Thin domains with doubly oscillatory boundary

Arrieta, José M.; Villanueva-Pesqueira, Manuel

doi:10.1002/mma.2875

Cited by 18 publications

(15 citation statements)

References 10 publications

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“…Recently we have considered in [6,8,9,10,36] many classes of oscillating thin domains discussing limit problems and convergence properties. We also mention [11] who deal with a linear elliptic problem in a thin domain presenting doubly oscillatory behavior which is related to the present one studied here but with constant profile, that means, assuming G (x) = g(x/ ) and H (x) = h(x/ ) for some periodic functions g and h. This situation is some times called as purely periodic case. Our goal here is to consider a semilinear parabolic problem in R also presenting doubly oscillatory behavior but now with variable profile generally called locally periodic case.…”

Section: R εmentioning

confidence: 85%

“…It is worth observing that is not an easy task. In order to do so, we first need to combine different techniques introduced in [9,10] and [11] to investigate the linear elliptic problem. We use extension operators and oscillating test functions from homogenization theory with boundary perturbation results to obtain the limit problem for the elliptic equation.…”

Section: R εmentioning

confidence: 99%

“…Now, let us to pass to the limit in the different functions that form the integrands of (3.13) to get the homogenized problem. It is worth to observe that we will combine here techniques from [9,10,11,43] establishing suitable oscillating test functions to accomplish our goal.…”

Section: The Piecewise Periodic Casementioning

confidence: 99%

“…Now we are in condition to get our main result concerned to the elliptic equation (2.2) under hypothesis (H). Using approximation arguments on functions G and H , the boundary perturbation result given by Proposition 2.4, and Lemma 3.1, we are able to accomplish our goal using techniques previously discussed in [9,10,11].…”

Section: The General Homogenized Limitmentioning

confidence: 99%

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Parabolic problems in highly oscillating thin domains

Pereira

2014

Annali di Matematica

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In this work we consider the asymptotic behavior of the nonlinear semigroup defined by a semilinear parabolic problem with homogeneous Neumann boundary conditions posed in a region of R 2 that degenerates into a line segment when a positive parameter goes to zero (a thin domain). Here we also allow that its boundary presents highly oscillatory behavior with different orders and variable profile. We take thin domains possessing the same order to the thickness and amplitude of the oscillations but assuming different order to the period of oscillations on the top and the bottom of the boundary. We combine methods from linear homogenization theory and the theory on nonlinear dynamics of dissipative systems to obtain the limit problem establishing convergence properties for the solutions. At the end we show the upper semicontinuity of the attractors and stationary states. 2010 Mathematics Subject Classification. 35R15, 35B27, 35B40, 35B41, 35B25, 74Q10. Key words and phrases. Partial differential equations on infinite-dimensional spaces, asymptotic behavior of solutions, attractors, singular perturbations, thin domains, oscillatory behavior, lower semicontinuity, homogenization. † Partially supported by CNPq 302847/2011-1, CAPES/DGU 267/2008 and FAPESP 2008/53094-4, Brazil.

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Section: R εmentioning

confidence: 85%

Section: R εmentioning

confidence: 99%

Section: The Piecewise Periodic Casementioning

confidence: 99%

Section: The General Homogenized Limitmentioning

confidence: 99%

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Parabolic problems in highly oscillating thin domains

Pereira

2014

Annali di Matematica

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“…fixed right-hand side f (x) ∈ L 2 (Q)) has been studied [22] and the case of perforated domains with jump was originally studied in [3] (see also [36], [20], [32] [40] and [21] for a wide bibliography). We refer to [1], [2], [14], [15], [38], [39], [41] (and references therein) for the homogenization in domains with an oscillating boundary when the amplitude of the oscillations goes to zero, and to [11], [12], [24] for the case of fixed amplitude. For transmission problem through an oscillating boundary of fixed amplitude see [11], [25] and for vanishing amplitude see [44].…”

mentioning

confidence: 99%

Existence and Homogenization for a Singular Problem Through Rough Surfaces

Donato¹,

Giachetti²

2016

SIAM J. Math. Anal.

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The paper deals with existence and homogenization for elliptic problems with lower order terms singular in the u-variable (u is the solution) in a cylinder Q in R N , so that the lower order term becomes infinite on the set {u = 0}. A rapidly oscillating interface inside Q separates the cylinder in two composite connected components. The interface has a periodic microstructure and it is situated in a small neighbourhood of a hyperplane which separates the two components of Q. At the interface we suppose the following transmission conditions: (i) the flux is continuous, (ii) the jump of a solution at the interface is proportional to the flux through the interface. This is a steady state model for the heat conduction in two heterogeneous electrically conducting materials with an imperfect contact between them. On the exterior boundary Dirichlet boundary conditions are prescribed.We also derive a corrector result for every values of the two parameters γ and κ which are related respectively to the microstructure period and to the amplitude of the interface oscillations.

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Elliptic equations in weak oscillatory thin domains: Beyond periodicity with boundary‐concentrated reaction terms

Barbosa,

Villanueva‐Pesqueira

2024

Math Methods in App Sciences

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In this paper, we analyze the limit behavior of a family of solutions of the Laplace operator with homogeneous Neumann boundary conditions, set in a two‐dimensional thin domain that presents weak oscillations on both boundaries and with terms concentrated in a narrow oscillating neighborhood of the top boundary. The aim of this problem is to study the behavior of the solutions as the thin domain presents oscillatory behaviors beyond the classical periodic assumptions,including scenarios such as quasi‐periodic or almost‐periodic oscillations. We then prove that the family of solutions converges to the solution of a one‐dimensional limit equation capturing the geometry and oscillatory behavior of boundary of the domain and the narrow strip where the concentration terms take place. In addition, we include a series of numerical experiments illustrating the theoretical results obtained in the quasi‐periodic context.

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Thin domains with doubly oscillatory boundary

Cited by 18 publications

References 10 publications

Parabolic problems in highly oscillating thin domains

Parabolic problems in highly oscillating thin domains

Existence and Homogenization for a Singular Problem Through Rough Surfaces

Elliptic equations in weak oscillatory thin domains: Beyond periodicity with boundary‐concentrated reaction terms

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