Page 230 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 230
ARIATIONAL AND EVOLUTIONARY MODELS INVOLVING LOCAL/NONLOCAL
INTERACTIONS (MS-58)
Minimal planar N-partitions for large N
Giovanni Alberti, giovanni.alberti@unipi.it
University of Pisa, Italy
By minimal N -partition of a given planar domain E we mean a partition which consists of N
sets (cells) with equal area, that minimize the total perimeter (that is, the length of the union of
the boundaries of all cells).
T. C. Hales proved in 2001 that if E is a flat 2-dimensional torus, then a regular hexagonal
N -partition (if there is any) is minimal. It is then interesting to understand what happens for a
planar domain E that does not admit a regular hexagonal N-partition; in particular the following
questions naturally arise: Are the cells asymptotically hexagonal as N tends to infinity, and to
which extent the partition looks locally hexagonal? Is the partition rigid, in the sense that the
orientation of the cells is (essentially) the same through the domain?
In this talk I will describe some results obtained in these directions together with Marco Caroc-
cia (Politecnico di Milano) and Giacomo Del Nin (University of Warwick).
Phase Separation in Nonlocal Multispecies Models
Martin Burger, martin.burger@fau.de
FAU Erlangen-Nürnberg, Germany
In this talk we discuss some nonlocal variational problems arising as macroscopic steady states
of some many particle systems with different species. We will demonstrate the the emergence
of phase separation effects, explained from relations to Cahn-Hilliard Systems, and network
formation, whose theoretical understanding is still rather open. We will also comment on cor-
responding evolution equations formulated in terms gradient flows, e.g. in Wasserstein spaces.
One dimensional multi-agent optimal control and Mean Field limits with
density constraints
Annalisa Cesaroni, annalisa.cesaroni@unipd.it
University of Padova, Italy
In this talk I will consider a deterministic system (in one dimension) of many evolving inter-
acting agents with constraints on the reciprocal distance between agents, in which each agent
chooses its speed in order to minimize an energy depending on the position of the other agents
through an aggregative potential (given in term of an interaction kernel and a coercive function).
I will focus on periodic (in time) patterns of this model, discussing their qualitative properties,
and its macroscopic mean-field limit as the number of agents tends to infinity. The talk is based
on joint works with Marco Cirant (Padova).
228
INTERACTIONS (MS-58)
Minimal planar N-partitions for large N
Giovanni Alberti, giovanni.alberti@unipi.it
University of Pisa, Italy
By minimal N -partition of a given planar domain E we mean a partition which consists of N
sets (cells) with equal area, that minimize the total perimeter (that is, the length of the union of
the boundaries of all cells).
T. C. Hales proved in 2001 that if E is a flat 2-dimensional torus, then a regular hexagonal
N -partition (if there is any) is minimal. It is then interesting to understand what happens for a
planar domain E that does not admit a regular hexagonal N-partition; in particular the following
questions naturally arise: Are the cells asymptotically hexagonal as N tends to infinity, and to
which extent the partition looks locally hexagonal? Is the partition rigid, in the sense that the
orientation of the cells is (essentially) the same through the domain?
In this talk I will describe some results obtained in these directions together with Marco Caroc-
cia (Politecnico di Milano) and Giacomo Del Nin (University of Warwick).
Phase Separation in Nonlocal Multispecies Models
Martin Burger, martin.burger@fau.de
FAU Erlangen-Nürnberg, Germany
In this talk we discuss some nonlocal variational problems arising as macroscopic steady states
of some many particle systems with different species. We will demonstrate the the emergence
of phase separation effects, explained from relations to Cahn-Hilliard Systems, and network
formation, whose theoretical understanding is still rather open. We will also comment on cor-
responding evolution equations formulated in terms gradient flows, e.g. in Wasserstein spaces.
One dimensional multi-agent optimal control and Mean Field limits with
density constraints
Annalisa Cesaroni, annalisa.cesaroni@unipd.it
University of Padova, Italy
In this talk I will consider a deterministic system (in one dimension) of many evolving inter-
acting agents with constraints on the reciprocal distance between agents, in which each agent
chooses its speed in order to minimize an energy depending on the position of the other agents
through an aggregative potential (given in term of an interaction kernel and a coercive function).
I will focus on periodic (in time) patterns of this model, discussing their qualitative properties,
and its macroscopic mean-field limit as the number of agents tends to infinity. The talk is based
on joint works with Marco Cirant (Padova).
228
