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

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 $Γ_S$, 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 $Γ_S$ 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.

💡 Research Summary

The paper addresses the first Steklov–Dirichlet eigenvalue problem on eccentric spherical shells in $\mathbb{R}^{n+2}$ for any $n\ge 1$. The domain $\Omega$ is the region between an inner ball $B_{t1}$ and an outer ball $B_{2}$; the inner boundary carries a Dirichlet condition $u=0$, while the outer boundary carries a Steklov condition $\partial_{\nu}u=\sigma u$. The authors first reformulate the Laplace equation in bispherical coordinates, which map the two spherical boundaries to constant‐$\xi$ surfaces. In this coordinate system the harmonic eigenfunction can be expanded as a Fourier–Gegenbauer series \