By Vladimir Maz'ya, Alexander Soloviev

ISBN-10: 3034601700

ISBN-13: 9783034601702

The function of this booklet is to provide a complete exposition of the idea of boundary quintessential equations for unmarried and double layer potentials on curves with external and inside cusps. the idea used to be constructed by way of the authors over the last two decades 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 ideas close to the cusps are got. Kernels and cokernels of the indispensable operators are explicitly defined. the strategy is predicated on a examine of auxiliary boundary price difficulties that is of curiosity in itself.

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**Vladimir Maz'ya, Alexander Soloviev's Boundary integral equations on contours with peaks PDF**

The aim of this e-book is to provide a finished exposition of the idea of boundary essential equations for unmarried and double layer potentials on curves with external and inside cusps. the idea used to be built by means of the authors over the past two decades and the current quantity relies on their effects.

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**Sample text**

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± .

89) does not exceed c σ p . 90) Lp,β+1 (Γ) holds. We pass to the estimates for I6 and I7 . 91) holds. Indeed, ∂ 1 1 + (κ+ (u)2 log ∂nq |z − q| 1 |z − q|2 1/2 x κ+ (τ )(x − u)dτ c xμ−1 u if q ∈ Γc+ (x), and ∂ 1 1 + (κ+ (u)2 log ∂nq |z − q| 1 |z − q| |κ± (u)| + 1/2 |κ± (u) − κ+ (x)| |q − z| c uμ + xμ x if q ∈ Γ± (x). 91) follows. Therefore, (1+ε)x |I6 (z)| + |I7 (z)| (|σ+ (u)| + |σ− (u)|)du . 22), we obtain I6 Lp,β−μ (Γ+ ∪Γ− ) + I7 Lp,β−μ (Γ+ ∪Γ− ) c σ Lp,β+1 (Γ) . It remains to consider I8 . 92) Chapter 1.

3. Dirichlet and Neumann problems for a domain with peak 47 where ck (ϕ) are linear continuous functionals in N1,+ p,β (Γ), 0 1 r0 = if 1/2 < β + 1/p < 1 , if 0 < β + 1/p < 1/2 , and g # satisﬁes g# N1,− p,β (Γ) c ϕ N1,+ p,β (Γ) with c independent of ϕ. Proof. (i) Let ϕ ∈ N1,+ p,β (Γ) vanish outside Γ+ ∪Γ− . We extend the function Φ(τ ) = (ϕ ◦ θ0 )(τ ) by zero outside a small neighborhood of O. 123) where Φ(+) (ξ) = (Φ(ξ) + Φ(−ξ))/2. Let r(ξ) be a measurable function on (0, ∞) subject to |r(ξ)| choose ∈ [0, 1] such that |ξ|2μ+1 .

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