临界点附近电场的各向同性可实现性
Isotropic realizability of electric fields around critical points
摘要：
本文研究了一个给定的规则梯度场$\nabla u$作为电场的各向同性可实现性，即当$\nabla u$是方程$÷\left(\si\nabla u\right)=0$对于某一各向同性电导率$\si>0$的解时.在\cite{BMT}中，利用梯度流方法研究了函数$u$无临界点的情况。临界点的存在需要根据该点周围的动力系统的行为进行特殊处理。鞍点的情况下是最有利的，并导致我们的局部各向同性的可实现性的表征，通过一些有界性条件涉及拉普拉斯算子的$u$沿梯度流。在相同的有界性条件下，汇或源的情形都蕴含强最大值原理。然而，当临界点不是双曲的各向同性的可实现性一般不满足，甚至在该点的邻域内分段。特别讨论了当梯度系统的轨迹有界时，周期梯度场在环面上的各向同性可实现性。
英文摘要：
In this paper we study the isotropic realizability of a given regular gradient field $\nabla u$ as an electric field, namely when $\nabla u$ is solution of the equation $÷\left(\si\nabla u\right)=0$ for some isotropic conductivity $\si>0$. The case of a function $u$ without critical point was investigated in \cite{BMT} thanks to a gradient flow approach. The presence of a critical point needs a specific treatment according to the behavior of the dynamical system around the point. The case of a saddle point is the most favorable and leads us to a characterization of the local isotropic realizability through some boundedness condition involving the laplacian of $u$ along the gradient flow. The case of a sink or a source implies a strong maximum principle under the same boundedness condition. However, when the critical point is not hyperbolic the isotropic realizability is not generally satisfied even piecewisely in the neighborhood of the point. The isotropic realizability in the torus for periodic gradient fields is also discussed in particular when the trajectories of the gradient system are bounded.
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