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

TL;DR

This paper studies how the optimal placement of a stiffening set within a domain affects the first Laplace eigenvalue, analyzing asymptotic behavior as the length constraint grows and as the parameter p approaches infinity.

Contribution

It introduces a novel optimization problem for the first eigenvalue involving a length-constrained set and analyzes its asymptotic behavior using $ ext{Gamma}$-convergence and explicit constructions.

Findings

Optimal sets become more extensive as length L increases.

Asymptotic configurations are explicitly constructed.

Connections established between eigenvalue maximization and maximum-distance problems.

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.