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 ${H}^{0}$-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.

Keywords: periodic homogenization, perforated domains, $H^0$-convergence, Poincaré-Wirtinger inequality, Jones domains, John domains

@article{COCV_2002__8__555_0, author = {Damlamian, Alain and Donato, Patrizia}, title = {Which sequences of holes are admissible for periodic homogenization with {Neumann} boundary condition?}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {555--585}, publisher = {EDP-Sciences}, volume = {8}, year = {2002}, doi = {10.1051/cocv:2002046}, mrnumber = {1932963}, zbl = {1073.35020}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv:2002046/} }

TY - JOUR AU - Damlamian, Alain AU - Donato, Patrizia TI - Which sequences of holes are admissible for periodic homogenization with Neumann boundary condition? JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2002 SP - 555 EP - 585 VL - 8 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2002046/ DO - 10.1051/cocv:2002046 LA - en ID - COCV_2002__8__555_0 ER -

%0 Journal Article %A Damlamian, Alain %A Donato, Patrizia %T Which sequences of holes are admissible for periodic homogenization with Neumann boundary condition? %J ESAIM: Control, Optimisation and Calculus of Variations %D 2002 %P 555-585 %V 8 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2002046/ %R 10.1051/cocv:2002046 %G en %F COCV_2002__8__555_0

Damlamian, Alain; Donato, Patrizia. Which sequences of holes are admissible for periodic homogenization with Neumann boundary condition?. ESAIM: Control, Optimisation and Calculus of Variations, Volume 8 (2002), pp. 555-585. doi : 10.1051/cocv:2002046. http://www.numdam.org/articles/10.1051/cocv:2002046/

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