Given the Steklov eigenvalueproblem on the ball $B_r\subset\mathbb{R}^n$ with radius $r$ \begin{align*} \Delta \varphi &= 0~~~~~\text{in } ~~~B_r \\ \partial_\nu \varphi &= \lambda \varphi ~~~\text{auf } \partial B_r. \end{align*} It is well know that the spectrum can be arranged via $$ 0=\lambda_0 < \lambda_1 \leq \lambda_2 \leq \lambda_3 \leq \dots \to \infty. $$ I now know that we have $$ \lambda_1 = \frac{1}{r}. $$ My question know is, if $\lambda_2$ is known for the ball $B_r$?
2026-04-06 04:14:11.1775448851
Second eigenvalue for the Steklov eigenvalue problem.
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See Example $1.3.2$ of this paper Spectral Geometry of the Steklov Problem by Girouard and Polterovich.