The Optimal Gradient Estimates for Perfect Conductivity Problem with C^{1,\alpha} inclusions

TL;DR

This paper investigates how the electric field concentrates between two close, perfectly conducting inclusions with less smooth boundaries, providing bounds on the gradient and revealing the influence of convexity on blow-up behavior.

Contribution

It extends previous results by weakening boundary smoothness assumptions and analyzing the impact of convexity on electric field concentration in high-contrast composites.

Findings

Derived upper and lower bounds for the gradient of solutions.

Showed blow-up occurs only with strictly convex inclusions.

Established the relationship between convexity degree and blow-up rate.

Abstract

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 the 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 an pointwise upper bound of the gradient, we follow an iteration technique developed by Bao-Li-Li (ARMA 2015), who mainly deal with the system of linear elasticity. 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. Moreover, 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.