Let \(\Omega \subset \mathbb {R}^d\) be a bounded open connected set with Lipschitz boundary. Let \(A^N\) and \(A^D\) be the Stokes Neumann operator and Stokes Dirichlet operator on \(\Omega \), respectively. We study the associated Stokes version of the Dirichlet-to-Neumann operator and show a Krein formula which relates these three Stokes version operators. We also prove a Stokes version of the Friedlander inequalities, which relates the Dirichlet eigenvalues and the Neumann eigenvalues.

中文翻译：

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斯托克斯狄利克雷到诺依曼算子

令\(\Omega \subset \mathbb {R}^d\)为具有 Lipschitz 边界的有界开连通集。设\(A^N\)和\(A^D\)分别为\(\Omega \)上的 Stokes Neumann 算子和 Stokes Dirichlet 算子。我们研究了狄利克雷到诺伊曼算子的相关斯托克斯版本，并展示了一个将这三个斯托克斯版本算子联系起来的 Kerin 公式。我们还证明了弗里德兰德不等式的斯托克斯版本，它将狄利克雷特征值和诺依曼特征值联系起来。