Page 233 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 233
IATIONAL AND EVOLUTIONARY MODELS INVOLVING LOCAL/NONLOCAL
INTERACTIONS (MS-58)
conditions on m. The associated thin–film ferromagnetic energy is
Eε[m] = ε ∇m 2 + 1 m · e2 2 + πλ ∇ · (m − M ) ,2 1
2 L2 2ε L2 2| ln ε| H˙ − 2
where M is an arbitrary fixed background field to ensure global neutrality of magnetic charges.
In the macroscopic limit ε → 0 we show that the energy Γ–converges to a limit energy where
jump discontinuities of the magnetization are penalized anisotropically. In particular, in the sub-
critical regime λ ≤ 1 one–dimensional charged domain walls are favorable, in the supercritical
regime λ > 1 the limit model allows for zigzaging two–dimensional domain walls.
Stability of nonlocal geometric evolutions
Matteo Novaga, matteo.novaga@unipi.it
University of Pisa, Italy
We introduce a notion of uniform convergence for local and nonlocal curvatures, and we pro-
pose an abstract method to prove the convergence of the corresponding geometric flows, within
the level set formulation. We apply such a general method to characterize the limits of fractional
mean curvature flows and Riesz curvature flows.
Torus-like solutions for the Landau-de Gennes model
Adriano Pisante, pisante@mat.uniroma1.it
University of Roma “La Sapienza”, Italy
We study global minimizers of a continuum Landau-De Gennes energy functional for nematic
liquid crystals in three-dimensional domains, possibly in a restricted class of axisymmetric
configurations. Assuming smooth and uniaxial (e.g. homeotropic) boundary conditions and a
corresponding physically relevant norm constraint (Lyuksyutov constraint) in the interior, we
discuss partial regularity of the minimizers away from a possible finite set of interior singular-
ities lying on the symmetry axis or even full regularity when no symmetry is imposed. As a
consequence, we discuss boundary data which yield as minimizers smooth configuration with
maximally biaxial set carrying nontrivial topology. In the axially symmetric case we show how
singular (split) solutions or smooth (torus) solutions (or even both) for the Euler-Lagrange equa-
tions do appear for boundary data and/or domains which are smooth deformation of the radial
hedgehog in a nematic droplet.
Optimal design spectral problems with repulsion
Berardo Ruffini, ber.ruffini@gmail.com
Università di Bologna, Italy
In the short talk I will quickly review the state of the art of some variational problems based
on physical models (the Rayleigh liquid drop model, the Gamow liquid drop and the reduced
Hartree equation). Then I will focus on recent results about an optimal design problem related to
the latter of these models. The talk will take into consideration collaborations with M. Goldman,
C. B. Muratov, M. Novaga and D. Mazzoleni.
231
INTERACTIONS (MS-58)
conditions on m. The associated thin–film ferromagnetic energy is
Eε[m] = ε ∇m 2 + 1 m · e2 2 + πλ ∇ · (m − M ) ,2 1
2 L2 2ε L2 2| ln ε| H˙ − 2
where M is an arbitrary fixed background field to ensure global neutrality of magnetic charges.
In the macroscopic limit ε → 0 we show that the energy Γ–converges to a limit energy where
jump discontinuities of the magnetization are penalized anisotropically. In particular, in the sub-
critical regime λ ≤ 1 one–dimensional charged domain walls are favorable, in the supercritical
regime λ > 1 the limit model allows for zigzaging two–dimensional domain walls.
Stability of nonlocal geometric evolutions
Matteo Novaga, matteo.novaga@unipi.it
University of Pisa, Italy
We introduce a notion of uniform convergence for local and nonlocal curvatures, and we pro-
pose an abstract method to prove the convergence of the corresponding geometric flows, within
the level set formulation. We apply such a general method to characterize the limits of fractional
mean curvature flows and Riesz curvature flows.
Torus-like solutions for the Landau-de Gennes model
Adriano Pisante, pisante@mat.uniroma1.it
University of Roma “La Sapienza”, Italy
We study global minimizers of a continuum Landau-De Gennes energy functional for nematic
liquid crystals in three-dimensional domains, possibly in a restricted class of axisymmetric
configurations. Assuming smooth and uniaxial (e.g. homeotropic) boundary conditions and a
corresponding physically relevant norm constraint (Lyuksyutov constraint) in the interior, we
discuss partial regularity of the minimizers away from a possible finite set of interior singular-
ities lying on the symmetry axis or even full regularity when no symmetry is imposed. As a
consequence, we discuss boundary data which yield as minimizers smooth configuration with
maximally biaxial set carrying nontrivial topology. In the axially symmetric case we show how
singular (split) solutions or smooth (torus) solutions (or even both) for the Euler-Lagrange equa-
tions do appear for boundary data and/or domains which are smooth deformation of the radial
hedgehog in a nematic droplet.
Optimal design spectral problems with repulsion
Berardo Ruffini, ber.ruffini@gmail.com
Università di Bologna, Italy
In the short talk I will quickly review the state of the art of some variational problems based
on physical models (the Rayleigh liquid drop model, the Gamow liquid drop and the reduced
Hartree equation). Then I will focus on recent results about an optimal design problem related to
the latter of these models. The talk will take into consideration collaborations with M. Goldman,
C. B. Muratov, M. Novaga and D. Mazzoleni.
231
