Browsing by Author "Zucco, Davide"

Now showing 1 - 3 of 3
  • Thumbnail Image
    Item
    Convex combinations of low eigenvalues, Fraenkel asymmetries and attainable sets
    (SISSA, 2015) Mazzoleni, Dario; Zucco, Davide; Mathematics
    We 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.
  • Thumbnail Image
    Item
    Transmission conditions obtained by homogenisation
    (2018-03) Dal Maso, Gianni; Franzina, Giovanni; Zucco, Davide; Mathematics
    We 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.
  • Thumbnail Image
    Item
    Where best to place a Dirichlet condition in an anisotropic membrane?
    (SISSA, 2014) Tilli, Paolo; Zucco, Davide
    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.