Seminar 1 on Steklov Eigenvalue Problem

What is Steklov Eigenvalue Problem

Let $\Omega$ be a compact Riemannian manifold (typically a bounded domain in $\mathbb{R}^n$) with smooth boundary $\partial \Omega$.

Formal Statement

Find real numbers $\sigma \in \mathbb{R}$ (the eigenvalues) and non-zero functions $u: \overline{\Omega} \to \mathbb{R}$ (the eigenfunctions) satisfying the following boundary value problem:

Notation Explained

• $\Delta$ is the Laplace–Beltrami operator (or simply the Laplacian $\sum_{i=1}^{n} \frac{\partial^2}{\partial x_i^2}$ in Euclidean space).
• $\frac{\partial u}{\partial \nu}$ denotes the outward normal derivative of $u$ on the boundary $\partial \Omega$.
• $\sigma$ is called a Steklov eigenvalue.

Key Interpretations

1. Harmonic Extension Problem: The eigenfunction $u$ is harmonic in the interior. The eigenvalue condition appears only on the boundary.
2. Dirichlet-to-Neumann Map: The problem is equivalent to finding the eigenvalues of the operator \(\Lambda: H^{1/2}(\partial\Omega) \to H^{-1/2}(\partial\Omega), \quad \Lambda f = \frac{\partial (\mathcal{H}f)}{\partial \nu},\) where $\mathcal{H}f$ is the harmonic extension of $f$ to $\Omega$.
3. Spectral Property: The Steklov eigenvalues form a discrete, non-decreasing sequence diverging to $+\infty$: \(0 = \sigma_0 < \sigma_1 \le \sigma_2 \le \sigma_3 \le \cdots \nearrow +\infty.\)

Rayleigh Quotient (Variational Characterization)

The eigenvalues can be characterized variationally as: \(\sigma_k = \min_{\substack{E \subset H^1(\Omega) \\ \dim E = k+1}} \max_{u \in E \setminus \{0\}} R(u), \quad \text{where } R(u) = \frac{\int_\Omega |\nabla u|^2 \, dV}{\int_{\partial \Omega} u^2 \, dS}.\)

Here, $R(u)$ measures the ratio of the Dirichlet energy in the interior to the $L^2$-norm on the boundary.

What we discuss

This is our first discussion on the Steklov problem. We are studying the Steklov eigenvalue problem in the discrete version.

We have read this paper, which mainly discusses the maximum and minimum values of Steklov eigenvalues on trees. The paper is available at this link.

And here is my Note on that paper.