We investigate asymptotically sharp upper and lower bounds for the approximation numbers of the compact Sobolev embeddings ${\stackrel{\circ }{W}}^m\left(\Omega \right)\hookrightarrow L_2\left(\Omega \right)$ and $W^m\left(\Omega \right)\hookrightarrow L_2\left(\Omega \right)$, defined on a bounded domain $\Omega \subset {\mathbb{R}}^d$, involving explicit constants depending on $m$ and $d$. The key of proof is to relate the approximation problems to certain Dirichlet and Neumann eigenvalue problems.

中文翻译：

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非周期Sobolev嵌入的近似数的精确估计

我们研究紧凑Sobolev嵌入的近似数的渐近尖锐上下界 ${\stackrel{\circ }{\mathrm{w\; ^}}}^{米}（\Omega ）\hookrightarrow {\mathrm{大号}}_2（\Omega ）$ 和 ${\mathrm{w\; ^}}^{米}（\Omega ）\hookrightarrow {\mathrm{大号}}_2（\Omega ）$，在有界域上定义 $\Omega \subset {\mathbb{[R}}^d$，涉及显式常量，具体取决于 $米$ 和 $d$。证明的关键是将逼近问题与某些Dirichlet和Neumann特征值问题联系起来。