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Γ-convergence approach to variational problems in perforated domains with Fourier boundary conditions

Published online by Cambridge University Press:  19 December 2008

Valeria Chiadò Piat  and
Andrey Piatnitski
Valeria Chiadò Piat
Affiliation:
Dipartimento di Matematica, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129 Torino, Italy.
Andrey Piatnitski
Affiliation:
Narvik University College, HiN, Postbox 385, 8505, Narvik, Norway and Lebedev Physical Institute RAS, Leninski prospect 53, Moscow 119991, Russia. andrey@sci.lebedev.ru
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Abstract

The work focuses on the Γ-convergence problem and the convergence of minimizers for a functional defined in a periodic perforated medium and combining the bulk (volume distributed) energy and the surface energy distributed on the perforation boundary. It is assumed that the mean value of surface energy at each level set of test function is equal to zero. Under natural coercivity and p-growth assumptions on the bulk energy, and the assumption that the surface energy satisfies p-growth upper bound, we show that the studied functional has a nontrivial Γ-limit and the corresponding variational problem admits homogenization.


Keywords

HomogenizationΓ-convergenceperforated medium
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 1 , January 2010 , pp. 148 - 175
Copyright
© EDP Sciences, SMAI, 2008

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References

Acerbi, E., Chiadò Piat, V., Dal Maso, G. and Percivale, D., An extension theorem from connected sets, and homogenization in general periodic domains. Nonlinear Anal. 18 (1992) 481–496. CrossRef
R.A. Adams, Sobolev spaces. Academic Press, New York (1975).
Belyaev, A.G., Chechkin, G.A. and Piatnitski, A.L., Asymptotic behavior of a solution to a boundary value problem in a perforated domain with oscillating boundary. Sib. Math. J. 39 (1998) 621–644. CrossRef
Belyaev, A.G., Chechkin, G.A. and Piatnitski, A.L., Homogenization of second-order elliptic operators in a perforated domain with oscillating Fourier boundary conditions. Sb. Math. 192 (2001) 933–949. CrossRef
A. Braides and A. Defranceschi, Homogenization of multiple integrals, Oxford Lecture Series in Mathematics and its Applications 12. The Clarendon Press, Oxford University Press, New York (1998).
Brillard, A., Asymptotic analysis of two elliptic equations with oscillating terms. RAIRO Modél. Math. Anal. Numér. 22 (1988) 187–216. CrossRef
D. Cioranescu and P. Donato, On a Robin problem in perforated domains, in Homogenization and applications to material sciences, D. Cioranescu et al. Eds., GAKUTO International Series, Mathematical Sciences and Applications 9, Tokyo, Gakkotosho (1997) 123–135.
D. Cioranescu and F. Murat, A strange term coming from nowhere, in Topics in the Mathematical Modelling of Composite Materials, Progr. Nonlinear Differential Equations Appl. 31, Birkhauser, Boston (1997) 45–93.
Cioranescu, D. and Saint Jean Paulin, J., Homogenization in open sets with holes. J. Math. Anal. Appl. 71 (1979) 590–607. CrossRef
D. Cioranescu and J. Saint Jean Paulin, Truss structures: Fourier conditions and eigenvalue problems, in Boundary control and boundary variation, J.P. Zolezio Ed., Lecture Notes Control Inf. Sci. 178, Springer-Verlag (1992) 125–141.
Conca, C., On the application of the homogenization theory to a class of problems arising in fluid mechanics. J. Math. Pures Appl. 64 (1985) 31–75.
G. Dal Maso, An introduction to Γ-convergence. Birkhauser, Boston (1993).
V.A. Marchenko and E.Y. Khruslov, Boundary value problems in domains with fine-grained boundaries. Naukova Dumka, Kiev (1974).
V.A. Marchenko and E.Y. Khruslov, Homogenization of partial differential equations. Birkhauser (2006).
Oleinik, O.A. and Shaposhnikova, T.A., On an averaging problem in a partially punctured domain with a boundary condition of mixed type on the boundary of the holes, containing a small parameter. Differ. Uravn. 31 (1995) 1150–1160, 1268. Translation in Differ. Equ. 31 (1995) 1086–1098.
Oleinik, O.A. and Shaposhnikova, T.A., On the homogenization of the Poisson equation in partially perforated domains with arbitrary density of cavities and mixed type conditions on their boundary. Rend. Mat. Acc. Linceis. IX 7 (1996) 129–146.
Pastukhova, S.E., Tartar's compensated compactness method in the averaging of the spectrum of a mixed problem for an elliptic equation in a punctured domain with a third boundary condition. Sb. Math. 186 (1995) 753–770. CrossRef
Pastukhova, S.E., On the character of the distribution of the temperature field in a perforated body with a given value on the outer boundary under heat exchange conditions on the boundary of the cavities that are in accord with Newton's law. Sb. Math. 187 (1996) 869–880. CrossRef
Pastukhova, S.E., Spectral asymptotics for a stationary heat conduction problem in a perforated domain. Mat. Zametki 69 (2001) 600–612 [in Russian]. Translation in Math. Notes 69 (2001) 546–558. CrossRef
W.P. Ziemer, Weakly differentiable functions. Springer-Verlag, New York (1989).