Optimal estimates for transmission problems including relative conductivities with different signs

TL;DR

This paper establishes uniform bounds for derivatives in transmission problems with inclusions of contrasting conductivities, including negative and zero conductivities, independent of their proximity, in two and higher dimensions.

Contribution

It provides new uniform derivative estimates for transmission problems with inclusions of different signs of conductivities, extending previous results to cases with negative and zero conductivities.

Findings

Gradient and higher order derivatives are bounded independently of inclusion distance in 2D with sign-changing conductivities.

Derivatives of any order are bounded independently of inclusion distance in higher dimensions when one inclusion is an insulator and the other a perfect conductor.

Results apply to smooth strictly convex inclusions, broadening the scope of transmission problem estimates.

Abstract

We study the gradient and higher order derivative estimates for the transmission problem in the presence of closely located inclusions. We show that in two dimensions, when relative conductivities of circular inclusions have different signs, the gradient and higher order derivatives are bounded independent of $\varepsilon$, the distance between the inclusions. We also show that for general smooth strictly convex inclusions, when one inclusion is an insulator and the other one is a perfect conductor, the derivatives of any order is bounded independent of $\varepsilon$ in any dimensions $n \ge 2$.