June  2010, 5(2): 189-215. doi: 10.3934/nhm.2010.5.189

Homogenization of variational functionals with nonstandard growth in perforated domains

1. 

Laboratoire de Mathématiques et de leurs Applications, CNRS-UMR 5142, Université de Pau, Av. de l’Université, 64000 Pau, France

2. 

Department of Mathematics, B.Verkin Institute for Low Temperature Physics and Engineering, 47, av. Lenin, 61103, Kharkov, Ukraine

3. 

Narvik University College, Postbox 385, Narvik, 8505, Norway

Received  November 2009 Revised  February 2010 Published  May 2010

The aim of the paper is to study the asymptotic behavior of solutions to a Neumann boundary value problem for a nonlinear elliptic equation with nonstandard growth condition of the form

-div(|$\nabla$ uε | pε (x)-2 $\nabla$ uε )+ (| uε | pε (x)-2 uε = f(x)

in a perforated domain Ωε , ε being a small parameter that characterizes the microscopic length scale of the microstructure. Under the assumption that the functions pε(x) converge uniformly to a limit function $p_0(x)$ and that $p_0$ satisfy certain logarithmic uniform continuity condition, it is shown that uε converges, as ε$ \to 0$, to a solution of homogenized equation whose coefficients are calculated in terms of local energy characteristics of the domain Ωε . This result is then illustrated with periodic and locally periodic examples.

Citation: Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. Homogenization of variational functionals with nonstandard growth in perforated domains. Networks & Heterogeneous Media, 2010, 5 (2) : 189-215. doi: 10.3934/nhm.2010.5.189
[1]

Gregory A. Chechkin, Tatiana P. Chechkina, Ciro D’Apice, Umberto De Maio. Homogenization in domains randomly perforated along the boundary. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 713-730. doi: 10.3934/dcdsb.2009.12.713

[2]

Hakima Bessaih, Yalchin Efendiev, Florin Maris. Homogenization of the evolution Stokes equation in a perforated domain with a stochastic Fourier boundary condition. Networks & Heterogeneous Media, 2015, 10 (2) : 343-367. doi: 10.3934/nhm.2015.10.343

[3]

Patrizia Donato, Florian Gaveau. Homogenization and correctors for the wave equation in non periodic perforated domains. Networks & Heterogeneous Media, 2008, 3 (1) : 97-124. doi: 10.3934/nhm.2008.3.97

[4]

Leonid Berlyand, Petru Mironescu. Two-parameter homogenization for a Ginzburg-Landau problem in a perforated domain. Networks & Heterogeneous Media, 2008, 3 (3) : 461-487. doi: 10.3934/nhm.2008.3.461

[5]

Mamadou Sango. Homogenization of the Neumann problem for a quasilinear elliptic equation in a perforated domain. Networks & Heterogeneous Media, 2010, 5 (2) : 361-384. doi: 10.3934/nhm.2010.5.361

[6]

Valeria Chiado Piat, Sergey S. Nazarov, Andrey Piatnitski. Steklov problems in perforated domains with a coefficient of indefinite sign. Networks & Heterogeneous Media, 2012, 7 (1) : 151-178. doi: 10.3934/nhm.2012.7.151

[7]

Li-Ming Yeh. Pointwise estimate for elliptic equations in periodic perforated domains. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1961-1986. doi: 10.3934/cpaa.2015.14.1961

[8]

T. A. Shaposhnikova, M. N. Zubova. Homogenization problem for a parabolic variational inequality with constraints on subsets situated on the boundary of the domain. Networks & Heterogeneous Media, 2008, 3 (3) : 675-689. doi: 10.3934/nhm.2008.3.675

[9]

Dorina Mitrea, Marius Mitrea, Sylvie Monniaux. The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1295-1333. doi: 10.3934/cpaa.2008.7.1295

[10]

Tomasz Adamowicz, Przemysław Górka. The Liouville theorems for elliptic equations with nonstandard growth. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2377-2392. doi: 10.3934/cpaa.2015.14.2377

[11]

Patrick Winkert, Rico Zacher. A priori bounds for weak solutions to elliptic equations with nonstandard growth. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 865-878. doi: 10.3934/dcdss.2012.5.865

[12]

VicenŢiu D. RǍdulescu, Somayeh Saiedinezhad. A nonlinear eigenvalue problem with $ p(x) $-growth and generalized Robin boundary value condition. Communications on Pure & Applied Analysis, 2018, 17 (1) : 39-52. doi: 10.3934/cpaa.2018003

[13]

Fanghua Lin, Xiaodong Yan. A type of homogenization problem. Discrete & Continuous Dynamical Systems - A, 2003, 9 (1) : 1-30. doi: 10.3934/dcds.2003.9.1

[14]

Paulo Cesar Carrião, Olimpio Hiroshi Miyagaki. On a class of variational systems in unbounded domains. Conference Publications, 2001, 2001 (Special) : 74-79. doi: 10.3934/proc.2001.2001.74

[15]

Stanislav Antontsev, Michel Chipot, Sergey Shmarev. Uniqueness and comparison theorems for solutions of doubly nonlinear parabolic equations with nonstandard growth conditions. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1527-1546. doi: 10.3934/cpaa.2013.12.1527

[16]

Renata Bunoiu, Claudia Timofte. Homogenization of a thermal problem with flux jump. Networks & Heterogeneous Media, 2016, 11 (4) : 545-562. doi: 10.3934/nhm.2016009

[17]

Sunghan Kim, Ki-Ahm Lee, Henrik Shahgholian. Homogenization of the boundary value for the Dirichlet problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 6843-6864. doi: 10.3934/dcds.2019234

[18]

Nicolas Forcadel, Wilfredo Salazar, Mamdouh Zaydan. Specified homogenization of a discrete traffic model leading to an effective junction condition. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2173-2206. doi: 10.3934/cpaa.2018104

[19]

Michel Chipot, Aleksandar Mojsic, Prosenjit Roy. On some variational problems set on domains tending to infinity. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3603-3621. doi: 10.3934/dcds.2016.36.3603

[20]

Hang-Chin Lai, Jin-Chirng Lee, Shuh-Jye Chern. A variational problem and optimal control. Journal of Industrial & Management Optimization, 2011, 7 (4) : 967-975. doi: 10.3934/jimo.2011.7.967

2018 Impact Factor: 0.871

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (2)

[Back to Top]