We study the insulated conductivity problem with closely spaced insulators
embedded in a homogeneous matrix where the current-electric field relation is
the power law J=∣E∣p−2E. The gradient of solutions may blow up as
ε, the distance between insulators, approaches to 0. In 2D, we
prove an upper bound of the gradient to be of order ε−α,
where α=1/2 when p∈(1,3] and any α>1/(p−1) when p>3.
We provide examples to show that this exponent is almost optimal. In dimensions
n≥3, we prove an upper bound of order ε−1/2+β for
some β>0, and show that β↗1/2 as n→∞.Comment: 39 pages. Theorem 1.3 is extended to all dimension