科研之友
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摘要
We 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 derivative 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 (Omega) over bar 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 -%26gt; infinity via Gamma-convergence, and we explicitly construct certain asymptotically optimal configurations. We also study the behavior as p -%26gt;infinity with L fixed, finding connections with maximum-distance problems related to the principal frequency of the infinity-Laplacian.
- 出版日期2013
