01539nas a2200121   4500008004100000245009200041210006900133260005100202520107800253100001701331700001801348856005101366       2013                        en  d00aAsymptotics of the first Laplace eigenvalue with Dirichlet regions of prescribed length0 aAsymptotics of the first Laplace eigenvalue with Dirichlet regio  bSociety for Industrial and Applied Mathematics3 aWe consider the problem of maximizing the first eigenvalue of the $p$-Laplacian (possibly with nonconstant 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 $\Sigma$, which 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.1 aTilli, Paolo1 aZucco, Davide  uhttp://urania.sissa.it/xmlui/handle/1963/35141