Homogenization of NonLinear Variational Problems with Thin LowConducting Layers
Abstract
This paper deals with the homogenization of a sequence of nonlinear conductivity energies in a bounded open set {omega} of R{sup d}, for d {>=}3. The energy density is of the same order as a{sub {epsilon}}(x/{epsilon})Du(x){sup p}, where {epsilon}{yields}0,a{sub {epsilon}} is periodic, u is a vectorvalued function in W{sup 1,p}({omega}{sup ;}R{sup m}) and p>1. The conductivity a{sub {epsilon}} is equal to 1 in the 'hard' phases composed by N{>=}2 two by two disjointclosure periodic sets while a{sub {epsilon}} tends uniformly to 0 in the 'soft' phases composed by periodic thin layers which separate the hard phases. We prove that the limit energy, according to {gamma}convergence, is a multiphase functional equal to the sum of the homogenized energies (of order 1) induced by the hard phases plus an interaction energy (of order 0) due to the soft phases. The number of limit phases is less than or equal to N and is obtained by evaluating the {gamma}limit of the rescaled energy of density {epsilon}{sup p}a{sub {epsilon}}(y)Dv(y){sup p} in the torus. Therefore, the homogenization result is achieved by a double {gamma}convergence procedure since the cell problem depends on {epsilon}.
 Authors:
 Dipartimento di Matematica, Universita di Roma 'Tor Vergata', Via della Ricerca Scientifica, 00133 (Italy), Email: braides@mat.uniroma2.it
 Centre de Mathematiques, I.N.S.A. de Rennes and I.R.M.A.R., 20 avenue des Buttes de Coesmes, 35043 (France), Email: mbriane@insarennes.fr
 Publication Date:
 OSTI Identifier:
 21067414
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Applied Mathematics and Optimization; Journal Volume: 55; Journal Issue: 1; Other Information: DOI: 10.1007/s0024500608616; Copyright (c) 2007 Springer; www.springerny.com; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CONVERGENCE; ENERGY DENSITY; FUNCTIONS; LAYERS; NONLINEAR PROBLEMS; PERIODICITY; THIN FILMS; VARIATIONAL METHODS; VECTORS
Citation Formats
Braides, Andrea, and Briane, Marc. Homogenization of NonLinear Variational Problems with Thin LowConducting Layers. United States: N. p., 2007.
Web. doi:10.1007/S0024500608616.
Braides, Andrea, & Briane, Marc. Homogenization of NonLinear Variational Problems with Thin LowConducting Layers. United States. doi:10.1007/S0024500608616.
Braides, Andrea, and Briane, Marc. Mon .
"Homogenization of NonLinear Variational Problems with Thin LowConducting Layers". United States.
doi:10.1007/S0024500608616.
@article{osti_21067414,
title = {Homogenization of NonLinear Variational Problems with Thin LowConducting Layers},
author = {Braides, Andrea and Briane, Marc},
abstractNote = {This paper deals with the homogenization of a sequence of nonlinear conductivity energies in a bounded open set {omega} of R{sup d}, for d {>=}3. The energy density is of the same order as a{sub {epsilon}}(x/{epsilon})Du(x){sup p}, where {epsilon}{yields}0,a{sub {epsilon}} is periodic, u is a vectorvalued function in W{sup 1,p}({omega}{sup ;}R{sup m}) and p>1. The conductivity a{sub {epsilon}} is equal to 1 in the 'hard' phases composed by N{>=}2 two by two disjointclosure periodic sets while a{sub {epsilon}} tends uniformly to 0 in the 'soft' phases composed by periodic thin layers which separate the hard phases. We prove that the limit energy, according to {gamma}convergence, is a multiphase functional equal to the sum of the homogenized energies (of order 1) induced by the hard phases plus an interaction energy (of order 0) due to the soft phases. The number of limit phases is less than or equal to N and is obtained by evaluating the {gamma}limit of the rescaled energy of density {epsilon}{sup p}a{sub {epsilon}}(y)Dv(y){sup p} in the torus. Therefore, the homogenization result is achieved by a double {gamma}convergence procedure since the cell problem depends on {epsilon}.},
doi = {10.1007/S0024500608616},
journal = {Applied Mathematics and Optimization},
number = 1,
volume = 55,
place = {United States},
year = {Mon Jan 15 00:00:00 EST 2007},
month = {Mon Jan 15 00:00:00 EST 2007}
}

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