01022nas a2200121   4500008004100000245005500041210005500096520063900151100002100790700002300811700001800834856004800852       2018                        en  d00aTransmission conditions obtained by homogenisation0 aTransmission conditions obtained by homogenisation3 aWe study the asymptotic behaviour of solutions to variational problems in perforated
domains with Neumann boundary conditions. We consider perforations that in the limit
concentrate on a smooth manifold. We characterise the limits of the solutions and show that they solve a variational problem with a transmission condition across the manifold. This is expressed through a measure on the manifold, vanishing on sets of capacity zero. Then, we prove that every such measure can be obtained by homogenising suitable perforations. Eventually, we provide an asymptotic formula for this measure by using some auxiliary minimum problems.1 aDal Maso, Gianni1 aFranzina, Giovanni1 aZucco, Davide  uhttp://preprints.sissa.it/handle/1963/3531001304nas a2200133   4500008004100000245008300041210006900124520084400193100002201037700002201059700002001081700001801101856005101119       2016                        en  d00aConfinement of dislocations inside a crystal with a prescribed external strain0 aConfinement of dislocations inside a crystal with a prescribed e3 aWe study screw dislocations in an isotropic crystal undergoing antiplane shear.
In the framework of linear elasticity, by fixing a suitable boundary condition for the strain
(prescribed non-vanishing boundary integral), we manage to confine the dislocations inside the material. More precisely, in the presence of an external strain with circulation equal to n times the lattice spacing, it is energetically convenient to have n distinct dislocations lying inside the crystal. The novelty of introducing a Dirichlet boundary condition for the tangential strain is crucial to the confinement: it is well known that, if Neumann boundary conditions are imposed, the dislocations tend to migrate to the boundary. The results are achieved using PDE techniques and Ƭ-convergence theory, in the framework of the so-called core radius approach.1 aLucardesi, Ilaria1 aMorandotti, Marco1 aScala, Riccardo1 aZucco, Davide  uhttp://urania.sissa.it/xmlui/handle/1963/3524701185nas a2200121   4500008004100000245008500041210006900126260001000195520076800205100002100973700001800994856005101012       2015                        en  d00aConvex combinations of low eigenvalues, Fraenkel asymmetries and attainable sets0 aConvex combinations of low eigenvalues Fraenkel asymmetries and   bSISSA3 aWe consider the problem of minimizing convex combinations of the first two eigenvalues of the Dirichlet-Laplacian among open set of $R^N$ of fixed measure. We show that, by purely elementary arguments, based on the minimality condition, it is possible to obtain informations on the geometry of the minimizers of convex combinations: we study, in particular, when these minimizers are no longer convex, and the optimality of balls. As an application of our results we study the boundary of the attainable set for the Dirichlet spectrum.

Our techniques involve symmetry results à la Serrin, explicit constants in quantitative inequalities, as well as a purely geometrical problem: the minimization of the Fraenkel 2-asymmetry among convex sets of fixed measure.1 aMazzoleni, Dario1 aZucco, Davide  uhttp://urania.sissa.it/xmlui/handle/1963/3514000995nas 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