We study a new link between the Steklov and Neumann eigenvalues of domains in Euclidean space. This is obtained through an homogenisation limit of the Steklov problem on a periodically perforated domain, converging to a family of eigenvalue problems with dynamical boundary conditions. For this problem, the spectral parameter appears both in the interior of the domain and on its boundary. This intermediary problem interpolates between Steklov and Neumann eigenvalues of the domain. As a corollary, we recover some isoperimetric type bounds for Neumann eigenvalues from known isoperimetric bounds for Steklov eigenvalues. The interpolation also leads to the construction of planar domains with first perimeter-normalized Stekov eigenvalue that is larger than any previously known example. The proofs are based on a modification of the energy method. It requires quantitative estimates for norms of harmonic functions. An intermediate step in the proof provides a homogenisation result for a transmission problem.

中文翻译：

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通过均质化从 Steklov 到 Neumann

我们研究了欧几里得空间中域的 Steklov 和 Neumann 特征值之间的新联系。这是通过周期性穿孔域上 Steklov 问题的均质化限制获得的，收敛到具有动态边界条件的特征值问题族。对于这个问题，光谱参数出现在域的内部和边界上。这个中间问题在域的 Steklov 和 Neumann 特征值之间进行插值。作为推论，我们从 Steklov 特征值的已知等周边界恢复了诺依曼特征值的一些等周类型边界。插值还导致构建平面域，其第一周长归一化 Stekov 特征值大于任何先前已知的示例。证明基于能量方法的修改。它需要对调和函数的范数进行定量估计。证明的中间步骤为传输问题提供了同质化结果。