Existence, uniqueness and homogenization results for a class of nonlinear PDE in perforated domains

par Bituin C. Cabarrubias

Thèse de doctorat en Mathématiques. Equations aux dérivées partielles

Sous la direction de Patrizia Donato et de Marian P. Roque.


  • Résumé

    This thesis is devoted to the existence, uniqueness and homogenization results for a quasilinear elliptic problem with oscillating coefficients and with nonlinear Robin boundary condition in a periodically perforated domain. A suitable frowth conditions are assumed on the nonlinear boundary term and on the quasilinear term, some assumptions on the modulus of continuity introduced in Chipot [17] and weaker than a Lipschitz condition, are prescribed. For the existence and uniqueness of a solution, we consider a more general framework which, in particular, will imply the existence and uniqueness of the solution of the problem. To deal with the existence of a solution, we prove first the weak continuity of the boundary nonlinear operator which is a difficult part. Together with this property, we use the Schauder's Fixed Point Theorem to show the existence. For the uniqueness, we adapt to our situation some arguments introduced in André-Chipot [5] (see also chapter 11 of [17] for Dirichlet conditions) and partially extended to linear Robin conditions in Bendib-Tcheugoué Tébou [11] and Bendib [10]. For the homogenization of the problem, we study the convergence to a limit problem using the Periodic Unfolding Method in perforated domains. Here, we proved related properties of the onfolding operators which are needed in the process. We also show the well-posedness of the limit system by proving that the homogenized operator inherits the modulus of continuity of the initial problem. As a consequence, the uniqueness of a solution of the homogenized quasilinear problem follows. A corrector result is also obtained using this method.

Consulter en bibliothèque

La version de soutenance existe sous forme papier

Informations

  • Détails : 1 vol. (105 p.)
  • Notes : Publication autorisée par le jury
  • Annexes : Bibliogr. 52 références

Où se trouve cette thèse\u00a0?

  • Bibliothèque : Université de Rouen Normandie. Service commun de la documentation. Section Sciences et Techniques (site du Madrillet).
  • Disponible pour le PEB
  • Cote : 12/ROUE/S046
  • Bibliothèque : Laboratoire de mathématiques Raphae͏̈l Salem. Bibliothèque de recherche en mathématiques.
  • Disponible pour le PEB
  • Cote : &Thèses CAB 17794
Voir dans le Sudoc, catalogue collectif des bibliothèques de l'enseignement supérieur et de la recherche.