Neumann problem for a quasilinear elliptic equation in a varying domain

Mamadou Sango

doi:10.1016/j.crma.2006.02.011

Partial Differential Equations

Neumann problem for a quasilinear elliptic equation in a varying domain
[Problème de Neumann pour une équation élliptique non lineaire dans un domaine perforé]

Présenté par : Philippe G. Ciarlet

Mamadou Sango ¹

¹ Department of Mathematics and Applied Mathematics, University of Pretoria/Mamelodi Campus, Pretoria 0002, South Africa

Comptes Rendus. Mathématique, Volume 342 (2006) no. 8, pp. 563-568.

Résumé
Abstract

Nous étudions le problème de Neumann pour un opérateur élliptique de type Leray–Lions dans un domaine $Ω^{(s)} = Ω \ F^{(s)}$ , $s = 1, 2, \dots$ , où Ω est un ouvert dans $R^{n}$ $(n ⩾ 3)$ , $F^{(s)}$ est un ensemble fermé situé au voisinage d'une variété differentiable Γ de dimension $(n - 1)$ à l'intérieur de Ω. Nous étudions the comportement asymptotique de $u^{(s)}$ quand $F^{(s)}$ converge vers Γ dans un sens approprié.

We investigate the Neumann problem for a nonlinear elliptic operator of Leray–Lions type in $Ω^{(s)} = Ω \ F^{(s)}$ , $s = 1, 2, \dots$ , where Ω is a domain in $R^{n}$ $(n ⩾ 3)$ , $F^{(s)}$ is a closed set located in the neighborhood of a $(n - 1)$ -dimensional manifold Γ lying inside Ω. We study the asymptotic behavior of $u^{(s)}$ as $s \to \infty$ , when the set $F^{(s)}$ tends to Γ.

Reçu le : 2005-04-13
Accepté le : 2006-02-07
Publié le : 2006-03-24

DOI : 10.1016/j.crma.2006.02.011

Affiliations des auteurs :

Mamadou Sango ¹

¹ Department of Mathematics and Applied Mathematics, University of Pretoria/Mamelodi Campus, Pretoria 0002, South Africa

@article{CRMATH_2006__342_8_563_0,
     author = {Mamadou Sango},
     title = {Neumann problem for a quasilinear elliptic equation in a varying domain},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {563--568},
     publisher = {Elsevier},
     volume = {342},
     number = {8},
     year = {2006},
     doi = {10.1016/j.crma.2006.02.011},
     language = {en},
}

TY  - JOUR
AU  - Mamadou Sango
TI  - Neumann problem for a quasilinear elliptic equation in a varying domain
JO  - Comptes Rendus. Mathématique
PY  - 2006
SP  - 563
EP  - 568
VL  - 342
IS  - 8
PB  - Elsevier
DO  - 10.1016/j.crma.2006.02.011
LA  - en
ID  - CRMATH_2006__342_8_563_0
ER  -

%0 Journal Article
%A Mamadou Sango
%T Neumann problem for a quasilinear elliptic equation in a varying domain
%J Comptes Rendus. Mathématique
%D 2006
%P 563-568
%V 342
%N 8
%I Elsevier
%R 10.1016/j.crma.2006.02.011
%G en
%F CRMATH_2006__342_8_563_0

Mamadou Sango. Neumann problem for a quasilinear elliptic equation in a varying domain. Comptes Rendus. Mathématique, Volume 342 (2006) no. 8, pp. 563-568. doi : 10.1016/j.crma.2006.02.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.02.011/

Bibliographie
Cité par

[1] L. Boccardo; F. Murat Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal., Volume 19 (1992) no. 6, pp. 581-597

[2] D. Cioranescu; S.J. Paulin Homogenization in open sets with holes, J. Math. Anal. Appl., Volume 71 (1979) no. 2, pp. 590-607

[3] A. Damlamian Le problème de la passoire de Neumann, Rend. Sem. Mat. Univ. Politec. Torino, Volume 43 (1985/86) no. 3, pp. 427-450

[4] A. Damlamian; P. Donato Which sequences of holes are admissible for periodic homogenization with Neumann boundary conditions, A tribute to J.L. Lions, ESAIM Control Optim. Calc. Var., Volume 8 (2002), pp. 555-585

[5] V.A. Marchenko; E.Ya. Khruslov Boundary Value Problems in Domains with a Fine-Grained Boundary, Naukova Dumka, Kiev, 1974 (in Russian)

[6] V.G. Mazya Sobolev Spaces, Springer-Verlag, New York, 1985

[7] F. Murat The Neumann sieve, Isola d'Elba, 1983 (Res. Notes in Math.), Volume vol. 127, Pitman, Boston, MA (1985), pp. 24-32

[8] O.A. Oleinik; A.S. Shamaev; G.A. Yosifian Mathematical Problems in Elasticity and Homogenization, North-Holland, Amsterdam, 1992

[9] E. Sanchez-Palencia Boundary value problems in domains containing perforated walls, Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar, vol. III, Paris, 1980/1981, Res. Notes in Math., vol. 70, Pitman, Boston, MA, London, 1982, pp. 309-325

Cité par Sources :

Commentaires - Politique