Which electric fields are realizable in conducting materials?
1 Institut de Recherche Mathématique de Rennes, INSA de Rennes, France
2 Department of Mathematics, University of Utah, USA
3 Department of Mathematics, University of Utah, USA
Received: 29 July 2013
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 σ∇u 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.
Mathematics Subject Classification: 35B27 / 78A30 / 37C10
Key words: Isotropic conductivity / electric field / gradient system / laminate
© EDP Sciences, SMAI, 2014