The perfect conductivity problem with arbitrary vanishing orders and non-trivial topology
The perfect conductivity problem with arbitrary vanishing orders and non-trivial topology
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DOI：
10.1016/j.jmaa.2023.127469
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发表时间：
2023-01
影响因子：
1.3
通讯作者：
Morgan Sherman;B. Weinkove
中科院分区：
文献类型：
--
作者：
Morgan Sherman;B. Weinkove
理想导电性问题涉及在几乎接触到理想导体的情况下电场大小的最优界。这归结为在导体之间的狭窄区域中获得具有Dirichlet边界条件的调和函数的梯度估计。在本文中，我们推广了Bao-Li-Yen估计，以处理当导体的边界由任意消失阶图给出时的情况。我们的估计允许我们处理具有可能不平凡的拓扑的全局定义的窄区域。我们还证明了我们的估计在完美导体之间的距离方面的敏锐性。我们给出的精确的最优性陈述是新的，即使在宝丽音的背景下也是如此。
The perfect conductivity problem concerns optimal bounds for the magnitude of an electric field in the presence of almost touching perfect conductors. This reduces to obtaining gradient estimates for harmonic functions with Dirichlet boundary conditions in the narrow region between the conductors. In this paper we extend estimates of Bao-Li-Yin to deal with the case when the boundaries of the conductors are given by graphs with arbitrary vanishing orders. Our estimates allow us to deal with globally defined narrow regions with possibly non-trivial topology.We also prove the sharpness of our estimates in terms of the distance between the perfect conductors. The precise optimality statement we give is new even in the setting of Bao-Li-Yin.