Sharp estimates for approximation numbers of non-periodic Sobolev embeddings

We investigate asymptotically sharp upper and lower bounds for the approximation numbers of the compact Sobolev embeddings $\overset{\circ}{W}{^m}(\Omega)\hookrightarrow L_2(\Omega)$ and $W^m(\Omega)\hookrightarrow L_2(\Omega)$, 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.