Homogenization of nonlinear Dirichlet problems in random perforated domains

Calvo‐Jurado, Carmen; Casado–Díaz, Juan; Luna-Laynez, Manuel

doi:10.1016/j.na.2015.12.008

Cited by 16 publications

(9 citation statements)

References 20 publications

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“…Many techniques and methods have been developed in order to understand the effect of the holes in perforated domains on the solutions of PDE problems with different boundary values. From pioneering works to recent ones we can still mention [1,6,8,11,17,29,30,36,33] and references therein that are concerned with elliptic and parabolic equations, nonlinear operators, as well as Stokes and Navier-Stokes equations from fluid mechanics. Note that this kind of problem is an "homogenization" problem, since the heterogeneous domain Ω is replaced by a homogeneous one, Ω, in the limit.…”

Section: Introductionmentioning

confidence: 99%

Nonlocal problems in perforated domains

Pereira

Rossi

2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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In this paper we analyze nonlocal equations in perforated domains. We consider nonlocal problems of the form f (x) = B J(x − y)(u(y) − u(x))dy with x in a perforated domain Ω ⊂ Ω. Here J is a non-singular kernel. We think about Ω as a fixed set Ω from where we have removed a subset that we call the holes. We deal both with the Neumann and Dirichlet conditions in the holes and assume a Dirichlet condition outside Ω. In the later case we impose that u vanishes in the holes but integrate in the whole R N (B = R N ) and in the former we just consider integrals in R N minus the holes (B = R N \ (Ω \ Ω )). Assuming weak convergence of the holes, specifically, under the assumption that the characteristic function of Ω has a weak limit, χ X weakly * in L ∞ (Ω), we analyze the limit as → 0 of the solutions to the nonlocal problems proving that there is a nonlocal limit problem. In the case in which the holes are periodically removed balls we obtain that the critical radius is of order of the size of the typical cell (that gives the period). In addition, in this periodic case, we also study the behavior of these nonlocal problems when we rescale the kernel in order to approximate local PDE problems.

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Section: Introductionmentioning

confidence: 99%

Nonlocal problems in perforated domains

Pereira

Rossi

2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…Starting with [6] and [21], there is a large amount of literature devoted to the homogenization of (1.1), both for deterministic and random holes H ε [3,20]; similar problems have also been studied in the case of the fractional laplacian (−∆) s , [2] or for nonlinear elliptic operators [4,24]. All the models considered in the deterministic case contain assumptions that ensure that, for ε small enough, the holes in H ε do not to overlap.…”

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confidence: 99%

Convergence rates for the homogenization of the Poisson problem in randomly perforated domains

Giunti¹

2021

NHM

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In this paper we provide converge rates for the homogenization of the Poisson problem with Dirichlet boundary conditions in a randomly perforated domain of R d , d 3. We assume that the holes that perforate the domain are spherical and are generated by a rescaled marked point process (Φ, R). The point process Φ generating the centres of the holes is either a Poisson point process or the lattice Z d ; the marks R generating the radii are unbounded i.i.d random variables having finite (d − 2 + β)-moment, for β > 0. We study the rate of convergence to the homogenized solution in terms of the parameter β. We stress that, for low values of β, the balls generating the holes may overlap with overwhelming probability.

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“…Let us explain our idea to go rigorously from (2) to (7). Given a smooth and compactly supported test field ϕ on R 3 , the point is to introduce a solution…”

Section: Strategy and Hypotheses For The Convergencementioning

confidence: 99%

“…As regards the analysis of (5) and of the more involved fluid problem (6), rigorous results were obtained in Levy and Sanchez-Palencia [35,27], Ammari et al [2], Haines and Mazzucato [22], as well as in the recent papers by Hillairet and Wu [24], Niethammer and Schubert [32] , or the author and Hillairet [18]. These references are by no mean exhaustive: one could further cite [26,6,7,30,15,21] and many more on related problems.…”

Section: Introductionmentioning

confidence: 99%

A simple justification of effective models for conducting or fluid media with dilute spherical inclusions

Gérard-Varet¹

2019

Preprint

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We present a gentle approach to the justification of effective media approximations, for PDE's set outside the union of n ≫ 1 spheres with low volume fraction. To illustrate our approach, we consider three classical examples: the derivation of the so-called strange term, made popular by Cioranescu and Murat, the derivation of the Brinkman term in the Stokes equation, and a scalar analogue of the effective viscosity problem. Under some separation assumption on the spheres, valid for periodic and random distributions of the centers, we recover effective models as n → +∞ by simple arguments.

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Homogenization of nonlinear Dirichlet problems in random perforated domains

Cited by 16 publications

References 20 publications

Nonlocal problems in perforated domains

Nonlocal problems in perforated domains

Convergence rates for the homogenization of the Poisson problem in randomly perforated domains

A simple justification of effective models for conducting or fluid media with dilute spherical inclusions

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