  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.
