A Strange Term Coming from Nowhere

Cioranescu, Doina; Murat, François

doi:10.1007/978-3-319-97184-1_4

Cited by 37 publications

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“…This is the same term that Cioranescu and Murat find in [6] for the periodic case. C 0 > 0 is some scaling factor for the size of holes that can be chosen arbitrarily in the definition of the geometry of .…”

Section: Introductionsupporting

confidence: 75%

“…Now we have the notation at hand to state our expansion of their Theorem 2.2 which also is the main result of this section. The first observation in our generalization of Cioranesu's and Murat's Theorem 2.2 in [6] is that the Dirichlet energy of w is monotone under refinement of the perforation. We put this important observation into the following Lemma.…”

Section: (19)mentioning

confidence: 90%

“…We perforate this tiled R N now by removing from each of the C i = C 2δ i (x i ) a closed ball T i = B a δ i (x i ) of radius a δ i concentric with C i . We define a δ i relative to δ i -or in other words: the size of holes relative to their distance -in the same way as Cioranescu and Murat do it in [6] as:…”

Section: Framework and Notationmentioning

confidence: 99%

“…The main work in Tartar's energy method is to find the 'oscillating test-functions' that converge nicely enough such that one can pass to the limit in the variational formulation of the problem on the perforated . In [6] Cioranescu and Murat construct for this very purpose the oscillatory family of auxiliary-functions w , that encodes the information of the holes and enjoys the following properties:…”

Section: The Dirichlet Problemmentioning

confidence: 99%

“…To calculate the limit of this we employ the same auxiliary function q as Cioranescu and Murat do in [6] in their proof of their Lemma 2.3. Be q the solution of the following Neumann problem in each of the (B i )…”

Section: (19)mentioning

confidence: 99%

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Explicit formulas for homogenization limits in certain non‐periodic problems including ramified domains

Eberle

2016

Z Angew Math Mech

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In this paper we give an explicit formula for the homogenization limit of Poisson's equation for a wide range of non‐periodic problems including self‐similarly ramified domains. This work was motivated by the modelling of the diffusion of medical sprays in lungs, which can be approximated by a self‐similarly ramified domain. Such motivation also led us to consider the influence that a continuous scaling of the size of holes towards a chosen direction (e.g. towards the fractal boundary) has on the homogenization limit. It turned out that our strategy to explicitly calculate a formula for the homogenization limit could be applied beyond self‐similar perforations as presented in this article.

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Section: Introductionsupporting

confidence: 75%

Section: (19)mentioning

confidence: 90%

Section: Framework and Notationmentioning

confidence: 99%

Section: The Dirichlet Problemmentioning

confidence: 99%

Section: (19)mentioning

confidence: 99%

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Explicit formulas for homogenization limits in certain non‐periodic problems including ramified domains

Eberle

2016

Z Angew Math Mech

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Nonlocal evolution equations in perforated domains

Pereira

2018

Math Methods in App Sciences

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In this paper, we study a nonlocal evolution equation posed in perforated domains. We consider problems of the form u t (t,We think about Ω as a fixed set Ω from where we have removed the subset A that we call the holes.Moreover, we take J as a nonsingular kernel. Assuming weak convergence of the holes, specifically, under the assumption that the characteristic function of Ω has a weak limit, ⇀  weakly * in L ∞ (Ω) as → 0, we analyze the limit of the solutions proving a nonlocal homogenized evolution equation.KEYWORDS asymptotic analysis, Neumann problem, nonlocal equations, perforated domains 6368

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Multiscale analysis of the acoustic scattering by many scatterers of impedance type

Challa

Sini

2016

Z. Angew. Math. Phys.

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We are concerned with the acoustic scattering problem, at a frequency κ, by many small obstacles of arbitrary shapes with impedance boundary condition. These scatterers are assumed to be included in a bounded domain Ω in R 3 which is embedded in an acoustic background characterized by an eventually locally varying index of refraction. The collection of the scatterers Dm, m = 1, ..., M is modeled by four parameters: their number M , their maximum radius a, their minimum distance d and the surface impedances λm, m = 1, ..., M . We consider the parameters M, d and λm's having the following scaling properties:≈ a t and λm := λm(a) = λm,0a −β , as a → 0, with non negative constants s, t and β and complex numbers λm,0's with eventually negative imaginary parts.We derive the asymptotic expansion of the farfields with explicit error estimate in terms of a, as a → 0. The dominant term is the Foldy-Lax field corresponding to the scattering by the point-like scatterers located at the centers zm's of the scatterers Dm's with λm|∂Dm| as the related scattering coefficients. This asymptotic expansion is justified under the following conditionsand the error of the approximation is C a 3−2β−s , as a → 0, where the positive constants a0, λ−, λ+ and C depend only on the a priori uniform bounds of the Lipschitz characters of the obstacles Dm's and the ones of M (a)a s and d(a) a t . We do not assume the periodicity in distributing the small scatterers. In addition, the scatterers can be arbitrary close since t can be arbitrary large, i.e. we can handle the mesoscale regime. Finally, for spherical scatterers, we can also allow the limit case β = 1 with a slightly better error of the approximation.

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A Strange Term Coming from Nowhere

Cited by 37 publications

References 0 publications

Explicit formulas for homogenization limits in certain non‐periodic problems including ramified domains

Explicit formulas for homogenization limits in certain non‐periodic problems including ramified domains

Nonlocal evolution equations in perforated domains

Multiscale analysis of the acoustic scattering by many scatterers of impedance type

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