Which sequences of holes are admissible for periodic homogenization with Neumann boundary condition?

¹ Laboratoire d'Analyse et de Mathématiques Appliquée, UMR 8050 du CNRS, Universités de Marne-la-Vallée et Paris 12 Val-de-Marne, Université Paris 12, 94010 Créteil Cedex France; damlamian@univ-paris12.fr.
² Laboratoire de Mathématiques Raphaël Salem, Université de Rouen, Site Colbert, 76821 Mont-Saint-Aignan Cedex, France; donato@univ-rouen.fr.
³ Université Paris VI, Laboratoire Jacques-Louis Lions, Boîte Courrier 187, 75252 Paris Cedex, France; donato@ann.jussieu.fr.

Received: 17 December 2001

Abstract

In this paper we give a general presentation of the homogenization of Neumann type problems in periodically perforated domains, including the case where the shape of the reference hole varies with the size of the period (in the spirit of the construction of self-similar fractals). We shows that H⁰-convergence holds under the extra assumption that there exists a bounded sequence of extension operators for the reference holes. The general class of Jones-domains gives an example where this result applies. When this assumption fails, another approach, using the Poincaré–Wirtinger inequality is presented. A corresponding class where it applies is that of John-domains, for which the Poincaré–Wirtinger constant is controlled. The relationship between these two kinds of assumptions is also clarified.