Averaging of variational inequalities for the Laplacian with nonlinear restrictions along manifolds

Gómez, D.; Лобо, М.; Pérez, M. E.; Шапошникова, Т. А.

doi:10.1080/00036811.2011.602635

Cited by 21 publications

(25 citation statements)

References 13 publications

(4 reference statements)

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“…[17] considers a boundary value problem for the p-Laplacian for the particular relation α = (n − 1)/(n − p) and κ = (p − 1)(n − 1)/(n − p). Here, we complete the results in [6,7] when p = 2 and extend them for the p-Laplacian and for all the possible relations between parameters. We refer to [5,7] for an extensive bibliography on variational inequalities in homogenization problems and to [4,5] for that on applications of the p-Laplacian to different models arising, e.g., in Newtonian fluids, glaciology and flows through porous media.…”

Section: Introductionmentioning

confidence: 78%

“…In fact, similar geometrical configurations for linear and nonlinear boundary value problems, as well as for variational inequalities, have been considered in many previous papers for operators different from the p-Laplacian (cf. [1,6,7,8,9,16,17,23] for further references). Comparing with the present paper, [23] studies variational inequalities for the biharmonic operator; [1,8,9,16,17] consider the Laplace operator and linear problems; [1] contains an extra advection term related with the flow velocity; [6,7] consider variational inequalities for the Laplace operator for certain relations between κ and α.…”

Section: Introductionmentioning

confidence: 99%

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Homogenization of Variational Inequalities for the p-Laplace Operator in Perforated Media Along Manifolds

et al. 2017

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We address homogenization problems of variational inequalities for the p-Laplace operator in a domain of R n (n ≥ 3, p ∈ [2, n)) periodically perforated by balls of radius O(ε α) where α > 1 and ε is the size of the period. The perforations are distributed along a (n − 1)dimensional manifold γ, and we impose constraints for solutions and their fluxes (associated with the p-Laplacian) on the boundary of the perforations. These constraints imply that the solution is positive and that the flux is bounded from above by a negative, nonlinear monotonic function of the solution multiplied by a parameter ε −κ , κ ∈ R and ε is a small parameter that we shall make to go to zero. We analyze different relations between the parameters p, n, ε, α and κ, and obtain homogenized problems which are completely new in the literature even for the case p = 2.

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Section: Introductionmentioning

confidence: 78%

Section: Introductionmentioning

confidence: 99%

Homogenization of Variational Inequalities for the p-Laplace Operator in Perforated Media Along Manifolds

et al. 2017

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“…We focus on the possible change of character of the partial differential equation from linear to nonlinear and from nonlinear with an averaged term containing the given function .x, u/ to the very particular case (the most critical situation) of the averaged term containing the nonlinear function H.x, u/ defined implicitly from through (35). These kinds of changes have been already detected in several papers for the case of perforations that are balls along N 1 dimensional manifolds with N 1, but with the averaged term arising in the equation on the manifold: in this connection, we refer to [2] and [3] for variational inequalities, [1] and [4] for nonlinear equations, and [1,5] and [6] for associated spectral problems.…”

Section: Introductionmentioning

confidence: 82%

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

Gómez

Лобо

Pérez

et al. 2014

Math Methods in App Sciences

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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.

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“…We also note that the above mentioned fact (on the double contribution for the strange term) has already been detected in [17,23] for variational inequalities for the Laplacian (p = 2) in perforated media depending on whether the perforations are placed over the whole domain or along a manifold. We mention [17] for an extensive bibliography on variational inequalities in homogenization problems.…”

Section: The Homogenized Problemsmentioning

confidence: 96%

“…We mention [17] for an extensive bibliography on variational inequalities in homogenization problems. Also, [21] should be mentioned as the first work in the literature where a nonlinear strange term appears defined implicitly via a functional equation, and [24,25] as works which consider for the first time homogenization problems for the Laplace operator and semilinear boundary conditions leaving as an open question the most critical case (namely, the one homologous to the big point in Figure 2 when p = 2), problem which remained unsolved for a long time even for the Laplace operator (cf.…”

Section: The Homogenized Problemsmentioning

confidence: 99%

Unilateral problems for thep-Laplace operator in perforated media involving large parameters

et al. 2018

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Abstract. We address homogenization problems for variational inequalities issue from unilateral constraints for the p-Laplacian posed in perforated domains of R n , with n ≥ 3 and p ∈ [2, n]. ε is a small parameter which measures the periodicity of the structure while aε ε measures the size of the perforations. We impose constraints for solutions and their fluxes (associated with the p-Laplacian) on the boundary of the perforations. These constraints imply that the solution is positive and that the flux is bounded from above by a negative, nonlinear monotonic function of the solution multiplied by a parameter βε which may be very large, namely, βε → ∞ as ε → 0. We first consider the case where p < n and the domains periodically perforated by tiny balls and we obtain homogenized problems depending on the relations between the different parameters of the problem: p, n, ε, aε and βε. Critical relations for parameters are obtained which mark important changes in the behavior of the solutions. Correctors which provide improved convergence are also computed. Then, we extend the results for p = n and the case of non periodically distributed isoperimetric perforations. We make it clear that in the averaged constants of the problem, the perimeter of the perforations appears for any shape.

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Averaging of variational inequalities for the Laplacian with nonlinear restrictions along manifolds

Cited by 21 publications

References 13 publications

Homogenization of Variational Inequalities for the p-Laplace Operator in Perforated Media Along Manifolds

Homogenization of Variational Inequalities for the p-Laplace Operator in Perforated Media Along Manifolds

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

Unilateral problems for thep-Laplace operator in perforated media involving large parameters

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