On homogenization of nonlinear Robin type boundary conditions for cavities along manifolds and associated spectral problems

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Let uϵ be the solution of the Poisson equation in a domain normalΩ⊂double-struckR3 perforated by thin tubes with a nonlinear Robin‐type boundary condition on the boundary of the tubes (the flux here being β(ϵ)σ(x,uϵ)), and with a Dirichlet condition on the rest of the boundary of Ω. ϵ is a small parameter that we shall make to go to zero; it denotes the period of a grid on a plane where the tubes/cylinders have their bases; the size of the transversal section of the tubes is O(aϵ) with aϵ≪ϵ. A certain nonperiodicity is allowed for the distribution of the thin tubes, although the perimeter is a fixed number a. Here, σ∈C1(falsenormalΩ¯×double-struckR) is a strictly monotonic function of the second argument, and the adsorption parameter β(ϵ) > 0 can converge toward infinity. Depending on the relations between the three parameters ϵ, aϵ, and β(ϵ), the effective equations in volume are obtained. Among the multiple possible relations, we provide critical relations, which imply different averages of the process ranging from linear to nonlinear. All this allows us to derive spectral convergence as ϵ→0 for the associated spectral problems in the case of σ a linear function of uϵ. Copyright © 2014 John Wiley & Sons, Ltd.

Section: Introductionmentioning

confidence: 82%

Section: State Of the Art In The Literaturementioning

confidence: 99%

“…See [1] and [4] for the case where the perforations are placed along manifolds. See [1] and [18], for example, for further references on the subject.…”

Section: State Of the Art In The Literaturementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On critical parameters in homogenization of perforated domains by thin tubes with nonlinear flux and related spectral problems

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Asymptotics for models of non‐stationary diffusion in domains with a surface distribution of obstacles

Gómez

Лобо

Pérez‐Martínez

2018

We consider a time-dependent model for the diffusion of a substance through an incompressible fluid in a perforated domain Ω , Ω ⊂ Ω ⊂ R n with n = 3, 4. The fluid flows in a domain containing a periodical set of "obstacles" (Ω∖Ω ) placed along an inner (n − 1)-dimensional manifold Σ ⊂ Ω. The size of the obstacles is much smaller than the size of the characteristic period . An advection term appears in the partial differential equation linking the fluid velocity with the concentration, while we assume a nonlinear adsorption law on the boundary of the obstacles. This law involves a monotone nonlinear function of the concentration and a large adsorption parameter. The "critical adsorption parameter" depends on the size of the obstacles , and, for different sizes, we derive the time-dependent homogenized models. These models contain a "strange term" in the transmission conditions on Σ, which is a nonlinear function and inherits the properties of . The case in which the fluid velocity and the concentration do not interact is also considered for n ≥ 3. CorrespondenceMaría-Eugenia Pérez-Martínez, ETSI Caminos, Canales y Puertos. Av. de los Castros s/n.,

Operator estimates for non‐periodically perforated domains with Dirichlet and nonlinear Robin conditions: Strange term

Borisov

2023

We consider a boundary value problem for a general second‐order linear equation in a domain with a fine perforation. The latter is made by small cavities; both the shapes of the cavities and their distribution are arbitrary. The boundaries of the cavities are subject either to a Dirichlet or a nonlinear Robin condition. On the perforation, certain rather weak conditions are imposed to ensure that under the homogenization, we obtain a similar problem in a non‐perforated domain with an additional potential in the equation usually called a strange term. Our main results state the convergence of the solution of the perturbed problem to that of the homogenized one in ‐ and ‐norms uniformly in ‐norm of the right hand side in the equation. The estimates for the convergence rates are established, and their order sharpness is discussed.