Coupled and uncoupled limits for a N -dimensional multidomain Neumann problem

Gaudiello, Antonio; Gustafsson, Björn; Lefter, Cătălin; Mossino, Jacqueline

doi:10.1016/s0764-4442(00)00311-6

Cited by 4 publications

(8 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A preliminary version of these results, concerning the model problem, but including oscillating coefficients, was published in [12] with sketch of proofs.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic analysis of a class of minimization problems in a thin multidomain

Gaudiello

Gustafsson

Lefter

et al. 2002

Calculus of Variations and Partial Differential Equations

Self Cite

View full text Add to dashboard Cite

We consider a quasilinear Neumann problem with exponent p ∈]1, +∞[, in a multidomain of R N , N ≥ 2, consisting of two vertical cylinders, one placed upon the other: the first one with given height and small cross section, the other one with small height and given cross section. Assuming that the volumes of the two cylinders tend to zero with same rate, we prove that the limit problem is well posed in the union of the limit domains, with respective dimension 1 and N − 1. Moreover, this limit problem is coupled if p > N − 1 and uncoupled if 1 < p ≤ N − 1. IntroductionLet N ≥ 2, let ω ⊂ R N −1 be a bounded open connected set with a smooth boundary such that the origin in R N −1 , denoted by 0 , belongs to ω, and let {r n } n∈N , {h n } n∈N be two sequences of positive numbers converging to 0. For every n ∈ N, consider the thin multidomain Ω n = Ω 1 n ∪ Ω 2 n , the union of two vertical cylinders with small volumes: Ω 1 n = r n ω × [0, 1[ with small cross section r n ω and constant height, Ω 2 n = ω×] − h n , 0[ with small height h n and constant cross section (see figure next page). This paper arises from the desire of studying the asymptotic behaviour, as n → +∞, of the following model problem:

show abstract

“…A preliminary version of these results, concerning the model problem, but including oscillating coefficients, was published in [12] with sketch of proofs.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic analysis of a class of minimization problems in a thin multidomain

Gaudiello

Gustafsson

Lefter

et al. 2002

Calculus of Variations and Partial Differential Equations

Self Cite

View full text Add to dashboard Cite

show abstract

“…Then, by making use of the method of oscillating test functions, introduced by L. Tartar in [17], by applying some results obtained by A. Gaudiello, B. Gustafsson, C. Lefter and J. Mossino in [6] and [7] and by adapting the techniques used by M. Vanninathan in [16], we derive the limit eigenvalue problem and the limit of the rescaled basis, as n → +∞, in the case h n ≃ r For the study of thin multi-structures we refer to [2], [3], [4], [11], [12], [13], [14] and the references quoted therein. For a thin multi-structure as considered in this paper, we refer to [5], [6], [7], [8], [9] and [10]. For the study of the spectrum of the Laplace operator in a thin tube with a Dirichlet condition on its boundary we refer to [1].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…6.2-1 and Th. 6.2-2 in [15]) for every n ∈ N, there exists an increasing diverging sequence of positive numbers {λ n,k } k∈N and a H n -Hilbert orthonormal basis {u n,k } k∈N , such that {λ n,k } k∈N forms the set of all the eigenvalues of the following problem: 5) and, for every k ∈ N, u n,k ∈ V n is an eigenvector of (2.5) with eigenvalue λ n,k . Moreover,…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Asymptotic analysis of the eigenvalues of a Laplacian problem in a thin multidomain

Gaudiello¹,

Sili²

2007

Indiana Univ. Math. J.

Self Cite

View full text Add to dashboard Cite

We consider a thin multidomain of R N , N ≥ 2, consisting of two vertical cylinders, one placed upon the other: the first one with given height and small cross section, the second one with small thickness and given cross section. In this multidomain we study the asymptotic behavior, when the volumes of the two cylinders vanish, of a Laplacian eigenvalue problem and of a L 2 -Hilbert orthonormal basis of eigenvectors. We derive the limit eigenvalue problem (which is well posed in the union of the limit domains, with respective dimension 1 and N − 1) and the limit basis. We discuss the limit models and we precise how these limits depend on the dimension N and on limit q of the ratio between the volumes of the two cylinders.

show abstract

“…The proofs of these results make use of the main ideas of Γ-convergence method introduced by E. De Giorgi (see [10]) and they develop in several steps: a priori estimates, construction of the recovery sequence, density results and l.s.c arguments (see Subsection 2.2). The main difficulty with respect to [11], where the asymptotic behavior of the Laplacian is studied when h n r 2 n , arises from the fact that the set of the admissible vector valued functions of Problem (1.1) is not a convex set, due to the constraint |V ((x 1 ,x 2 ,x 3 ))| = 1. This difficulty is overcome by working with a projection from R 3 into S 2 = {(x 1 ,x 2 ,x 3 ) ∈ R 3 : |(x 1 ,x 2 ,x 3 )| =1}, introduced in [4] (see also [1]), and by using the Sard's Lemma.…”

Section: Introductionmentioning

confidence: 99%

“…This difficulty is overcome by working with a projection from R 3 into S 2 = {(x 1 ,x 2 ,x 3 ) ∈ R 3 : |(x 1 ,x 2 ,x 3 )| =1}, introduced in [4] (see also [1]), and by using the Sard's Lemma. Moreover, point out that the cases h n << r 2 n and h n >> r 2 n are not treated in [11]. Remark that it is not necessary that the two cylinders are scaled to the same one or that the first cylinder has height 1.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic analysis, in a thin multidomain, of minimizing maps with values in \( S^{2} \)

Gaudiello

Hadiji

2009

Ann. Inst. H. Poincaré C Anal. Non Linéaire

Self Cite

View full text Add to dashboard Cite

We consider a thin multidomain of R 3 consisting of two vertical cylinders, one placed upon the other: the first one with given height and small cross section, the second one with small thickness and given cross section. The first part of this paper is devoted to analyze, in this thin multidomain, a "static Landau-Lifshitz equation", when the volumes of the two cylinders vanish. We derive the limit problem, which decomposes into two uncoupled problems, well posed on the limit cylinders (with dimensions 1 and 2, respectively). We precise how the limit problem depends on limit of the ratio between the volumes of the two cylinders. In the second part of this paper, we study the asymptotic behavior of the two limit problems, when the exterior limit fields increase. We show that in some cases, contrary to the initial problem, the energies of the limit problems diverge and we find the order of these energies.Résumé. Nous considérons un multi-domaine mince de R 3 se composant de deux cylindres verticaux, superposés l'un sur l'autre : le premier possède une taille donnée et une petite section transversale, le second a une petiteépaisseur et une section transversale donnée. La première partie de cet article est consacréeà analyser, dans ce multi-domaine, uneéquation stationnaire de type Landau-Lifshitz, quand les volumes des deux cylindres tendent vers 0. Nous montrons que le problème limite, se décompose en deux probèmes découplés, bien posés sur le domaine limite. Ensuite, nous précisons comment le problème limite dépend de la limite du rapport des volumes des deux cylindres. Dans la deuxième partie de cet article, nousétudions le comportement asymptotique des deux problèmes limites, quand les champs extérieurs limites augmentent. Nous prouvons que dans certains cas, contrairement au problème initial, lesénergies des problèmes limites divergent et nous précisons l'ordre de cesénergies.

show abstract

Coupled and uncoupled limits for a N -dimensional multidomain Neumann problem

Cited by 4 publications

References 1 publication

Asymptotic analysis of a class of minimization problems in a thin multidomain

Asymptotic analysis of a class of minimization problems in a thin multidomain

Asymptotic analysis of the eigenvalues of a Laplacian problem in a thin multidomain

Asymptotic analysis, in a thin multidomain, of minimizing maps with values in \( S^{2} \)

Contact Info

Product

Resources

About