在⾼对⽐复合材料中，当两个内含物之间的距离很⼩时，物理场 （如电场或应⼒）通常会在窄区域内发⽣集中现象。从⼯程学的⾓度， 研究物理场与相邻内含物之间距离的依赖关系是⾮常重要的。

在本⽂中，我们⾸先考虑传导问题，当超导体彼此之间的距离很⼩ 时，获得了 ⽅程解的梯度的上下界逐点估计。具体地，我们从两个⽅⾯ 推⼴了 Bao-Li-Yin（ARMA 2009）的⼯作：1、我们把内含物边界的光滑 性由 减弱为 。当内含物边界的光滑性为 时，我们不能应 ⽤椭圆⽅程的 估计。为了克服这⼀新的困难，我们借助 De GiorgiNash 估计和 Campanato ⽅法应⽤于能量估计的迭代技术中，得到了上 下界估计，进⽽说明了梯度估计的最优性。2、当两个内含物并不是严 格凸时，我们证明了梯度不会发⽣爆破。

其次，我们研究了部分系数退化的线性弹性系统，即 Lamé 系统， 梯度的爆破估计，进⽽揭⽰了应⼒集中的程度与 边界的内含物之间 距离的依赖关系，推⼴ 了Bao-Li-Li (ARMA 2015)中假设条件下的结 果。我们利⽤了对右端含有散度项的椭圆⽅程解的估计，并结合 Campanato ⽅法，建⽴了包含上下界的最优梯度估计。另外，我们也给出了梯度在爆破点附近的渐近展开公式。

最后，我们也给出了当 的假设条件下内含物靠近基体边界时， 这两个问题解的梯度估计， 推⼴了 Bao-Ju-Li（Adv. Math. 2017）的结果。我们给出了具体的边界数据，以保证在所有维数下梯度的下界发⽣爆破，进⽽说明梯度关于内含物与基体界⾯之间距离的爆破速率是最优的。

In high-contrast composite materials, the concentration of the physical fields (electric field or stress) 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 physical fields on the distance between two adjacent inclusions. In this report, we first derive upper and lower bounds of the gradient of solutions for 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: 1. We weaken the smoothness of the inclusions from C2,↵ to C1,↵. However, when the inclusions are of C1,↵, we can not use W2,p estimates for elliptic equations any more. In order to overcome this new diculty, 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 is also obtained to show the optimality of the blow-up rate. 2. When two inclusions are only convex but not strictly convex, we prove that blow-up does not occur any more. Secondly, we investigate the gradient blow-up estimates for the Lam´e system of linear elasticity with partially infinite coecients to show the dependence of the degree of stress enhancement on the distance between inclusions in a composite containing densely placed sti↵ C1,↵ inclusions, weaker than the previous C2,↵ assumptions in BaoLi-Li (ARMA 2015). We make use of W1,p estimates for elliptic system with right hand side in divergence form and combine with the Campanato’s approach to establish the optimal gradient estimates, including upper and lower bounds. Moreover, we obtain an asymptotic formula of the gradient near the blow-up point. Finally, we establish boundary gradient estimates for both two problems under the C1,↵ assumptions, weaker than the previous work by Bao-Ju-Li (Adv. Math 2017). When the inclusion is located close to the boundary of matrix domain, we give the specific examples of boundary data to obtain the lower bound gradient estimates in all dimensions, which guarantee the blow-up occurs and indicate that the blow-up rates of the gradients with respect to the distance between the interfacial surfaces are optimal.