and double layer potentials on curves with external and inside cusps. the idea used to be constructed through 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 taken care of. Theorems on solvability in quite a few functionality areas and asymptotic representations for options close to the cusps are received. Kernels and cokernels of the critical operators are explicitly defined. the strategy relies on a research of auxiliary boundary worth difficulties that is of curiosity in itself.

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**Additional info for Boundary Integral Equations on Contours with Peaks**

**Example text**

Let r be a measurable function on (0, ∞) subject to |r(τ )| 1. 106) 0 ∞ τ +1 ∞ p |Φ± (ν)|dν c1 τ (1−α)p dτ τ −1 0 |Φ± (τ )|p τ (1−α)p dτ. 104). 105). Let h be a measurable function on [0, δ] such that |h(x)| xμ+1 . 107) δ p xβp dx |ϕ (t)|p t(β+1)p dt. 103), we obtain that for ζ, ζ1 ∈ G with |ζ − ζ1 | < 1 the distance between ω(ζ) and ω(ζ1 ) does not exceed c xμ+1 . 105) holds. 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.

Let us estimate the norm of (πI − T )σ(z+ ) + (πI − T )σ(z− ) in Lp,β−μ (Γ ∩ {|q| < δ/2}). We represent (πI − T )σ(z) for z = x + iκ± (x) ∈ Γ± in the form ∂ 1 dsq ± π[σ(z+ ) − σ(z− )] − σ(q) log ∂nq |z − q| Γ+ (x)∪Γ− (x) − σ(q) Γc± (x) − ∂ 1 dsq + πσ(z∓ ) − log ∂nq |z − q| σ(q) Γr+ (x)∪Γr− (x) ∂ 1 dsq − log ∂nq |z − q| σ(q) Γc∓ (x) σ(q) Γ\(Γ+ ∪Γ− ) ∂ 1 dsq log ∂nq |z − q| ∂ 1 dsq = log ∂nq |z − q| 6 Ik (z) . k=1 Chapter 1. 91) holds. Therefore, (1+ε)x |I2 (z)| + |I3 (z)| (|σ+ (u)| + |σ− (u)|)du .

P Lp,β+1 (Γ) . 2. 15. 17. Let Ω+ have an inward peak and let 0 < β + p−1 < min{μ, 1}. Then the operator (πI + T ) : L1p,β+1 (Γ) −→ N1,− p,β (Γ) is continuous. 5 Operator πI − T on a contour with inward peak On contours with inward peaks, the operator πI − T has better properties than T . 14). 18. Let Ω+ have an inward peak and let 0 < β + p−1 < min{μ, 1}. Then the operator L1p,β+1 (Γ) σ −→ (πI − T )σ ∈ M1,+ p,β (Γ) is continuous. Proof. Let us estimate the norm of (πI − T )σ(z+ ) + (πI − T )σ(z− ) in Lp,β−μ (Γ ∩ {|q| < δ/2}).