Optimal gradient estimates for the insulated conductivity problem with general convex inclusions case

TL;DR

This paper establishes sharp upper and lower bounds on the gradient blow-up rates for solutions to the insulated conductivity problem with general convex inclusions in dimensions three and higher, extending previous results from spherical cases.

Contribution

It provides the first sharp gradient estimates for general convex insulators, revealing the blow-up rate tied to the eigenvalues of a geometry-dependent elliptic operator.

Findings

Established pointwise upper bounds for the gradient.

Derived matching lower bounds for optimal blow-up rates.

Extended previous spherical inclusion results to general convex shapes.

Abstract

We study the insulated conductivity problem which involves two adjacent convex insulators embedded in a bounded domain. It is known that the gradient of solutions may blow up as the distance between the two inclusions tends to zero. However, the sharpness of the blow up rate for general convex insulator case in dimension $n\geq3$ has remained open. The novelty of this paper is that we answer this problem affirmatively by establishing a pointwise upper bound of the gradient for general convex insulators, along with a corresponding lower bound that achieves optimal blow up rates. These rates are associated with the first nonzero eigenvalue of an elliptic operator determined by the geometry of insulators. Our results improve and make complete the previous result for ball insulators case studied in \cite{DLY}.