Spectral analysis of the Neumann-Poincar\'e operator and characterization of the gradient blow-up

TL;DR

This paper characterizes the blow-up of the gradient of solutions to the conductivity equation near closely spaced inclusions using spectral analysis of the Neumann-Poincaré operator, providing explicit quantitative descriptions.

Contribution

It introduces a spectral characterization of gradient blow-up via eigenfunctions of the Neumann-Poincaré operator, linking boundary behavior to eigenvalues.

Findings

Gradient blow-up is characterized by a singular function related to the Neumann-Poincaré eigenfunction.

Explicit functions are derived to quantify the gradient blow-up.

Comparison with two disks provides a precise quantitative description.

Abstract

When perfectly conducting or insulating inclusions are closely located, stress which is the gradient of the solution to the conductivity equation can be arbitrarily large as the distance between two inclusions tends to zero. It is important to precisely characterize the blow-up of the gradient. In this paper we show that the blow-up of the gradient can be characterized by a singular function defined by the single layer potential of an eigenfunction corresponding to the eigenvalue ${1}/{2}$ of a Neumann-Poincar\'e type operator defined on the boundaries of the inclusions. By comparing the singular function with the one corresponding to two disks osculating to the inclusions, we quantitatively characterize the blow-up of the gradient in terms of explicit functions.