By Vladimir Maz'ya, Alexander Soloviev

The objective of this e-book is to offer a finished exposition of the idea of boundary crucial equations for unmarried and double layer potentials on curves with external and inside cusps. the idea was once built by means of the authors over the past 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 options close to the cusps are got. Kernels and cokernels of the crucial operators are explicitly defined. the tactic relies on a learn of auxiliary boundary price difficulties that's of curiosity in itself.

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**Additional resources for Boundary integral equations on contours with peaks**

**Example text**

5). In this subsection, we prove a property of πI − T which is stronger than the continuity of T in L1p,β+1 (cf. 14). 16. Let Ω+ have an outward 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. 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± .

Dirichlet and Neumann problems for a domain with peak 35 The following lemma is a particular case of more general results of the article [11]. 1. The operator k Wp,α (G) k−2 k−1/p u → {Δu, u|∂G } ∈ Wp,α (G) × Wp,α (∂G) performs an isomorphism for every real α and k = 1, 2. The same is true for k = 1, 2, . . if G is replaced by the strip Π. We need Kellogg’s theorem (see [10]) on boundary properties of a conformal mapping of the unit disk onto a bounded domain with smooth boundary. 2 (Kellogg).

98) does not exceed c σ p . Lp,β+1 (Γ) We turn to the integral I5 . Obviously, ∂ 1 1 1 = −Re cos(nq , 0x) + Im cos(nq , 0y) . 99) Here (nq , 0x) and (nq , 0y) are the angles between the normal vector nq and the coordinate axes. 99) in the form m −Re k=0 zk q k+1 cos(nq , 0x) − Re z m+1 − z) q m+1 (q cos(nq , 0x). Chapter 1. Lp -theory of Boundary Integral Equations 32 It is clear that z m+1 − z) Re c xm+1 uμ−m−2 . 99). 100) 1 1 1 1 + Re m+1 Im z m+1 Re . Re z m+1 Im q m+1 q−z q q−z Since Im z k Re q −k−1 = O xμ u−1 and Re z k Im q −k−1 = xk Im q −k−1 + O xμ u−1 , we arrive at m zk Im k=0 q k+1 m = xk Im q −k−1 + O xμ u−1 .