TY - JOUR
T1 - Asymptotics of the first Laplace eigenvalue with Dirichlet regions of prescribed length
Y1 - 2013
A1 - Paolo Tilli
A1 - Davide Zucco
AB - 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 $\partial\Omega$ 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 $\overline{\Omega}$ 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.
PB - Society for Industrial and Applied Mathematics
UR - http://urania.sissa.it/xmlui/handle/1963/35141
U1 - 35379
U2 - Physics
U4 - 1
U5 - MAT/05
ER -