TITLE

# Homogenization of the Neumann problem in perforated domains: an alternative approach

AUTHOR(S)
Barchiesi, Marco; Focardi, Matteo
PUB. DATE
September 2011
SOURCE
Calculus of Variations & Partial Differential Equations;Sep2011, Vol. 42 Issue 1/2, p257
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
The main result of this paper is a compactness theorem for families of functions in the space SBV (Special functions of Bounded Variation) defined on periodically perforated domains. Given an open and bounded set $${\Omega\subseteq\mathbb{R}^n}$$, and an open, connected, and (âˆ’1/2, 1/2)-periodic set $${P\subseteq\mathbb{R}^n}$$, consider for any Îµ > 0 the perforated domain Î© := Î© âˆ© Îµ P. Let $${(u_\varepsilon)\subset SBV^p(\Omega_{\varepsilon})}$$, p > 1, be such that $${\int_{\Omega_{\varepsilon}}\left|{\nabla{u}_\varepsilon}\right|^pdx+\mathcal{H}^{n-1}(S_{u_\varepsilon}\,\cap\,\Omega_{\varepsilon}) +\left\Vert{u_\varepsilon}\right\Vert_{L^p(\Omega_{\varepsilon})}}$$ is bounded. Then, we prove that, up to a subsequence, there exists $${u\in GSBV^p\,\cap\, L^p(\Omega)}$$ satisfying $${\lim_\varepsilon\left\Vert{u-u_\varepsilon}\right\Vert_{L^1(\Omega_{\varepsilon})}=0}$$. Our analysis avoids the use of any extension procedure in SBV, weakens the hypotheses on P to the minimal ones and simplifies the proof of the results recently obtained in Focardi et al. (Math Models Methods Appl Sci 19:2065-2100, ) and Cagnetti and Scardia (J Math Pures Appl (9), ). Among the arguments we introduce, we provide a localized version of the PoincarÃ©-Wirtinger inequality in SBV. As a possible application we study the asymptotic behavior of a brittle porous material represented by the perforated domain Î©. Finally, we slightly extend the well-known homogenization theorem for Sobolev energies on perforated domains.
ACCESSION #
62869040

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