[00356] Recent progress in variational problems with nonlocality
Session Date & Time :
00356 (1/2) : 3C (Aug.23, 13:20-15:00)
00356 (2/2) : 3D (Aug.23, 15:30-17:10)
Type : Proposal of Minisymposium
Abstract : This minisymposium will discuss some recent developments in the analysis of variational problems from science and engineering in which nonlocal interactions have a pronounced effect. Examples will include geometric variational problems with long-range repulsion, topologically non-trivial spin configurations in magnetic materials, long-range interactions in phase transitions, capillary theory and theory of dislocations.
Serena Dipierro (University of Western Australia )
Enrico Valdinoci (University of Western Australia )
Lucia Scardia (Heriot Watt University)
Anne Bernand-Mantel (INSA)
Annalisa Cesaroni (University of Padua)
Massimiliano Morini (University of Parma)
Theresa Simon (University of Muenster)
Adriana Garroni (University of Rome 1)
Talks in Minisymposium :
[03751] Skyrmion theory in magnetic thin films: the role of non-local magnetic dipolar interaction
Author(s) :
Anne Bernand-Mantel
Cyrill Muratov (University of Pisa)
Theresa Simon (Muenster University)
Valeriy Slastikov (Bristol University)
Abstract : Compact magnetic skyrmion are potential bit-encoding states for spintronic memory and logic applications that have been the subject of a rapidly growing number of studies in recent years. We will present our recent work where we used rigorous mathematical analysis to develop a skyrmion theory that takes into account the full dipolar energy in the thin film regime and provides analytical formulas for compact skyrmion radius, rotation angle and energy.
[03755] Minimal partitions for local and nonlocal energies
Author(s) :
Annalisa Cesaroni (University of Padova)
Abstract : The Kelvin problem, posed by Lord Kelvin in 1887, is the problem of finding a partition of $\mathbb{R}^3$ into cells
of equal volume, so that the total area of the surfaces separating them is as small as possible.
I will discuss some related problems in $\mathbb{R}^n$, in particular the problem of finding the foam whose
cell minimizes a general perimeter functional among all periodic partitions given by lattice tilings.
Moreover I will present some qualitative results in low dimension.
[03768] Nonlocal capillarity theory
Author(s) :
Enrico Valdinoci (University of Western Australia)
Abstract : We describe some recent results motivated by a nonlocal theory of capillarity, as related to the formation of droplets due to long-range interaction potentials. We will discuss the notion of contact angle in this setting, considering a nonlocal version of the classical Young's Law, together with some regularity and asymptotic properties.
[03769] Long-range phase transition equations
Author(s) :
Serena Dipierro (University of Western Australia)
Abstract : Phase transitions are a classical topic of investigation. They represent a complex phenomenon which needs to be attacked with different methodologies and different perspectives. I will discuss some rigidity and symmetry results for a long-range phase coexistence equation, their close relation with surfaces of minimal perimeter and a famous conjecture by Ennio De Giorgi.
[03830] The elastica functional as the critical Gamma-limit of the screened Gamow model
Author(s) :
Theresa Simon (University of Münster)
Cyrill Muratov (University of Pisa)
Matteo Novaga (University of Pisa)
Abstract : I will consider the large mass limit of a nonlocal isoperimetric problem in two dimensions with screened Coulomb repulsion. In this regime, the nonlocal interaction localizes on the boundary of the sets. It turns out that in the case of exactly cancelled surface area, the problem changes from length to curvature minimization: The next-order Gamma limit is given by the elastica functional, i.e., the integral over the squared curvature over the boundary.
[03839] Asymptotics of phase field models for crystal defects
Author(s) :
Adriana Garroni (Sapienza, University of Rome)
Sergio Conti (University of Bonn)
Stefan Mueller (University of Bonn)
Abstract : We consider Nabarro Peierls type model for line defects in crystals. We study the asymptotics in scaling regime which allows for the number of dislocations to diverge and results, in the limit as the lattice spacing tends to zero, in a macroscopic model for plasticity where the relevant variable is a diffuse quantity that represents the dislocation density.
[04003] A distributional approach to nonlocal curvature motions
Author(s) :
Massimiliano Morini (University of Parma)
Abstract : After reviewing the new distributional approach recently developed to provide a well-posed formulation of the crystalline mean curvature flow, we show how to extend it to some nonlocal motions. Applications include the fractional mean curvature flow and the Minkowski flow; i.e., the geometric flow generated by the (n-1)-dimensional Minkowski pre-content. This is a work in collaboration with F. Cagnettti (University of Sussex) and D. Reggiani (Scuola Superiore Meridionale).
[04447] Minimisers of anisotropic Coulomb energies in 3d
Author(s) :
Lucia Scardia (Heriot-Watt University)
Abstract : Nonlocal energies are continuum models for large systems of particles with long-range interactions.
Under the assumption that the interaction potential is radially symmetric, several authors have investigated qualitative properties of energy minimisers.
But what can be said in the case of anisotropic kernels?
I will present some results and partial answers in this direction obtained in a long-standing collaboration with
Maria Giovanna Mora and Luca Rondi, and with Jose’ Antonio Carrillo, Joan Mateu and Joan Verdera.
