The transmission problem with imperfect interfaces of small resistance

TL;DR

This paper studies the transmission problem with imperfect interfaces characterized by positive resistance, constructing solutions via layer potentials and analyzing their convergence to perfect interface solutions as resistance diminishes.

Contribution

It introduces a layer potential method for solving the imperfect interface transmission problem and proves convergence of solutions to the perfect interface case as resistance approaches zero.

Findings

Solutions converge in Sobolev spaces as interface resistance tends to zero.

Gradient convergence in uniform norm for sufficiently regular boundaries.

Layer potential construction of solutions for imperfect interfaces.

Abstract

We consider the transmission problem in presence of interfaces with imperfect bonding. The imperfect bonding condition is characterized by the positive resistance along the interface, which causes discontinuity of the potential across the interface while the flux is continuous. If the interface resistance is zero, then the interface is of perfect bonding, where both the potential and the flux of the solution are continuous across the interface. In this paper, we first construct using layer potentials the solution to the transmission problem with imperfect interfaces. We then prove that the solutions converge in various Sobolev spaces to the solution to the transmission problem with perfect interfaces as the interface resistance tends to zero. In particular, it is shown that the gradient of the solution converges in the uniform norm if the boundary is sufficiently regular.