Which electric fields are realizable in conducting materials?
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 2, p. 307-323

In this paper we study the realizability of a given smooth periodic gradient field ∇u defined in Rd, in the sense of finding when one can obtain a matrix conductivity σ such that σu is a divergence free current field. The construction is shown to be always possible locally in Rd provided that ∇u is non-vanishing. This condition is also necessary in dimension two but not in dimension three. In fact the realizability may fail for non-regular gradient fields, and in general the conductivity cannot be both periodic and isotropic. However, using a dynamical systems approach the isotropic realizability is proved to hold in the whole space (without periodicity) under the assumption that the gradient does not vanish anywhere. Moreover, a sharp condition is obtained to ensure the isotropic realizability in the torus. The realizability of a matrix field is also investigated both in the periodic case and in the laminate case. In this context the sign of the matrix field determinant plays an essential role according to the space dimension. The present contribution essentially deals with the realizability question in the case of periodic boundary conditions.

DOI : https://doi.org/10.1051/m2an/2013109
Classification:  35B27,  78A30,  37C10
Keywords: isotropic conductivity, electric field, gradient system, laminate
@article{M2AN_2014__48_2_307_0,
author = {Briane, Marc and Milton, Graeme W. and Treibergs, Andrejs},
title = {Which electric fields are realizable in conducting materials?},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {48},
number = {2},
year = {2014},
pages = {307-323},
doi = {10.1051/m2an/2013109},
mrnumber = {3177847},
language = {en},
url = {http://www.numdam.org/item/M2AN_2014__48_2_307_0}
}

Briane, Marc; Milton, Graeme W.; Treibergs, Andrejs. Which electric fields are realizable in conducting materials?. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 2, pp. 307-323. doi : 10.1051/m2an/2013109. http://www.numdam.org/item/M2AN_2014__48_2_307_0/

[1] G. Alessandrini and V. Nesi, Univalent σ-harmonic mappings. Arch. Ration. Mech. Anal. 158 (2001) 155-171. | MR 1838656 | Zbl 0977.31006

[2] G. Allaire, Shape Optimization by the Homogenization Method, vol. 146 of Appl. Math. Sci. Springer-Verlag, New-York (2002) 456. | MR 1859696 | Zbl 0990.35001

[3] A. Ancona, Some results and examples about the behavior of harmonic functions and Green's functions with respect to second order elliptic operators. Nagoya Math. J. 165 (2002) 123-158. | MR 1892102 | Zbl 1028.31003

[4] V.I. Arnold, Ordinary differential equations, translated from the third Russian edition by R. Cooke, Springer Textbook. Springer-Verlag, Berlin (1992) 334. | MR 1162307 | Zbl 0744.34001

[5] N. Bakhvalov and G. Panasenko, Homogenisation: Averaging Processes in Periodic Media, Mathematical Problems in the Mechanics of Composite Materials, translated from the Russian by D. Leĭtes, vol. 36 of Math. Appl. (Soviet Series). Kluwer Academic Publishers Group, Dordrecht (1989) 366. | MR 1112788 | Zbl 0692.73012

[6] P. Bauman, A. Marini and V. Nesi, Univalent solutions of an elliptic system of partial differential equations arising in homogenization. Indiana Univ. Math. J. 50 (2001) 747-757. | MR 1871388 | Zbl pre01780879

[7] A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, in vol. 5 of Stud. Math. Appl. North-Holland Publishing Co., Amsterdam-New York (1978) 700. | MR 503330 | Zbl 0404.35001

[8] M. Briane, Correctors for the homogenization of a laminate. Adv. Math. Sci. Appl. 4 (1994) 357-379. | MR 1294225 | Zbl 0829.35009

[9] M. Briane, G.W. Milton and V. Nesi, Change of sign of the corrector's determinant for homogenization in three-dimensional conductivity. Arch. Ration. Mech. Anal. 173 (2004) 133-150. | MR 2073507 | Zbl 1118.78009

[10] M. Briane, and V. Nesi, Is it wise to keep laminating? ESAIM: COCV 10 (2004) 452-477. | Numdam | MR 2111075 | Zbl 1072.74057

[11] A. Cherkaev and Y. Zhang, Optimal anisotropic three-phase conducting composites: Plane problem. Int. J. Solids Struct. 48 (2011) 2800-2813.

[12] B. Dacorogna, Direct Methods in the Calculus of Variations, in vol. 78 of Appl. Math. Sci. Springer-Verlag, Berlin-Heidelberg (1989) 308. | MR 990890 | Zbl 0703.49001

[13] V.V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals, translated from the Russian by G.A. Yosifian. Springer-Verlag, Berlin (1994) 570. | MR 1329546 | Zbl 0838.35001

[14] G.W. Milton, Modelling the properties of composites by laminates, Homogenization and Effective Moduli of Materials and Media, in vol. 1 of IMA Math. Appl. Springer-Verlag, New York (1986) 150-174. | MR 859415 | Zbl 0631.73011

[15] G.W. Milton, The Theory of Composites, Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2002) 719. | MR 1899805 | Zbl 0993.74002

[16] F. Murat and L. Tartar, H-convergence, Topics in the Mathematical Modelling of Composite Materials, in vol. 31 of Progr. Nonlinear Differ. Equ. Appl., edited by L. Cherkaev and R.V. Kohn. Birkhaüser, Boston (1997) 21-43. | MR 1493039 | Zbl 0920.35019

[17] V. Nesi, Bounds on the effective conductivity of two-dimensional composites made of n ≥ 3 isotropic phases in prescribed volume fraction: the weighted translation method. Proc. Roy. Soc. Edinburgh Sect. A 125 (1995) 1219-1239. | MR 1363001 | Zbl 0852.35016

[18] U. Raitums, On the local representation of G-closure. Arch. Rational Mech. Anal. 158 (2001) 213-234. | MR 1842345 | Zbl 1123.35320