  We consider the problem of maximizing the first eigenvalue of the
$p$-laplacian (possibly with non-constant coefficients) over a fixed domain
$\Omega$, with Dirichlet conditions along $\partial\Omega$ and along a
supplementary set $\Sigma$, which is the unknown of the optimization problem.
The set $\Sigma$, that plays the role of a supplementary stiffening rib for a
membrane $\Omega$, is a compact connected set (e.g. a curve or a connected
system of curves) that can be placed anywhere in $\overline{\Omega}$, and is
subject to the constraint of an upper bound $L$ to its total length
(one-dimensional Hausdorff measure). This upper bound prevents $\Sigma$ from
spreading throughout $\Omega$ and makes the problem well-posed. We investigate
the behavior of optimal sets $\Sigma_L$ as $L\to\infty$ via
$\Gamma$-convergence, and we explicitly construct certain asymptotically
optimal configurations. We also study the behavior as $p\to\infty$ with $L$
fixed, finding connections with maximum-distance problems related to the
principal frequency of the $\infty$-laplacian.
