Page 232 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 232
ARIATIONAL AND EVOLUTIONARY MODELS INVOLVING LOCAL/NONLOCAL
INTERACTIONS (MS-58)
Minimisers of a fractional seminorm and nonlocal minimal surfaces
Serena Dipierro, serena.dipierro@uwa.edu.au
University of Western Australia, Australia
The recent literature has intensively studied two classes of nonlocal variational problems, namely
the ones related to the minimisation of energy functionals that act on functions in suitable
Sobolev-Gagliardo spaces, and the ones related to the minimisation of fractional perimeters
that act on measurable sets of the Euclidean space. In this talk, we relate these two types of
variational problems. In particular, we investigate the connection between the nonlocal mini-
mal surfaces and the minimisers of a Gagliardo seminorm, showing that a function is a min-
imiser for the fractional seminorm if and only if its level sets are minimisers for the fractional
perimeter, and that the characteristic function of a nonlocal minimal surface is a minimiser for
the fractional seminorm. We also discuss an existence result for minimisers of the fractional
seminorm, an explicit non-uniqueness example for nonlocal minimal surfaces, and a Yin-Yang
result describing the full and void patterns of nonlocal minimal surfaces. This is a joint work
with Claudia Bucur, Luca Lombardini and Enrico Valdinoci.
Striped patterns for generalized antiferromagnetic functionals with power
law kernels of exponent smaller than d + 2
Alicja Kerschbaum, kerschbaum@math.fau.de
FAU Erlangen-Nürnberg, Germany
In this talk I will consider a class of continuous sharp interface generalized antiferromagnetic
models previously studied by Daneri, Goldman and Runa. The functional consists of a perime-
ter term (retaining discrete symmetry) and a repulsive nonlocal term with a power law kernel. In
a suitable regime the two terms enter in competition and symmetry breaking with formation of
periodic striped patterns is expected to occur. We will show that the results obtained by Daneri
and Runa showing striped pattern formation for power law kernels with exponents p ≥ d + 2
can be extended to power law kernels within a range of exponents strictly smaller than d + 2
and strictly larger than d + 1, being d the dimension of the underlying space. Notice that the
exponent p = d + 1 corresponds to an anisotropic version (retaining discrete symmetry) of the
model for pattern formation in thin magnetic films.
Gamma–limit for zigzag walls
Hans Knüpfer, knuepfer@uni-heidelberg.de
Universität Heidelberg, Germany
Coauthor: Wenhui Shi
Ferromagnets typically exhibit the formation of magnetic domains with uniform magnetization
separated by thin transition layers. The Zigazag wall is one type of such transition layers which
particularly appears in thin ferromagnetic films. In order to investigate this transition layer, we
consider a sample in the form a thin strip and enforce a transition layer by suitable boundary
230
INTERACTIONS (MS-58)
Minimisers of a fractional seminorm and nonlocal minimal surfaces
Serena Dipierro, serena.dipierro@uwa.edu.au
University of Western Australia, Australia
The recent literature has intensively studied two classes of nonlocal variational problems, namely
the ones related to the minimisation of energy functionals that act on functions in suitable
Sobolev-Gagliardo spaces, and the ones related to the minimisation of fractional perimeters
that act on measurable sets of the Euclidean space. In this talk, we relate these two types of
variational problems. In particular, we investigate the connection between the nonlocal mini-
mal surfaces and the minimisers of a Gagliardo seminorm, showing that a function is a min-
imiser for the fractional seminorm if and only if its level sets are minimisers for the fractional
perimeter, and that the characteristic function of a nonlocal minimal surface is a minimiser for
the fractional seminorm. We also discuss an existence result for minimisers of the fractional
seminorm, an explicit non-uniqueness example for nonlocal minimal surfaces, and a Yin-Yang
result describing the full and void patterns of nonlocal minimal surfaces. This is a joint work
with Claudia Bucur, Luca Lombardini and Enrico Valdinoci.
Striped patterns for generalized antiferromagnetic functionals with power
law kernels of exponent smaller than d + 2
Alicja Kerschbaum, kerschbaum@math.fau.de
FAU Erlangen-Nürnberg, Germany
In this talk I will consider a class of continuous sharp interface generalized antiferromagnetic
models previously studied by Daneri, Goldman and Runa. The functional consists of a perime-
ter term (retaining discrete symmetry) and a repulsive nonlocal term with a power law kernel. In
a suitable regime the two terms enter in competition and symmetry breaking with formation of
periodic striped patterns is expected to occur. We will show that the results obtained by Daneri
and Runa showing striped pattern formation for power law kernels with exponents p ≥ d + 2
can be extended to power law kernels within a range of exponents strictly smaller than d + 2
and strictly larger than d + 1, being d the dimension of the underlying space. Notice that the
exponent p = d + 1 corresponds to an anisotropic version (retaining discrete symmetry) of the
model for pattern formation in thin magnetic films.
Gamma–limit for zigzag walls
Hans Knüpfer, knuepfer@uni-heidelberg.de
Universität Heidelberg, Germany
Coauthor: Wenhui Shi
Ferromagnets typically exhibit the formation of magnetic domains with uniform magnetization
separated by thin transition layers. The Zigazag wall is one type of such transition layers which
particularly appears in thin ferromagnetic films. In order to investigate this transition layer, we
consider a sample in the form a thin strip and enforce a transition layer by suitable boundary
230
