By Vladimir Maz'ya, Alexander Soloviev

This e-book is a complete exposition of the idea of boundary quintessential 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.

**Read or Download Boundary Integral Equations on Contours with Peaks (Operator Theory: Advances and Applications) PDF**

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**Additional info for Boundary Integral Equations on Contours with Peaks (Operator Theory: Advances and Applications)**

**Example text**

144). The coeﬃcients ck (ϕ) and the function g # satisfy m |ck (ϕ)| + g # k=1 c ϕ N1,− p,β (Γ) N1,+ p,β (Γ) and the conjugate function g = G(1/θ−1 ) is a harmonic extension of ϕ onto Ω+ with the normal derivative in Lp,β+1 (Γ). 8. We represent HΦ(ξ) for a positive ξ in the form 1 HΦ(ξ) = π ∞ −∞ 1 dt = Φ(t) ξ−t π 2ξ 1 + π ξ Φ(ξ − t) − Φ(ξ + t) 0 ∞ 2 dt + Φ(−t) ξ+t π 0 dt t Φ (odd) 2ξ tdt 2 (t) 2 + ξ ξ − t2 π ∞ Φ(odd) (t) 2ξ dt ξ 2 − t2 4 = Ik (ξ), k=1 where Φ(odd) (ξ) = 1 1 Φ(ξ) − Φ(−ξ) , Φ(odd) (ξ) = Φ(ξ) + Φ(−ξ) .

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 .

88) ∞ −1/μ ξ p(1−α) c R δ −μ −1 p σ− (u) x2μ+1 du (x − u)2 + x2μ+2 μ |σ− (τ )| dτ (ξ − τ )2 + μ2 τ p dξ , −1 where β + p = μ(α − p ). 88) is majorized by ∞ δ τ −pα |σ− (τ −1/μ |σ− (u)|p up(β+1) du . p )| dτ δ −μ 0 Thus, it suﬃces to estimate (1+ε)x πσ− (x) − (1−ε)x σ− (u)ρ(x) du . (x − u)2 + (ρ(x))2 Chapter 1. 51) that γr (ξ) πσ− (h(ξ)) − σ− (h(τ ))ρ(h(ξ))h (τ ) dτ (h(ξ) − h(τ ))2 + (ρ(h(ξ)))2 γ (ξ) σ− (h(τ )) dτ + I(ξ). (ξ − τ )2 + 1 = πσ− (h(ξ)) − R Here the limits of integration γ (ξ) = h−1 ((1 + ε)x)) and γr (ξ) = h−1 (1 − ε)x) are comparable with ξ and the last term admits the estimate |I(ξ)| ξ c ξ dτ |σ− (h(τ ))| +c τ h−1 (δ) ∞ dτ |σ− (h(τ ))| 2 + c τ ∞ h−1 (δ) ξ |σ− (h(τ ))| dτ .