  We consider a spectral optimal design problem involving the Neumann traces of
the Dirichlet-Laplacian eigenfunctions on a smooth bounded open subset $\Omega$
of $\R^n$. The cost functional measures the amount of energy that Dirichlet
eigenfunctions concentrate on the boundary and that can be recovered with a
bounded density function. We first prove that, assuming a $L^1$ constraint on
densities, the so-called {\it Rellich functions} maximize this
functional.Motivated by several issues in shape optimization or observation
theory where it is relevant to deal with bounded densities, and noticing that
the $L^\infty$-norm of {\it Rellich functions} may be large, depending on the
shape of $\Omega$, we analyze the effect of adding pointwise constraints when
maximizing the same functional. We investigate the optimality of {\it
bang-bang} functions and {\it Rellich densities} for this problem. We also deal
with similar issues for a close problem, where the cost functional is replaced
by a spectral approximation.Finally, this study is completed by the
investigation of particular geometries and is illustrated by several numerical
simulations.
