TY - RPRT
T1 - Transmission conditions obtained by homogenisation
Y1 - 2018
A1 - Gianni Dal Maso
A1 - Giovanni Franzina
A1 - Davide Zucco
AB - 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.
UR - http://preprints.sissa.it/handle/1963/35310
U1 - 35618
U2 - Mathematics
U4 - 1
U5 - MAT/05
ER -
TY - JOUR
T1 - Confinement of dislocations inside a crystal with a prescribed external strain
Y1 - 2016
A1 - Ilaria Lucardesi
A1 - Marco Morandotti
A1 - Riccardo Scala
A1 - Davide Zucco
AB - We 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.
UR - http://urania.sissa.it/xmlui/handle/1963/35247
N1 - Preprint SISSA 20/2016/MATE
U1 - 35558
U2 - Mathematics
ER -
TY - RPRT
T1 - Convex combinations of low eigenvalues, Fraenkel asymmetries and attainable sets
Y1 - 2015
A1 - Dario Mazzoleni
A1 - Davide Zucco
AB - 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.
PB - SISSA
UR - http://urania.sissa.it/xmlui/handle/1963/35140
U1 - 35378
U2 - Mathematics
U4 - 1
U5 - MAT/05
ER -
TY - RPRT
T1 - Where best to place a Dirichlet condition in an anisotropic membrane?
Y1 - 2014
A1 - Paolo Tilli
A1 - Davide Zucco
AB - 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.
PB - SISSA
UR - http://urania.sissa.it/xmlui/handle/1963/7481
U1 - 7592
ER -
TY - JOUR
T1 - Asymptotics of the first Laplace eigenvalue with Dirichlet regions of prescribed length
Y1 - 2013
A1 - Paolo Tilli
A1 - Davide Zucco
AB - 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.
PB - Society for Industrial and Applied Mathematics
UR - http://urania.sissa.it/xmlui/handle/1963/35141
U1 - 35379
U2 - Physics
U4 - 1
U5 - MAT/05
ER -