@article {2014,
title = {Where best to place a Dirichlet condition in an anisotropic membrane?},
number = {SISSA;61/2014/MATE},
year = {2014},
institution = {SISSA},
abstract = {We study a shape optimization problem for the first eigenvalue of an elliptic operator in divergence form, with non constant coefficients, over a fixed domain $\Omega$.
Dirichlet conditions are imposed along $\partial\Omega$ and, in addition, along a set $\Sigma$ of prescribed length ($1$-dimensional Hausdorff measure).
We look for the best shape and position for the supplementary Dirichlet region $\Sigma$ in order to maximize the first eigenvalue. We characterize the limit distribution of the optimal sets, as their prescribed length tends to infinity, via $\Gamma$-convergence.},
url = {http://urania.sissa.it/xmlui/handle/1963/7481},
author = {Paolo Tilli and Davide Zucco}
}
@article {2013,
title = {Asymptotics of the first Laplace eigenvalue with Dirichlet regions of prescribed length},
number = {SIAM Journal on Mathematical Analysis;volume 45; issue 6; pp. 3266-3282},
year = {2013},
publisher = {Society for Industrial and Applied Mathematics},
abstract = {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.},
doi = {10.1137/130916825},
url = {http://urania.sissa.it/xmlui/handle/1963/35141},
author = {Paolo Tilli and Davide Zucco}
}