Spectral properties of the Neumann-Poincar\'e operator and uniformity of estimates for the conductivity equation with complex coefficients

TL;DR

This paper investigates the spectral properties of the Neumann-Poincaré operator to establish uniform estimates for solutions of the conductivity equation with complex coefficients, including multiple inclusions, ensuring well-posedness and stability.

Contribution

It provides a spectral characterization of solvability and uniform estimates for the conductivity problem with complex coefficients, extending to multiple inclusions and boundary perturbations.

Findings

Solutions are uniformly bounded in $H^1$-norm away from a specific interval on the negative real axis.

The solution depends Lipschitz continuously on the complex conductivity parameter $k$.

Uniform boundary perturbation formulas are valid for small inclusions, independent of $k$ within certain bounds.

Abstract

We consider well-posedness of the boundary value problem in presence of an inclusion with complex conductivity $k$. We first consider the transmission problem in $\mathbb{R}^d$ and characterize solvability of the problem in terms of the spectrum of the Neumann-Poincar\'e operator. We then deal with the boundary value problem and show that the solution is bounded in its $H^1$-norm uniformly in $k$ as long as $k$ is at some distance from a closed interval in the negative real axis. We then show with an estimate that the solution depends on $k$ in its $H^1$-norm Lipschitz continuously. We finally show that the boundary perturbation formula in presence of a diametrically small inclusion is valid uniformly in $k$ away from the closed interval mentioned before. The results for the single inclusion case are extended to the case when there are multiple inclusions with different complex conductivities: We first obtain a complete characterization of solvability when inclusions consist of two disjoint disks and then prove solvability and uniform estimates when imaginary parts of conductivities have the same signs. The results are obtained using the spectral property of the associated Neumann-Poincar\'e operator and the spectral resolution.