R^3中电流场的各向同性可实现性
Isotropic realizability of current fields in R^3
摘要：
本文讨论了R^3中正则发散自由场j作为电流场的各向同性可实现性，即对于各向同性的电导率σ和梯度场Du，j何时可以写成σ Du.利用Frobenius定理得到了R^3中的局部各向同性可实现性，条件是j和旋度j在R^3中正交.一个反例表明，弗罗贝纽斯条件不足以导出R^3中的全局各向同性可实现性。然而，假设（j，curl j，j x curl j）是R^3的正交基，则可从沿方向j/ |j|、curl j/|卷曲j|和（j/ |j| ^2）x curl j。当场j是周期性的时，环面中的各向同性可实现性还需要由沿第三方向的流（j/ |j| ^2）x \curl j满足的有界性假设。
英文摘要：
This paper deals with the isotropic realizability of a given regular divergence free field j in R^3 as a current field, namely to know when j can be written as sigma Du for some isotropic conductivity sigma, and some gradient field Du. The local isotropic realizability in R^3 is obtained by Frobenius' theorem provided that j and curl j are orthogonal in R^3. A counter-example shows that Frobenius' condition is not sufficient to derive the global isotropic realizability in R^3. However, assuming that (j, curl j, j x curl j) is an orthogonal basis of R^3, an admissible conductivity sigma is constructed from a combination of the three dynamical flows along the directions j/|j|, curl j/|curl j| and (j/|j|^2) x curl j. When the field j is periodic, the isotropic realizability in the torus needs in addition a boundedness assumption satisfied by the flow along the third direction (j/|j|^2) x \curl j. Several examples illustrate the sharpness of the realizability conditions.
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