00995nas a2200121 4500008004300000245007400043210006900117260001000186520059200196100001700788700001800805856005000823 2014 en_Ud 00aWhere best to place a Dirichlet condition in an anisotropic membrane?0 aWhere best to place a Dirichlet condition in an anisotropic memb bSISSA3 aWe 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.1 aTilli, Paolo1 aZucco, Davide uhttp://urania.sissa.it/xmlui/handle/1963/748101539nas a2200121 4500008004100000245009200041210006900133260005100202520107800253100001701331700001801348856005101366 2013 en d00aAsymptotics of the first Laplace eigenvalue with Dirichlet regions of prescribed length0 aAsymptotics of the first Laplace eigenvalue with Dirichlet regio bSociety for Industrial and Applied Mathematics3 aWe 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.1 aTilli, Paolo1 aZucco, Davide uhttp://urania.sissa.it/xmlui/handle/1963/35141