Exact relations for composites: Towards a complete solution.
Which electric fields are realizable in conducting materials?
In this paper we study the realizability of a given smooth periodic gradient field ∇ defined in R, in the sense of finding when one can obtain a matrix conductivity such that ∇ is a divergence free current field. The construction is shown to be always possible locally in R provided that ∇ 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...
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