The electric field increases toward infinity in the narrow region between closely adjacent perfect conductors as they approach each other. Much attention has been devoted to the blow-up estimate, especially in two dimensions, for the practical relevance to high stress concentration in fiber-reinforced elastic composites. In this paper, we establish optimal estimates for the electric field associated with the distance between two spherical conductors in $n-${dimensional spaces for $n \geq 2$}. {The novelty of these estimates is that they explicitly describe the dependency of the blow-up rate on the geometric parameters: the radii of the conductors.}
Mathematical Analysis of Subwavelength Resonances and Gradient Blow-up for Two Close-to-Touching Inclusions within the Two-Dimensional Elasticity
Yau Mathematical Sciences Center, Tsinghua University（丘成桐数学科学中心，清华大学）；Academy for Multidisciplinary Studies, Capital Normal University（首都师范大学多学科研究院）；Qiuzhen College, Tsinghua University（清华大学求真学院）
Optimal estimates for the perfect conductivity problem with inclusions close to the boundary
当一个凸形理想导电夹杂靠近矩阵域边界时，一个较大的包含该夹杂的凸区域，电场可以任意大。利用能量方法，我们建立了该理想电导率问题梯度估计的逐点上界和下界。这些结果给出了任意形状和任意尺寸导体的最佳电场爆破率。Ammari，Kang，Lee，Lee and Lim（2007）研究了一个特殊的情况，当圆形夹杂靠近二维圆形矩阵域的边界时。从方法论的角度来看，本文所发展的技术与以往局限于圆形情形的技术有很大的不同，这使得我们可以进一步研究具有散度型的一般椭圆型方程.
