Approximation of the first Steklov-Dirichlet eigenvalue on eccentric spherical shells in general dimensions

TL;DR

This paper develops a finite section method to approximate the first Steklov-Dirichlet eigenvalue on eccentric spherical shells in any dimension, proving exponential convergence and extending previous 2D results.

Contribution

It introduces a new finite section approach for the Dirichlet-to-Neumann operator and extends numerical schemes from 2D to higher dimensions with proven convergence.

Findings

The method converges exponentially to the true eigenvalue.

Numerical schemes are successfully extended to general dimensions.

Numerical examples demonstrate the effectiveness of the approach.

Abstract

We study the first Steklov-Dirichlet eigenvalue on eccentric spherical shells in $\mathbb{R}^{n+2}$ with $n\geq 1$, imposing the Steklov condition on the outer boundary sphere, denoted by $\Gamma_S$, and the Dirichlet condition on the inner boundary sphere. The first eigenfunction admits a Fourier--Gegenbauer series expansion via the bispherical coordinates, where the Dirichlet-to-Neumann operator on $\Gamma_S$ can be recursively expressed in terms of the expansion coefficients arXiv:2309.09587. In this paper, we develop a finite section approach for the Dirichlet-to-Neumann operator to approximate the first Steklov--Dirichlet eigenvalue on eccentric spherical shells. We prove the exponential convergence of this approach by using the variational characterization of the first eigenvalue. Furthermore, based on the convergence result, we propose a numerical computation scheme as an extension of the two-dimensional result in [Hong et al., Ann. Mat. Pura Appl., 2022] to general dimensions. We provide numerical examples of the first Steklov-Dirichlet eigenvalue on eccentric spherical shells with various geometric configurations.