Asymptotics of the first Laplace eigenvalue with Dirichlet regions of prescribed length

Abstract

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.