American Institute of Mathematical Sciences

Advanced
March  2009, 4(1): 127-152. doi: 10.3934/nhm.2009.4.127

Homogenization of interfacial energies and construction of plane-like minimizers in periodic media through a cell problem

Antonin Chambolle 1, and  Gilles Thouroude 2,

1. 

CMAP, Ecole Polytechnique, CNRS, 91128 Palaiseau, France
2. 

Ecole Normale Supérieure, 45 rue d'Ulm, 75230 Paris Cedex 05, France

Received  July 2008 Revised  January 2009 Published  February 2009

We consider the homogenization of a periodic interfacial energy, such as considered in recents papers by Caffarelli and De La Llave [14], or Dirr, Lucia and Novaga [16]. In particular, we include the case where an external forcing field (which is unbounded in the limit) is present, and suggest two different ways to take care of this additional perturbation. We provide a proof of a $\Gamma$-limit, however, we also observe that thanks to the coarea formula, in many cases such a result is already known in the framework of $BV$ homogenization. This leads to an interesting new construction for the plane-like minimizers in periodic media of Caffarelli and De La Llave, through a cell problem.
Keywords: Homogenization, Minimal Surfaces, BV functions..
Mathematics Subject Classification: 35B27, 74Q05, 49Q20, 53A1.
Citation: Antonin Chambolle, Gilles Thouroude. Homogenization of interfacial energies and construction of plane-like minimizers in periodic media through a cell problem. Networks & Heterogeneous Media, 2009, 4 (1) : 127-152. doi: 10.3934/nhm.2009.4.127
[1]

Stefano Bianchini, Daniela Tonon. A decomposition theorem for $BV$ functions. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1549-1566. doi: 10.3934/cpaa.2011.10.1549
[2]

José Ginés Espín Buendía, Daniel Peralta-salas, Gabriel Soler López. Existence of minimal flows on nonorientable surfaces. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4191-4211. doi: 10.3934/dcds.2017178
[3]

Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. I: Numerical tests and examples. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 41-74. doi: 10.3934/dcdsb.2010.14.41
[4]

Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. II: Analytical error estimates. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 75-109. doi: 10.3934/dcdsb.2010.14.75
[5]

Francisco Brito, Maria Luiza Leite, Vicente de Souza Neto. Liouville's formula under the viewpoint of minimal surfaces. Communications on Pure & Applied Analysis, 2004, 3 (1) : 41-51. doi: 10.3934/cpaa.2004.3.41
[6]

Eva Glasmachers, Gerhard Knieper, Carlos Ogouyandjou, Jan Philipp Schröder. Topological entropy of minimal geodesics and volume growth on surfaces. Journal of Modern Dynamics, 2014, 8 (1) : 75-91. doi: 10.3934/jmd.2014.8.75
[7]

Francesco Maggi, Salvatore Stuvard, Antonello Scardicchio. Soap films with gravity and almost-minimal surfaces. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-36. doi: 10.3934/dcds.2019236
[8]

Dominik Hafemeyer, Florian Mannel, Ira Neitzel, Boris Vexler. Finite element error estimates for one-dimensional elliptic optimal control by BV-functions. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2019041
[9]

Filippo Morabito. Singly periodic free boundary minimal surfaces in a solid cylinder of $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4987-5001. doi: 10.3934/dcds.2015.35.4987
[10]

Xi-Nan Ma, Jiang Ye, Yun-Hua Ye. Principal curvature estimates for the level sets of harmonic functions and minimal graphs in $R^3$. Communications on Pure & Applied Analysis, 2011, 10 (1) : 225-243. doi: 10.3934/cpaa.2011.10.225
[11]

Alberto Bressan, Wen Shen. BV estimates for multicomponent chromatography with relaxation. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 21-38. doi: 10.3934/dcds.2000.6.21
[12]

Alfredo Lorenzi, Eugenio Sinestrari. Identifying a BV-kernel in a hyperbolic integrodifferential equation. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1199-1219. doi: 10.3934/dcds.2008.21.1199
[13]

Marita Thomas. Quasistatic damage evolution with spatial $\mathrm{BV}$-regularization. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 235-255. doi: 10.3934/dcdss.2013.6.235
[14]

Haïm Brezis. Remarks on some minimization problems associated with BV norms. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-17. doi: 10.3934/dcds.2019242
[15]

Lluís Alsedà, David Juher, Pere Mumbrú. Minimal dynamics for tree maps. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 511-541. doi: 10.3934/dcds.2008.20.511
[16]

Gabriel Ponce, Ali Tahzibi, Régis Varão. Minimal yet measurable foliations. Journal of Modern Dynamics, 2014, 8 (1) : 93-107. doi: 10.3934/jmd.2014.8.93
[17]

Virginie Bonnaillie-Noël, Corentin Léna. Spectral minimal partitions of a sector. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 27-53. doi: 10.3934/dcdsb.2014.19.27
[18]

Kristian Bjerklöv, Russell Johnson. Minimal subsets of projective flows. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 493-516. doi: 10.3934/dcdsb.2008.9.493
[19]

Nancy Guelman, Jorge Iglesias, Aldo Portela. Examples of minimal set for IFSs. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5253-5269. doi: 10.3934/dcds.2017227
[20]

Pavel Krejčí, Thomas Roche. Lipschitz continuous data dependence of sweeping processes in BV spaces. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 637-650. doi: 10.3934/dcdsb.2011.15.637

2018 Impact Factor: 0.871

Tools

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences