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: homogenization, Γ-convergence, perforated medium

@article{COCV_2010__16_1_148_0, author = {Chiad\`o Piat, Valeria and Piatnitski, Andrey}, title = {$\Gamma $-convergence approach to variational problems in perforated domains with {Fourier} boundary conditions}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {148--175}, publisher = {EDP-Sciences}, volume = {16}, number = {1}, year = {2010}, doi = {10.1051/cocv:2008073}, mrnumber = {2598093}, zbl = {1188.35015}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv:2008073/} }

TY - JOUR AU - Chiadò Piat, Valeria AU - Piatnitski, Andrey TI - $\Gamma $-convergence approach to variational problems in perforated domains with Fourier boundary conditions JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2010 SP - 148 EP - 175 VL - 16 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2008073/ DO - 10.1051/cocv:2008073 LA - en ID - COCV_2010__16_1_148_0 ER -

%0 Journal Article %A Chiadò Piat, Valeria %A Piatnitski, Andrey %T $\Gamma $-convergence approach to variational problems in perforated domains with Fourier boundary conditions %J ESAIM: Control, Optimisation and Calculus of Variations %D 2010 %P 148-175 %V 16 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv:2008073/ %R 10.1051/cocv:2008073 %G en %F COCV_2010__16_1_148_0

Chiadò Piat, Valeria; Piatnitski, Andrey. $\Gamma $-convergence approach to variational problems in perforated domains with Fourier boundary conditions. ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 1, pp. 148-175. doi : 10.1051/cocv:2008073. http://archive.numdam.org/articles/10.1051/cocv:2008073/

[1] An extension theorem from connected sets, and homogenization in general periodic domains. Nonlinear Anal. 18 (1992) 481-496. | Zbl

, , and ,[2] Sobolev spaces. Academic Press, New York (1975). | Zbl

,[3] Asymptotic behavior of a solution to a boundary value problem in a perforated domain with oscillating boundary. Sib. Math. J. 39 (1998) 621-644. | Zbl

, and ,[4] Homogenization of second-order elliptic operators in a perforated domain with oscillating Fourier boundary conditions. Sb. Math. 192 (2001) 933-949. | Zbl

, and ,[5] Homogenization of multiple integrals, Oxford Lecture Series in Mathematics and its Applications 12. The Clarendon Press, Oxford University Press, New York (1998). | Zbl

and ,[6] Asymptotic analysis of two elliptic equations with oscillating terms. RAIRO Modél. Math. Anal. Numér. 22 (1988) 187-216. | Numdam | Zbl

,[7] 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. | Zbl

and ,[8] 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. | Zbl

and ,[9] Homogenization in open sets with holes. J. Math. Anal. Appl. 71 (1979) 590-607. | Zbl

and ,[10] 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. | Zbl

and ,[11] On the application of the homogenization theory to a class of problems arising in fluid mechanics. J. Math. Pures Appl. 64 (1985) 31-75. | Zbl

,[12] An introduction to Γ-convergence. Birkhauser, Boston (1993). | Zbl

,[13] Boundary value problems in domains with fine-grained boundaries. Naukova Dumka, Kiev (1974). | Zbl

and ,[14] Homogenization of partial differential equations. Birkhauser (2006). | Zbl

and ,[15] 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. | Zbl

and ,[16] 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. | Zbl

and ,[17] 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. | Zbl

,[18] 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. | Zbl

,[19] 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. | Zbl

,[20] Weakly differentiable functions. Springer-Verlag, New York (1989). | Zbl

,*Cited by Sources: *