论文标题
在维度大于两个以上的绝缘电导率问题解决方案的梯度估计值
Gradient estimates of solutions to the insulated conductivity problem in dimension greater than two
论文作者
论文摘要
我们研究了绝缘的电导率问题，其中包含在$ \ mathbb {r}^n $中的包含域中的内含物。解决方案的梯度可能会爆炸为$ \ varepsilon $，夹杂物之间的距离，即$ 0 $。事实证明，爆炸率的上限为$ \ varepsilon^{ - 1/2} $。已知上限在尺寸$ n = 2 $。但是，该上限是否在尺寸上呈锋利$ n \ ge 3 $保持开放。在本文中，对于某些$β> 0 $，我们将上限$ n \ ge 3 $改进了$ \ varepsilon^{ - 1/2 +β} $的顺序。
We study the insulated conductivity problem with inclusions embedded in a bounded domain in $\mathbb{R}^n$. The gradient of solutions may blow up as $\varepsilon$, the distance between inclusions, approaches to $0$. An upper bound for the blow up rate was proved to be of order $\varepsilon^{-1/2}$. The upper bound was known to be sharp in dimension $n = 2$. However, whether this upper bound is sharp in dimension $n \ge 3$ has remained open. In this paper, we improve the upper bound in dimension $n \ge 3$ to be of order $\varepsilon^{-1/2 + β}$, for some $β> 0$.
