# American Institute of Mathematical Sciences

September  2008, 3(3): 509-522. doi: 10.3934/nhm.2008.3.509

## Duality results in the homogenization of two-dimensional high-contrast conductivities

 1 Centre de Mathématiques INSA de Rennes & IRMAR, 20 ave. des Buttes de Coësmes, 35043 Rennes Cedex 2 I.R.M.A.R., Université de Rennes 2, Rennes Cedex, France

Received  January 2008 Published  June 2008

The paper deals with some extensions of the Keller-Dykhneduality relations arising in the classical homogenization of two-dimensional uniformly bounded conductivities, to the case of high-contrast conductivities. Only assuming a $L^1$-bound on the conductivity we prove that the conductivity and its dual converge respectively, in a suitable sense, to the homogenized conductivity and its dual. In the periodic case a similar duality result is obtained under a less restrictive assumption.
Citation: Marc Briane, David Manceau. Duality results in the homogenization of two-dimensional high-contrast conductivities. Networks and Heterogeneous Media, 2008, 3 (3) : 509-522. doi: 10.3934/nhm.2008.3.509
 [1] Hiroshi Matano, Ken-Ichi Nakamura, Bendong Lou. Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit. Networks and Heterogeneous Media, 2006, 1 (4) : 537-568. doi: 10.3934/nhm.2006.1.537 [2] Mohamed Camar-Eddine, Laurent Pater. Homogenization of high-contrast and non symmetric conductivities for non periodic columnar structures. Networks and Heterogeneous Media, 2013, 8 (4) : 913-941. doi: 10.3934/nhm.2013.8.913 [3] Florian Kogelbauer. On the symmetry of spatially periodic two-dimensional water waves. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 7057-7061. doi: 10.3934/dcds.2016107 [4] Jong-Shenq Guo, Chang-Hong Wu. Front propagation for a two-dimensional periodic monostable lattice dynamical system. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 197-223. doi: 10.3934/dcds.2010.26.197 [5] Tong Peng. Designing prorated lifetime warranty strategy for high-value and durable products under two-dimensional warranty. Journal of Industrial and Management Optimization, 2021, 17 (2) : 953-970. doi: 10.3934/jimo.2020006 [6] Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5495-5508. doi: 10.3934/dcdsb.2020355 [7] Guillaume Bal. Homogenization in random media and effective medium theory for high frequency waves. Discrete and Continuous Dynamical Systems - B, 2007, 8 (2) : 473-492. doi: 10.3934/dcdsb.2007.8.473 [8] Giovanni Scilla. Motion of discrete interfaces in low-contrast periodic media. Networks and Heterogeneous Media, 2014, 9 (1) : 169-189. doi: 10.3934/nhm.2014.9.169 [9] Fioralba Cakoni, Houssem Haddar, Isaac Harris. Homogenization of the transmission eigenvalue problem for periodic media and application to the inverse problem. Inverse Problems and Imaging, 2015, 9 (4) : 1025-1049. doi: 10.3934/ipi.2015.9.1025 [10] François Murat, Ali Sili. A remark about the periodic homogenization of certain composite fibered media. Networks and Heterogeneous Media, 2020, 15 (1) : 125-142. doi: 10.3934/nhm.2020006 [11] Lars Lamberg, Lauri Ylinen. Two-Dimensional tomography with unknown view angles. Inverse Problems and Imaging, 2007, 1 (4) : 623-642. doi: 10.3934/ipi.2007.1.623 [12] Elissar Nasreddine. Two-dimensional individual clustering model. Discrete and Continuous Dynamical Systems - S, 2014, 7 (2) : 307-316. doi: 10.3934/dcdss.2014.7.307 [13] Jerzy Gawinecki, Wojciech M. Zajączkowski. Global regular solutions to two-dimensional thermoviscoelasticity. Communications on Pure and Applied Analysis, 2016, 15 (3) : 1009-1028. doi: 10.3934/cpaa.2016.15.1009 [14] Ibrahim Fatkullin, Valeriy Slastikov. Diffusive transport in two-dimensional nematics. Discrete and Continuous Dynamical Systems - S, 2015, 8 (2) : 323-340. doi: 10.3934/dcdss.2015.8.323 [15] Min Chen. Numerical investigation of a two-dimensional Boussinesq system. Discrete and Continuous Dynamical Systems, 2009, 23 (4) : 1169-1190. doi: 10.3934/dcds.2009.23.1169 [16] Fumihiko Nakamura, Michael C. Mackey. Asymptotic (statistical) periodicity in two-dimensional maps. Discrete and Continuous Dynamical Systems - B, 2022, 27 (8) : 4285-4303. doi: 10.3934/dcdsb.2021227 [17] Jinjing Jiao, Guanghua Shi. Quasi-periodic solutions for the two-dimensional systems with an elliptic-type degenerate equilibrium point under small perturbations. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5157-5180. doi: 10.3934/cpaa.2020231 [18] Vsevolod Laptev. Deterministic homogenization for media with barriers. Discrete and Continuous Dynamical Systems - S, 2015, 8 (1) : 29-44. doi: 10.3934/dcdss.2015.8.29 [19] Antonin Chambolle, Gilles Thouroude. Homogenization of interfacial energies and construction of plane-like minimizers in periodic media through a cell problem. Networks and Heterogeneous Media, 2009, 4 (1) : 127-152. doi: 10.3934/nhm.2009.4.127 [20] Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. An improved homogenization result for immiscible compressible two-phase flow in porous media. Networks and Heterogeneous Media, 2017, 12 (1) : 147-171. doi: 10.3934/nhm.2017006

2021 Impact Factor: 1.41