We obtain upper bounds for the Steklov eigenvalues ${\sigma }_k(M)$ of a smooth, compact, n-dimensional submanifold M of Euclidean space with boundary Σ that involve the intersection indices of M and of Σ. One of our main results is an explicit upper bound in terms of the intersection index of Σ, the volume of Σ and the volume of M as well as dimensional constants. By also taking the injectivity radius of Σ into account, we obtain an upper bound that has the optimal exponent of k with respect to the asymptotics of the Steklov eigenvalues as $k\to \infty$.

中文翻译：

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欧几里得空间中子流形的 Steklov 特征值的上界通过交集索引

我们获得了 Steklov 特征值的上限 ${\sigma }_{克}(米)$具有边界 Σ 的欧几里得空间的光滑、紧凑、n维子流形M，涉及M和 Σ的交集索引。我们的主要结果之一是 Σ 的交集指数、Σ 的体积和M的体积以及尺寸常数的显式上限。通过还考虑 Σ 的注入半径，我们获得了一个上限，该上限具有关于 Steklov 特征值渐近的k的最佳指数为$克\to \infty$.