Page 234 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 234
ARIATIONAL AND EVOLUTIONARY MODELS INVOLVING LOCAL/NONLOCAL
INTERACTIONS (MS-58)
Pattern formation in local/non-local models interaction functionals
Eris Runa, eris.runa@gmail.com
Deutsche Bank, Germany
In this talk I will review some recent results on the one-dimensionality of the minimizers of
a family of continuous local/nonlocal interaction func- tionals in general dimension. Such
functionals have a local term, typically a perimeter term or its Modica-Mortola approxima-
tion, which penalizes inter- faces, and a nonlocal term favouring oscillations which are high in
frequency and in amplitude. The competition between the two terms is expected by experiments
and simulations to give rise to periodic patterns at equilibrium. Functionals of this type are used
to model pattern formation, either in mate- rial science or in biology. One of the main difficulties
in proving the emergence of such regular structures, together with nonlocality, is due to the fact
that the functionals retain more symmetries (in this case symmetry with respect to permutation
of coordinates) than the minimizers. We will present new techniques and results showing that
for two classes of functionals (used to model generalized anti-ferromagnetic systems, respec-
tively colloidal suspensions), both in sharp interface and in diffuse interface models, minimizers
are (in general dimension) one-dimensional and periodic.
Convexity properties of the isoperimetric profile
Giorgio Saracco, gsaracco@sissa.it
Scuola Internazionale Superiore di Studi Avanzati, Italy
Coauthor: Gian Paolo Leonardi
Given an open, bounded set Ω we consider the isoperimetric profile J that to each volume
V ∈ [0, |Ω|] associates the least perimeter P (E) among Borel subsets E of Ω needed to enclose
the given volume. We shall prove that for a wide class of planar sets, which encompasses
convex sets, there exists a threshold V¯ such that J is concave below it and convex above it.
Moreover, J 2 is globally convex. In order to prove these properties, a full characterization
of the isoperimetric sets will be provided. Some comments on the n-dimensional case will be
given.
Stripe formation in Ising models with competing interactions
Robert Seiringer, rseiring@ist.ac.at
IST Austria, Austria
We consider Ising models in two and three dimensions, with short range ferromagnetic and long
range, power-law decaying, antiferromagnetic interactions. The competition between these two
kinds of interactions induces the system to form domains of minus spins in a background of
plus spins, or vice versa. If the decay exponent of the long range interaction is large enough,
this happens if the ratio J between the strength of the ferromagnetic and antiferromagnetic
interactions is smaller than a critical value Jc, beyond which the ground state is homogeneous.
We give a characterization of the infinite volume ground states of the system, for J in a left
neighborhood of Jc. In particular, we prove that the quasi-one-dimensional states consisting
of infinite stripes (d = 2) or slabs (d = 3), all of the same optimal width and orientation, and
232
INTERACTIONS (MS-58)
Pattern formation in local/non-local models interaction functionals
Eris Runa, eris.runa@gmail.com
Deutsche Bank, Germany
In this talk I will review some recent results on the one-dimensionality of the minimizers of
a family of continuous local/nonlocal interaction func- tionals in general dimension. Such
functionals have a local term, typically a perimeter term or its Modica-Mortola approxima-
tion, which penalizes inter- faces, and a nonlocal term favouring oscillations which are high in
frequency and in amplitude. The competition between the two terms is expected by experiments
and simulations to give rise to periodic patterns at equilibrium. Functionals of this type are used
to model pattern formation, either in mate- rial science or in biology. One of the main difficulties
in proving the emergence of such regular structures, together with nonlocality, is due to the fact
that the functionals retain more symmetries (in this case symmetry with respect to permutation
of coordinates) than the minimizers. We will present new techniques and results showing that
for two classes of functionals (used to model generalized anti-ferromagnetic systems, respec-
tively colloidal suspensions), both in sharp interface and in diffuse interface models, minimizers
are (in general dimension) one-dimensional and periodic.
Convexity properties of the isoperimetric profile
Giorgio Saracco, gsaracco@sissa.it
Scuola Internazionale Superiore di Studi Avanzati, Italy
Coauthor: Gian Paolo Leonardi
Given an open, bounded set Ω we consider the isoperimetric profile J that to each volume
V ∈ [0, |Ω|] associates the least perimeter P (E) among Borel subsets E of Ω needed to enclose
the given volume. We shall prove that for a wide class of planar sets, which encompasses
convex sets, there exists a threshold V¯ such that J is concave below it and convex above it.
Moreover, J 2 is globally convex. In order to prove these properties, a full characterization
of the isoperimetric sets will be provided. Some comments on the n-dimensional case will be
given.
Stripe formation in Ising models with competing interactions
Robert Seiringer, rseiring@ist.ac.at
IST Austria, Austria
We consider Ising models in two and three dimensions, with short range ferromagnetic and long
range, power-law decaying, antiferromagnetic interactions. The competition between these two
kinds of interactions induces the system to form domains of minus spins in a background of
plus spins, or vice versa. If the decay exponent of the long range interaction is large enough,
this happens if the ratio J between the strength of the ferromagnetic and antiferromagnetic
interactions is smaller than a critical value Jc, beyond which the ground state is homogeneous.
We give a characterization of the infinite volume ground states of the system, for J in a left
neighborhood of Jc. In particular, we prove that the quasi-one-dimensional states consisting
of infinite stripes (d = 2) or slabs (d = 3), all of the same optimal width and orientation, and
232
