%0 Report
%D 2014
%T Where best to place a Dirichlet condition in an anisotropic membrane?
%A Paolo Tilli
%A Davide Zucco
%X 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.
%I SISSA
%G en_US
%U http://urania.sissa.it/xmlui/handle/1963/7481
%1 7592
%$ Submitted by dzucco@sissa.it (dzucco@sissa.it) on 2014-11-07T16:09:28Z
No. of bitstreams: 1
Tilli_Zucco_varifold.pdf: 379650 bytes, checksum: 341cfa2d9d25e41789652de4c99c22fe (MD5)
%0 Journal Article
%D 2013
%T Asymptotics of the first Laplace eigenvalue with Dirichlet regions of prescribed length
%A Paolo Tilli
%A Davide Zucco
%X 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.
%I Society for Industrial and Applied Mathematics
%G en
%U http://urania.sissa.it/xmlui/handle/1963/35141
%1 35379
%2 Physics
%4 1
%# MAT/05
%$ Approved for entry into archive by Maria Pia Calandra (calapia@sissa.it) on 2015-12-09T16:12:39Z (GMT) No. of bitstreams: 0
%R 10.1137/130916825