三维球形绝缘子间最短线段上电场的最优估计
An optimal estimate for electric fields on the shortest line segment between two spherical insulators in three dimensions
摘要：
我们考虑一个梯度估计的电导率问题，其内容是两个相邻的绝缘体在三维空间。当具有极端导电性的夹杂物（绝缘体或完美导体）靠近时，梯度可以集中在夹杂物之间，然后随着夹杂物之间的距离接近零而变得任意大。三维绝缘体之间的梯度估计一直被认为是一个具有挑战性的问题，而在距离方面的最佳爆破率已成功地获得了其他极端的电导率问题在二维和三维，并在夹杂物之间的最短线段。本文建立了三维空间中两个绝缘单位球之间最短线段上梯度的上、下界。这些界给出了梯度在线段上的最优爆破速率，这与其它问题中的爆破速率有很大的不同。
英文摘要：
We consider a gradient estimate for a conductivity problem whose inclusions are two neighboring insulators in three dimensions. When inclusions with an extreme conductivity (insulators or perfect conductors) are closely located, the gradient can be concentrated in between inclusions and then becomes arbitrarily large as the distance between inclusions approaches zero. The gradient estimate in between insulators in three dimensions has been regarded as a challenging problem, while the optimal blow-up rates in terms of the distance were successfully obtained for the other extreme conductivity problems in two and three dimensions, and are attained on the shortest line segment between inclusions. In this paper, we establish upper and lower bounds of gradients on the shortest line segment between two insulating unit spheres in three dimensions. These bounds present the optimal blow-up rate of gradient on the line segment which is substantially different from the rates in the other problems.
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