Optimal gradient estimates of solutions to the insulated conductivity problem in dimension greater than two

Source abstract

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 . It was known that the optimal blow-up rate in dimension n = 2 is of order \varepsilon^{-1/2} . It has recently been proved that in dimensions n \ge 3 , an upper bound of the gradient is of order \varepsilon^{-1/2 + \beta} for some \beta > 0 . On the other hand, optimal values of \beta have not been identified. In this paper, we prove that when the inclusions are balls, the optimal value of \beta is [-(n-1)+\sqrt{(n-1)^{2}+4(n-2)}]/4 \in (0,1/2) in dimensions n \ge 3 .