Abstract : This mini-symposium focuses on recent developments in the calculus of variations
with application to problems in nonlinear elasticity, plasticity, liquid crystals,
and foams, with an emphasis on topological defects.
Defects play a prominent role, for example, in superfluidity, superconductivity,
and plasticity. Low-energy configurations exhibit lower-dimensional concentration
patterns; examples include dislocations, vortices, grain boundaries, interfaces,
and phase transitions.
The aim of the mini-symposium is to bring together experts focusing on
different aspects of this common big picture, to promote exchange of ideas,
and identify new ways of tackling open problems.
Organizer(s) : Georg Dolzmann, Adriana Garroni, Lucia Scardia
[04293] Ginzburg-Landau with Oblique Anchoring and Boojums
Format : Talk at Waseda University
Author(s) :
Lia Bronsard (McMaster University)
Stan Alama (McMaster University)
Dmitry Golovaty (The University of Akron)
Abstract : We study the Ginzburg-Landau functional with oblique angle condition via boundary penalization. We consider the singular limit and for strong anchoring strength, defects will occur in the interior, but for weaker anchoring strength all defects will occur on the boundary. These `boojums'' defects carry a fractional winding number and will occur in ordered pairs along the boundary. For the ``light" boojums, we prove an asymptotic convergence. S. Alama, D. Golovaty, P. Mironescu are collaborators.
[04158] Dipole removal for discrete energy minimizers
Format : Talk at Waseda University
Author(s) :
Mircea Petrache (Pontificia Catolica Universidad de Chile)
Adriana Garroni (University of Rome La Sapienza)
Emanuele Spadaro (University of Rome La Sapienza)
Abstract : We consider a minimization problem for vector fields in the plane, allowing discrete vortex-like singularities, and we find conditions on the boundary datum on the boundary of a ball, under which the minimum-energy optimizer must have a single interior singularity. A consequence is that the screw dislocation energy minimizers with continuous boundary datum close enough to $u(\theta)=\theta$ will have exactly one charge.
The approach passes by a discretized version of the problem of independent interest, connected to the initial problem via the Smirnov decomposition of 1-currents, and proved by a discussion based on the MaxFlow - MinCut theorem. The same strategy may apply to a larger class of problems with integer-degree topological singularities.
Extension of the 1-charge result are given for cases where the minimum possible number of interior singularities is larger than 1, and we give counterexamples to further extensions of the result.
[03999] From Volterra's dislocations to strain-gradient plasticity
Format : Talk at Waseda University
Author(s) :
Raz Kupferman (The Hebrew University)
Cy Maor (The Hebrew university)
Abstract : Dislocations, first classified by Volterra, can be viewed as generated by cut and weld procedures. It is only in the last 15 years that plasticity models have been derived rigorously as limits of models of finitely-many dislocations. In most of these works, the elemental dislocation is modeled as an “admissible” strain field, which is in a sense, a pre-assumed linearization of Volterra's model. I will show how strain gradient plasticity are obtained from Volterra's model.
[01900] Harmonic dipoles in elasticity
Format : Talk at Waseda University
Author(s) :
Duvan Henao (Universidad de O'Higgins)
Marco Barchiesi (Università degli Studi di Trieste)
Carlos Mora-Corral (Universidad Autónoma de Madrid)
Rémy Rodiac (Université Paris Saclay)
Abstract : Malý (1993) proved that the relaxation of the neoHookean energy coincides with the neoHookean energy at diffeomorphisms, thus establishing the first existence result for neoHookean materials in 3D. We present some progress on the more explicit understanding of what deformations can fall into the weak closure of regular (injective, orientation-preserving, controlled Jacobian) maps and of the relaxed energy evaluated at deformations with singularities. For the pathological example of Conti & De Lellis (2003) we show that the singular energy is precisely twice the length of the dipole times the area of the bubble across which two portions of the elastic body which were separated in the reference configuration are now in contact in the deformed configuration. This, in turns, coincides with twice the total variation of the singular part of the derivative of the inverse map. We show that in the weak closure all maps have inverses with BV regularity, and in the axisymmetric case establish the Sobolev regularity for the first two components. In this axisymmetric case we obtain, as a lower bound for the singular energy, precise twice the variation of the singular part of the inverse. In the case of map with further SBV regularity for the inverse, we show that the singularities are dipoles, showing that the example of Conti & De Lellis is very generic.
[03156] effective geometric motions of Ginzburg--Landau equations with potentials of high-dimensional wells
Format : Talk at Waseda University
Author(s) :
Yuning Liu (NYU shanghai)
Abstract : We study the co-dimensional one interface limit and geometric motions of parabolic Ginzburg--Landau systems with potentials of high-dimensional wells. In particular combining modulated energy methods and weak convergence methods, we derive a sharp interface limit and the limiting harmonic heat flows in the inner and outer bulk regions segregated by the sharp interface.
[02113] Quasistatic evolution problems for models of geomaterials coupling plasticity and damage
Format : Talk at Waseda University
Author(s) :
Vito Crismale (Sapienza Università di Roma)
Abstract : I will discuss existence of quasistatic evolutions for a model proposed by Kazymyrenko and Marigo in 2019, which uses a suitable coupling between plasticity and damage to study the behavior of geomaterials under compression.
[04281] Liquid crystal colloids: from the electrostatic analogy to interaction energies
Format : Talk at Waseda University
Author(s) :
Raghavendra Venkatraman (Courant Institute )
Abstract : We discuss some recent progress on nematic liquid crystal colloids, with the goal of deriving simplified descriptions of colloidal suspensions in a liquid crystal matrix. The first half of the talk will provide justification of the so-called "electrostatic analogy" frequently used in physics (and proposed by Brochard and De Gennes in the 70s). The second half will be about interaction energies for paranematic colloids.
[03855] Evolution of vector fields on flexible curves and surfaces
Format : Talk at Waseda University
Author(s) :
Georg Dolzmann (University of Regensburg)
Abstract : We discuss some recent progress on a model system consisting of a flexible surface and a vector field
defined on the surface in the case in which an interaction between the vector field and the conformation of the surface
is present. Recent approaches towards the existence of solutions will be reviewed and short time existence will be
established.
The lecture is based on joint work with Christopher Brand (Regensburg), Julia Menzel (Regensburg) and Alessandra
Pluda (Pisa).
