By Jeffrey A.
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Extra resources for Applied partial differential equations. An introduction
To conclude this section, we make one more remark. 18. We leave the details to the reader. 54 3. 3. 1. Problem Formulation. , let h ∈ C 1 (∂D) and ∂D be given by ∂D = x ˜:x ˜ = x + h(x)ν(x), x ∈ ∂D . 34) ⎧ ∆u + ω 2 u = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ω2 ⎪ ⎪ u =0 ∆u + ⎪ ⎪ k ⎪ ⎨ u |+ − u |− = 0 ⎪ ∂u ∂u ⎪ ⎪ −k =0 ⎪ ⎪ ⎪ ∂ν + ∂ν − ⎪ ⎪ ⎪ ⎪ ⎩ ∂u = 0 ∂ν in Ω \ D , in D , on ∂D , on ∂D , on ∂Ω. 34) to the calculation of the asymptotic expressions of the characteristic values of the operator-valued function A (ω) given by ⎛ 1 ω I − KΩ ⎜ 2 ⎜ ω ω → A (ω) := ⎜ DΩ ⎜ ⎝ ∂ Dω ∂ν Ω ⎞ ω −SD ω SD 1 ω ∗ I + (KD ) 2 0 ω √ k −SD ω √ 1 −k(− I + (KDk )∗ ) 2 ⎟ ⎟ ⎟.
A0 (ω)−1 Anp (ω)ω n . 10). 2. Leading-Order Terms. As a simplest case, let us now ﬁnd the leadingorder term in the asymptotic expansion of µj − µj as → 0. 11) Recalling that 0 SB 2 A0 (ω)−1 A1 (ω)ωdω. 14). It is now ⎛ 1 ω −1 ) 0 ( I − KΩ ⎜ −1 2 A0 (ω) = ⎝ ω 1 ω −1 0 −1 −CDΩ ( I − KΩ ) [·](z) ϕe (SB ) 2 where C := cap(∂B). 31) that (A0 )(ω)−1 A1 (ω) ω (A0 )(ω)−1 A1 (ω) ω (A0 )(ω) −1 A1 (ω) ω = 0, 11 21 22 (A0 )(ω)−1 A1 (ω) ω 12 easy to see that ⎞ ⎟ ⎠, = NΩω (x, z) 0 −1 ω = (SB ) [∇DΩ [·](z) · x], ω = −CDΩ [NΩω (·, z)](z)ϕe · dσ(y) ∂B √ −1ω Cϕe · dσ(y).
3. Radiation Condition. Let us formulate the radiation conditions for the elastic waves when Im ω ≥ 0 and ω = 0. 49) kT = ω ω =√ cT µ and kL = ω ω = √ . 50) ( + kT2 )u(p) = 0, ∇ × u(p) = 0, ( 2 )u(s) = 0, + kL ∇ · u(s) = 0. 21) for solutions of the Helmholtz equation by requiring that √ ∂r u(p) (x) − −1kT u(p) (x) = o(r −1 ), as r = |x| → +∞. 51). By a straightforward calculation, one can see that the single- and double-layer potentials satisfy the radiation condition. We refer to [1, 159] for details.