Abstracts

We consider the sharp interface Ohta-Kawasaki functional related to the modeling of diblock copolymers. After reviewing the model and a local minimality criterion recently established, we present several applications to the study of the stability and the global minimality of striped patterns.

This is a joint work with V. Rybalko.

The search for second order sufficient minimality conditions is a very well studied issue in the Calculus of Variations. Nevertheless, in the framework of free-discontinuity problems, this kind of study has been started only in recent years.

In this talk, I will present a second variation approach to a nonlocal variant of the isoperimetric problem, leading to some new local and, in some cases, global minimality results.

This is joint work with Marco Bonacini.

I will present several homogenization results, dimensional reduction ones and existence of minimizers for integral and supremal functionals which find application in different fields such as elasticity, fracture mechanics, optimal design, electrology.

The results have been obtained in collaborations with members of the project "Thin Structures, Homogenization and Multi-Phase Problems".

It is well known that the Dirichlet intergal \(\int_{\mathbb{R}^n} |\nabla u|^p\) decreases under the Steiner symmetrization about a hyperplane or the Schwarz spherical symmetrization about a point. This is indeed a consequence of a Pòlya-Szegő type inequality. Recently severlas studies have been devoted to the issue of characterizing the extremals in this type of inequalities and this led the research through the natural questions of proving quantitative versions.

The aim of the talk is to present a stability result for the Pòlya-Szegő inequality. In particular we will discuss some natural geometric conditions needed to deal with the fact that, even when the Dirichlet integral of a function \(u\) and of its symmetral coincide, \(u\) can be very different from its symmetral.

The talk is based on a recent result obtained in collaboration with M. Barchiesi, N. Fusco, and G.M. Capriani.

Mathematical settings in which heterogeneous structures affect electron transport through a tube-shaped quantum waveguide are studied, highlighting some interactions between heterogeneities and geometric parameters like curvature and torsion.

Epitaxially grown heterogeneous nanowires present dislocations at the interface between the phases if their radius is big. We consider a corresponding variational discrete model with quadratic pairwise atomic interaction energy.

By employing the notion of Gamma-convergence and a geometric rigidity estimate, we perform a discrete to continuum limit and a dimension reduction to a one-dimensional system. Moreover, we compare a defect-free model and models with dislocations at the interface and show that the latter are energetically convenient if the thickness of the wire is sufficiently large.

From joint work with M. Palombaro and A. Schlömerkemper.

Some aspects of dislocation modelling at the continuum scale is the framework of linear and nonlinear elasticity will be discussed. In particular, a variational model and modelling by means od Cartesian currents will be proposed. The incompatibility operator will be introduced.

We study the analytic properties of the deformation gradient of a single crystal around dislocations. We consider a variational problem with energies linked both to the elastic response of the bulk and to the defect induced by dislocations. By the use of geometric and analytic tools as currents and the theory of harmonic and cartesian maps it is possible to obtain theorems of existence of minimizers among a large class of competitors.

Dislocations are geometric lattice defects which are the carriers of plastic deformation in crystals, and the classical approach to modelling them has been to use continuum linear elasticity theory. Such an approach introduces singularities with infinite elastic energy however, which indicate the breakdown of the assumption that the body is well-approximated by a continuum. To better understand the behaviour of dislocations at the microscopic level, this talk studies a lattice model in which these singularities are not present, proving the existence of stable configurations of screw dislocations in a variety of domains with arbitrary net Burgers vector. Along the way, we show that continuum linear elasticity does provide a good approximation of the equilibrium configurations.

The talk will focus on the ground state (first eigenpair) asymptotics for a singularly perturbed second order elliptic operator with rapidly oscillating locally periodic coefficients. The corresponding spectral problem is stated in a regular bounded domain with homogeneous Dirichlet boundary condition.

It will be shown that, under the ellipticity assumptions, the limit of principal eigenvalue and the logarithmic asymptotics of principal eigenfunction can be described in terms of a viscosity solution to "homogenized" Hamilton-Jacobi type equation with state constrint boundary condition.

A viscosity solution of homogenized problem need not be unique. In certain particular cases we can construct the second term of the eigenpair asymptotics and choose a particular solution of homogenized problem which is responsible for the asymptotics in question.

This is a joint work with V. Rybalko.

Integral functionals acting on functions of bounded variation arise in the Calculus of Variations in the study of variational problems with linear growth, in image processing, and in the mathematical theory of phase transitions. Questions of interest include: given an integral functional defined on smooth functions, what is the "right" way to extend it to the space $BV$? How do these extensions behave under weak* convergence in $BV$? In this talk, I will answer the first question by describing a recent result which shows how to continuously extend integral functionals defined on smooth functions to the space $BV$. Since this extension is continuous with respect to a topology under which smooth functions are dense, it is unique. I will also describe my current topic of research, which aims to answer the second question by developing a theory of Young measures for $BV$ functions and their derivatives.

In this talk we discuss the shape optimization problem of how to distribute a given volume fraction of elastic material in a cylindrical design region of infinitesimal cross section in order to maximize the resistance to a twisting load. We enlighten how the possible presence of homogenized regions in an optimal design is related to a nonstandard free boundary problem with a gradient obstacle settled on the rod's cross section. A full understanding of this problem, in particular in connection with the geometry of the underlying domain, is still missing. We present some insights and partial results.

Working with variational principles subject to linear PDE constraints conveyed by a constant-rank operator \(\mathcal{A}\) allows us to treat a number of problems in continuum mechanics and electromagnetism in a unified way. The topic of this talk, which reports on a joint work with Filip Rindler (University of Warwick), is 3d-2d dimension reduction within this general framework. We study the effective behavior of integral functionals as the thickness \(\varepsilon\) of the domain tends to zero. Under certain onditions we show that the \(\Gamma\)-limit is an integral functional and give an explicit formula. The limit functional turns out to be constrained to \(\mathcal{A}_o\)-free vector fields, where the limit opertor \(\mathcal{A}_0\) is in general not of constant rank. As applications, we characterize a thin-film \(\Gamma\)-limit in micromagnetics and recover the energy of a membrane model with bending moment in nonlinear elasticity.

Using the calculus of variations it is shown that important qualitative features of the equilibrium shape of a material void in a linearly elastic solid may be deduced from smoothness and convexity properties of the interfacial energy.

In addition, short time existence, uniqueness, and regularity for an anisotropic surface diffusion evolution equation with curvature regularization are proved in the context of epitaxially strained two-dimensional films. This is achieved by using the \(H^{-1}\)-gradient flow structure of the evolution law, via De Giorgi's minimizing movements. This seems to be the first short time existence result for a surface diffusion type geometric evolution equation in the presence of elasticity.

In this talk I will present some joint works with P. Bella and B. Zwicknagl on the study of a variational model describing the epitaxial deposition of a thin crystalline film on a rigid substrate when there is a mismatch between the crystal lattices. Due to this misfit, a strain is induced in the film during the deposition. The resulting functional is a non-local isoperimetric functional which accounts for the competition between surface and elastic energy.

In the talk I will present a local minimality sufficiency criterion, based on the strict positivity of the second variation, in the context of a variational model for the epitaxial growth of elastic films. Our result holds also in the three-dimensional case and for a general class of nonlinear elastic energies and anisotropic surface energies. Applications to the study of the local minimality of flat morphologies are also shown.

We obtain a cohesive fracture model as a \(\Gamma\)-limit of damage models, in which the elastic coefficient is computed from the damage variable \(v\) through a function \(f_k\) of the form \(f_k(v)=\min(1,\epsilon_k^{1/2}f(v))\), with \(f\) diverging for \(v\) close to the value describing undamaged material.

The resulting fracture energy is linear in the opening at small values of \(s\), has a finite limit as \(s\) tends to infinity, and can be determined by solving a one-dimensional vectorial optimal profile problem.

Further, we extend the result to the Dugdale's and Griffith's fracture models, and to models with surface energy density having a power-law growth at small openings.