Γ-convergence approach to variational problems in perforated domains with Fourier boundary conditions
Valeria Chiadò Piat; Andrey Piatnitski
ESAIM: Control, Optimisation and Calculus of Variations (2010)
- Volume: 16, Issue: 1, page 148-175
- ISSN: 1292-8119
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topChiadò Piat, Valeria, and Piatnitski, Andrey. "Γ-convergence approach to variational problems in perforated domains with Fourier boundary conditions." ESAIM: Control, Optimisation and Calculus of Variations 16.1 (2010): 148-175. <http://eudml.org/doc/250721>.
@article{ChiadòPiat2010,
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.
},
author = {Chiadò Piat, Valeria, Piatnitski, Andrey},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Homogenization; Γ-convergence; perforated medium; periodic homogenization; surface term; bulk term},
language = {eng},
month = {1},
number = {1},
pages = {148-175},
publisher = {EDP Sciences},
title = {Γ-convergence approach to variational problems in perforated domains with Fourier boundary conditions},
url = {http://eudml.org/doc/250721},
volume = {16},
year = {2010},
}
TY - JOUR
AU - Chiadò Piat, Valeria
AU - Piatnitski, Andrey
TI - Γ-convergence approach to variational problems in perforated domains with Fourier boundary conditions
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/1//
PB - EDP Sciences
VL - 16
IS - 1
SP - 148
EP - 175
AB -
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.
LA - eng
KW - Homogenization; Γ-convergence; perforated medium; periodic homogenization; surface term; bulk term
UR - http://eudml.org/doc/250721
ER -
References
top- E. Acerbi, V. Chiadò Piat, G. Dal Maso and D. Percivale, An extension theorem from connected sets, and homogenization in general periodic domains. Nonlinear Anal.18 (1992) 481–496.
- R.A. Adams, Sobolev spaces. Academic Press, New York (1975).
- A.G. Belyaev, G.A. Chechkin and A.L. Piatnitski, Asymptotic behavior of a solution to a boundary value problem in a perforated domain with oscillating boundary. Sib. Math. J.39 (1998) 621–644.
- A.G. Belyaev, G.A. Chechkin and A.L. Piatnitski, Homogenization of second-order elliptic operators in a perforated domain with oscillating Fourier boundary conditions. Sb. Math.192 (2001) 933–949.
- A. Braides and A. Defranceschi, Homogenization of multiple integrals, Oxford Lecture Series in Mathematics and its Applications12. The Clarendon Press, Oxford University Press, New York (1998).
- A. Brillard, Asymptotic analysis of two elliptic equations with oscillating terms. RAIRO Modél. Math. Anal. Numér.22 (1988) 187–216.
- 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 Applications9, 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.
- D. Cioranescu and J. Saint Jean Paulin, Homogenization in open sets with holes. J. Math. Anal. Appl.71 (1979) 590–607.
- 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.
- C. Conca, 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).
- O.A. Oleinik and T.A. Shaposhnikova, 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.
- O.A. Oleinik and T.A. Shaposhnikova, 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. IX7 (1996) 129–146.
- S.E. Pastukhova, 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.
- S.E. Pastukhova, 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.
- S.E. Pastukhova, Spectral asymptotics for a stationary heat conduction problem in a perforated domain. Mat. Zametki69 (2001) 600–612 [in Russian]. Translation in Math. Notes69 (2001) 546–558.
- W.P. Ziemer, Weakly differentiable functions. Springer-Verlag, New York (1989).
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