The insulated conductivity problem, effective gradient estimates and the maximum principle

Abstract: We consider the insulated conductivity problem with two unit balls as insulating inclusions, a distance of order

v a r e p s i l o n

apart. The solution

u

represents the electric potential. In dimensions

n g e 3

it is an open problem to find the optimal bound on the gradient of

u

, the electric field, in the narrow region between the insulating bodies. Li-Yang recently proved a bound of order

v a r e p s i l o n^{- (1 - g a m m a) / 2}

for some

g a m m a > 0

. In this paper we use a direct maximum principle argument to sharpen the Li-Yang estimate for

n g e 4

. Our method gives effective lower bounds on the best constant

g a m m a

, which in particular approach

1

as

n

tends to infinity.

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