caa a22 4500 445843802 CHVBK 20180317145352.0 cr unu---uuuuu 170323e20110901xx s 000 0 eng 10.1007/s00526-010-0387-2 doi (NATIONALLICENCE)springer-10.1007/s00526-010-0387-2 Homogenization of the Neumann problem in perforated domains: an alternative approach [Elektronische Daten] [Marco Barchiesi, Matteo Focardi] 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 (−1/2, 1/2) n -periodic set $${P\subseteq\mathbb{R}^n}$$ , consider for any ε>0 the perforated domain Ω ε :=Ω∩ε 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 etal. (Math Models Methods Appl Sci 19:2065-2100, 2009) and Cagnetti and Scardia (J Math Pures Appl (9), to appear). Among the arguments we introduce, we provide a localized version of the Poincaré-Wirtinger inequality in SBV. As a possible application we study the asymptotic behavior of a brittle porous material represented by the perforated domain Ω ε . Finally, we slightly extend the well-known homogenization theorem for Sobolev energies on perforated domains. Springer-Verlag, 2010 Barchiesi Marco BCAM, Bizkaia Technology Park, Building 500, 48160, Derio, Spain aut Focardi Matteo Dipartimento di Matematica "U. Dini”, Università di Firenze, Viale Morgagni 67/A, 50134, Firenze, Italy aut Calculus of Variations and Partial Differential Equations Springer-Verlag 42/1-2(2011-09-01), 257-288 0944-2669 42:1-2<257 2011 42 526 https://doi.org/10.1007/s00526-010-0387-2 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00526-010-0387-2 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Barchiesi Marco BCAM, Bizkaia Technology Park, Building 500, 48160, Derio, Spain aut NATIONALLICENCE 700 1- Focardi Matteo Dipartimento di Matematica "U. Dini”, Università di Firenze, Viale Morgagni 67/A, 50134, Firenze, Italy aut NATIONALLICENCE 773 0- Calculus of Variations and Partial Differential Equations Springer-Verlag 42/1-2(2011-09-01), 257-288 0944-2669 42:1-2<257 2011 42 526 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer