Application of Uniform Distribution to Homogenization of a Thin Obstacle Problem with<i>p</i> − Laplacian

Karakhanyan, Aram; Strömqvist, Martin

doi:10.1080/03605302.2014.895013

Cited by 5 publications

(13 citation statements)

References 9 publications

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“…Once we have established the existence of these correctors, the proof of the Theorem 3 is identical to the planar case treated in [4].…”

Section: Correctorsmentioning

confidence: 88%

“…The proof of Theorem 3 from H 1 -H 3 is given in section 4 of [4] when is a hyper plane, and remains the same for the present case when is a convex surface.…”

Section: Proof Of Theoremmentioning

confidence: 89%

“…they are translates of P and the translates have the same distribution. Using the proof of Lemma 4 of [4], we conclude that…”

Section: Proof Of Theoremmentioning

confidence: 89%

“…These are the assumptions required for using the framework from [4], though the (A 4 ) is stricter here.…”

Section: Letmentioning

confidence: 99%

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Estimates for capacity and discrepancy of convex surfaces in sieve-like domains with an application to homogenization

Karakhanyan

Strömqvist

2016

Calc. Var.

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We consider the intersection of a convex surface with a periodic perforation of R d , which looks like a sieve, given by T ε = k∈Z d {εk + a ε T } where T is a given compact set and a ε ε is the size of the perforation in the ε-When ε tends to zero we establish uniform estimates for p-capacity, 1 < p < d, of the set ∩ T ε . Additionally, we prove that the intersections ∩ {εk + a ε T } k are uniformly distributed over and give estimates for the discrepancy of the distribution. As an application we show that the thin obstacle problem with the obstacle defined on the intersection of and the perforations, in a given bounded domain, is homogenizable when p < 1 +

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“…Once we have established the existence of these correctors, the proof of the Theorem 3 is identical to the planar case treated in [4].…”

Section: Correctorsmentioning

confidence: 88%

“…The proof of Theorem 3 from H 1 -H 3 is given in section 4 of [4] when is a hyper plane, and remains the same for the present case when is a convex surface.…”

Section: Proof Of Theoremmentioning

confidence: 89%

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Estimates for capacity and discrepancy of convex surfaces in sieve-like domains with an application to homogenization

Karakhanyan

Strömqvist

2016

Calc. Var.

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“…Note that, in this paper, we give results for the p-Laplacian on perforated domains, by tiny cavities, with constraints for solutions and their normal derivatives on the boundary of the cavities, which are completly new in the literature. Dealing with unilateral constraints for the p-Laplacian and the homogenization of perforated media, we mention very different problems and results in [36] for Signorini conditions (when α = 1) and [26,41] for obstacle problems. For different constraints and sizes of perforations, we provide a map of all possible homogenized problems and construct the corresponding correctors (see Figures 2 and 3).…”

Section: Final Commentsmentioning

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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A Basic Homogenization Problem for the p-Laplacian in $$\mathbb {R}^d$$ Perforated along a Sphere: $$L^\infty $$ Estimates

Gordon,

Nazarov,

Peres

2024

Potential Anal

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Application of Uniform Distribution to Homogenization of a Thin Obstacle Problem withp − Laplacian

Cited by 5 publications

References 9 publications

Estimates for capacity and discrepancy of convex surfaces in sieve-like domains with an application to homogenization

Estimates for capacity and discrepancy of convex surfaces in sieve-like domains with an application to homogenization

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

A Basic Homogenization Problem for the p-Laplacian in $$\mathbb {R}^d$$ Perforated along a Sphere: $$L^\infty $$ Estimates

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