In high-contrast composite materials, the electric field concentration is a common phenomenon when two inclusions are close to touch. It is important from an engineering point of view to study the dependence of the electric field on the distance between two adjacent inclusions. In this paper, we derive upper and lower bounds of the gradient of solutions to the conductivity problem where two perfectly conducting inclusions are located very close to each other. To be specific, we extend the known results of Bao-Li-Yin (ARMA 2009) in two folds: First, we weaken the smoothness of the inclusions from $C^{2,\alpha }$ to $C^{1,\alpha }$. To obtain a pointwise upper bound of the gradient, we follow an iteration technique which is first used to deal with elliptic systems in a narrow domain by Li-Li-Bao-Yin (QAM 2014). However, when the inclusions are of $C^{1,\alpha }$, we can not use $W^{2,p}$ estimates for elliptic equations any more. In order to overcome this new difficulty, we take advantage of De Giorgi-Nash estimates and Campanato's approach to apply an adapted version of the iteration technique with respect to the energy. A lower bound in the shortest line between two inclusions is also obtained to show the optimality of the blow-up rate. Second, when two inclusions are only convex but not strictly convex, we prove that blow-up does not occur any more. The establishment of the relationship between the blow-up rate of the gradient and the order of the convexity of the inclusions reveals the mechanism of such concentration phenomenon.

在高对比度复合材料中，当两个夹杂物靠近接触时，电场集中是一种普遍现象。从工程的角度来看，研究电场对两个相邻夹杂物之间距离的依赖性很重要。在本文中，我们推导出了两个完美导电的夹杂物彼此非常靠近的电导率问题解的梯度的上限和下限。具体来说，我们将 Bao-Li-Yin (ARMA 2009) 的已知结果扩展到两个方面：首先，我们从$C^{2,\alpha }$ 至 $C^{1,\alpha }$. 为了获得梯度的逐点上限，我们遵循了 Li-Li-Bao-Yin (QAM 2014) 首次用于处理窄域中的椭圆系统的迭代技术。然而，当夹杂物是$C^{1,\alpha }$，我们不能使用 ${宽}^{2,磷}$不再对椭圆方程进行估计。为了克服这个新的困难，我们利用 De Giorgi-Nash 估计和 Campanato 的方法来应用与能量相关的迭代技术的适应版本。还获得了两个夹杂物之间最短线的下限以显示爆破率的最优性。其次，当两个夹杂物只是凸而不是严格凸时，我们证明不会再发生爆破。梯度吹胀率与夹杂物凸度顺序关系的建立揭示了这种浓缩现象的发生机制。