By Vladimir V. Mityushev, Michael Ruzhansky

ISBN-10: 3319121472

ISBN-13: 9783319121475

ISBN-10: 3319121480

ISBN-13: 9783319121482

The booklet comprises lectures given through the plenary and key audio system on the ninth overseas ISAAC Congress held 2013 in Krakow, Poland. The contributions deal with contemporary advancements in research and surrounding parts, referring to issues from the speculation of partial differential equations, functionality areas, scattering, likelihood idea, and others, in addition to purposes to biomathematics, queueing versions, fractured porous media and geomechanics.

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Craster, Homogenisation for hexagonal lattices and honeycomb structures, under review (2014) 24. D. V. I. Adamou, Trapped modes in curved elastic plates. Proc. R. Soc. Lond. A 461, 1181–1197 (2005) 25. D. A. E. Tovstik, Localized vibration in elastic structures with slowly varying thickness. Quart. J. Mech. Appl. Math. 58, 645–664 (2005) 26. V. M. Joseph, J. Kaplunov, Long-wave asymptotic theories: the connection between functionally graded waveguides and periodic media, Wave Motion 51, 581–588 (2014) 27.

2 [a(ξ )U02 ]ξ2 =1 dξ1 + S The function V is given in [16] and is a particular solution to the first order problem. This is entirely on the macroscale with the microstructure built in through integrated quantities. Thus the medium is “homogenized”, but valid at high frequencies. As noted above one can have repeated roots and the theory can still be developed with coupled PDEs emerging and the theory follows through again. Dynamic Homogenization 47 3 Illustrative Examples We now take a couple of illustrative examples starting with a square array of split ring resonators from [38], these are slits with Neumann boundary conditions upon them.

Let x ∈ S be arbitrary and choose f ∈ L 1 (S) ∩ L p (S) with f = 0. As in the proof of Lemma 1 we split Iα f (x) = Jα f (x)+ K α f (x) where the integrals on the right hand side range from 1 to δ and δ to ∞ (respectively). Again arguing as in the proof of Lemma 1, we find that α 1 2 f ∗ (x)δ 2 , α Γ (α/2) |Jα f (x)| ≤ Now using (2) we obtain ∞ |K α f (x)| ≤ C p,n,α α t2 δ α ≤ C p,n,α δ 2 − 2np −1 − 2np || f || p || f || p , so that α α |Iα f (x)| ≤ C p,n,α ( f ∗ (x)δ 2 + δ 2 − 2np || f || p ). Picking || f || p f ∗ (x) δ= 2 p/n to minimize the right hand side gives |Iα f (x)| ≤ C p,n,α f ∗ (x) Thus for 1 < p < n α 1−αp/n and using (1), αp/n || f || p = C p,n,α f ∗ (x) p/q αp/n || f || p .

### Analytic Methods in Interdisciplinary Applications by Vladimir V. Mityushev, Michael Ruzhansky

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