Asymptotics of the First Laplace Eigenvalue with Dirichlet Regions of Prescribed Length

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