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.
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.
[03751] Skyrmion theory in magnetic thin films: the role of non-local magnetic dipolar interaction
Format : Talk at Waseda University
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.
[05516] Normalized solutions and limit profiles of the Gross-Pitaevskii-Poisson equation
Format : Online Talk on Zoom
Author(s) :
Vitaly Moroz (Swansea University)
Abstract : Gross-Pitaevskii-Poisson (GPP) equation is a nonlocal modification of the Gross-Pitaevskii equation with an attractive Coulomb-like term. It appears in the models of self-gravitating Bose-Einstein condensates proposed in cosmology and astrophysics to describe Cold Dark Matter and Boson Stars. We investigate the existence of prescribed mass (normalised) solutions to the GPP equation, paying special attention to the shape and asymptotic behaviour of the associated mass-energy relation curves and to the limit profiles of solutions at the endpoints of these curves. In particular, we show that after appropriate rescalings, the constructed normalized solutions converge either to a ground state of the Choquard equation, or to a compactly supported radial ground state of the integral Thomas-Fermi equation. In different regimes the constructed solutions include global minima, local but not global minima and unstable mountain-pass type solutions. This is a joint work with Riccardo Molle and Giuseppe Riey.
[04003] A distributional approach to nonlocal curvature motions
Format : Online Talk on Zoom
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).
[03830] The elastica functional as the critical Gamma-limit of the screened Gamow model
Format : Talk at Waseda University
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.
[04447] Reduced energies for thin ferromagnetic films with perpendicular anisotropy
Format : Talk at Waseda University
Author(s) :
Cyrill Muratov (University of Pisa)
Abstract : We derive four reduced two-dimensional models that describe, at different spatial scales, the micromagnetics of ultrathin ferromagnetic materials of finite spatial extent featuring perpendicular magnetic anisotropy and interfacial Dzyaloshinskii-Moriya interaction. Starting with a microscopic model that regularizes the stray field near the material’s lateral edges, we carry out an asymptotic analysis of the energy by means of Γ-convergence. Depending on the scaling assumptions on the size of the material domain vs. the strength of dipolar interaction, we obtain a hierarchy of the limit energies that exhibit progressively stronger stray field effects of the material edges. These limit energies feature, respectively, a renormalization of the out-of-plane anisotropy, an additional local boundary penalty term forcing out-of-plane alignment of the magnetization at the edge, a pinned magnetization at the edge, and, finally, a pinned magnetization and an additional field-like term that blows up at the edge, as the sample’s lateral size is increased. The pinning of the magnetization at the edge restores the topological protection and enables the existence of magnetic skyrmions in bounded samples.
[03755] Minimal partitions for local and nonlocal energies
Format : Talk at Waseda University
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.
[03839] Asymptotics of phase field models for crystal defects
Format : Talk at Waseda University
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.
