The Insulated Conductivity Problem with p-Laplacian

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. We prove an upper bound of the gradient to be of order ε^-α , where α= 1 / 2 when p∈ (1 , n+ 1] and any α> n/ (2 (p- 1)) when p> n+ 1 . We provide examples to show that this exponent is almost optimal in 2D. Additionally, in dimensions n≧ 3 , for any p> 1 , we prove another upper bound of order ε^-1/2+β for some β> 0 , and show that β↗ 1 / 2 as n→ ∞ .

Duke Scholars

Author Hanye Zhu Mathematics