In this paper we study the realizability of a given smooth periodic gradient field ∇u defined in R^{d}, in the sense of finding when one can obtain a matrix conductivity σ such that σ∇u is a divergence free current field. The construction is shown to be always possible locally in R^{d} 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.

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 }, pages = {307--323}, publisher = {EDP-Sciences}, volume = {48}, number = {2}, year = {2014}, doi = {10.1051/m2an/2013109}, mrnumber = {3177847}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2013109/} }

TY - JOUR AU - Briane, Marc AU - Milton, Graeme W. AU - Treibergs, Andrejs TI - Which electric fields are realizable in conducting materials? JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2014 SP - 307 EP - 323 VL - 48 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2013109/ DO - 10.1051/m2an/2013109 LA - en ID - M2AN_2014__48_2_307_0 ER -

%0 Journal Article %A Briane, Marc %A Milton, Graeme W. %A Treibergs, Andrejs %T Which electric fields are realizable in conducting materials? %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2014 %P 307-323 %V 48 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2013109/ %R 10.1051/m2an/2013109 %G en %F 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 , Volume 48 (2014) no. 2, pp. 307-323. doi : 10.1051/m2an/2013109. http://www.numdam.org/articles/10.1051/m2an/2013109/

[1] Univalent σ-harmonic mappings. Arch. Ration. Mech. Anal. 158 (2001) 155-171. | MR | Zbl

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

,[3] 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 | Zbl

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

,[5] 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 | Zbl

and ,[6] Univalent solutions of an elliptic system of partial differential equations arising in homogenization. Indiana Univ. Math. J. 50 (2001) 747-757. | MR

, and ,[7] Asymptotic Analysis for Periodic Structures, in vol. 5 of Stud. Math. Appl. North-Holland Publishing Co., Amsterdam-New York (1978) 700. | MR | Zbl

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

,[9] Change of sign of the corrector's determinant for homogenization in three-dimensional conductivity. Arch. Ration. Mech. Anal. 173 (2004) 133-150. | MR | Zbl

, and ,[10] Is it wise to keep laminating? ESAIM: COCV 10 (2004) 452-477. | EuDML | Numdam | MR | Zbl

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

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

,[13] Homogenization of Differential Operators and Integral Functionals, translated from the Russian by G.A. Yosifian. Springer-Verlag, Berlin (1994) 570. | MR | Zbl

, and ,[14] 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 | Zbl

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

,[16] 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 | Zbl

and ,[17] 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 | Zbl

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

,*Cited by Sources: *