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Asymptotics of the first laplace eigenvalue with dirichlet regions of prescribed length

SIAM Journal on Mathematical Analysis, ISSN: 0036-1410, Vol: 45, Issue: 6, Page: 3266-3282
2013
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  • Citations
    9
    • Citation Indexes
      9

Article Description

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. © 2013 Society for Industrial and Applied Mathematics.

Bibliographic Details

Paolo Tilli; Davide Zucco

Society for Industrial & Applied Mathematics (SIAM)

Mathematics

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