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Andrea Kubin

Publications and source records attributed to Andrea Kubin.

16 recordsLinked to original sources
arXiv2026

The anisotropic surface diffusion with elasticity in three dimensions via the Cahn-Taylor minimizing movement scheme

In this paper, we introduce a suitable notion of flat solutions for the anisotropic surface diffusion equation with elasticity in three-dimensions, based on a minimizing movement scheme inspired by that introduced by Cahn and Taylor. Using this scheme, we prove the existence of classical solutions to the equation without using a curvature regularization term. Moreover we establish the consistency of the approximation method.

math.AP↗
arXiv2026

A Bochner-type integration theory for random normed modules

We develop a measure and integration theory for random normed modules. Given a probability space $({\rm X},Σ,\mathfrak m)$, we introduce and study measures taking values into the space $L^0(\mathfrak m)$ of $\mathfrak m$-measurable functions quotiented up to $\mathfrak m$-a.e. equality. Moreover, we develop a Bochner-type integration theory with respect to an $L^0(\mathfrak m)$-valued measure $μ$, for maps whose target ${\rm M}$ is a complete random normed module with base $({\rm X},Σ,\mathfrak m)$, or equivalently an $L^0(\mathfrak m)$-Banach $L^0(\mathfrak m)$-module. Inter alia, we prove versions of the Radon-Nikodým theorem and of the Riesz-Markov-Kakutani representation theorem for $L^0(\mathfrak m)$-valued measures. We also outline several applications of our integration theory: we introduce a notion of martingale with values in a complete random normed module, we propose a definition of random Radon-Nikodým property and we discuss random sets of finite perimeter.

math.FA↗
arXiv2025

On the $Γ$-limit of weighted fractional energies

Given $p\in[1,\infty)$ and a bounded open set $Ω\subset\mathbb R^d$ with Lipschitz boundary, we study the $Γ$-convergence of the weighted fractional seminorm \[ [u]_{s,p,f}^p = \int_{\mathbb R^d} \int_{\mathbb R^d} \frac{|\tilde{u}(x)- \tilde{u}(y)|^p}{\|x-y\|^{d+sp}}\,f(x)\,f(y)\,\mathrm{d} x\,\mathrm{d} y \] as $s\to1^-$ for $u\in L^p(Ω)$, where $\tilde{u}=u$ on $Ω$ and $\tilde{u}=0$ on $\mathbb R^d\setminusΩ$. Assuming that $(f_s)_{s\in(0,1)}\subset L^\infty(\mathbb R^d;[0,\infty))$ and $f\in\mathrm{Lip}_b(\mathbb R^d;(0,\infty))$ are such that $f_s\to f$ in $L^\infty(\mathbb R^d)$ as $s\to1^-$, we show that $(1-s)[u]_{s,p,f_s}$ $Γ$-converges to the Dirichlet $p$-energy weighted by $f^2$. In the case $p=2$, we also prove the convergence of the corresponding gradient flows.

math.AP↗
arXiv2025

A variational approach to the volume-preserving anisotropic mean curvature flow in 2D

In this article, we introduce a variational algorithm, in the spirit of the minimizing movements scheme, to model the volume-preserving anisotropic mean curvature flow in 2D. We show that this algorithm can be used to prove the existence of classical solutions. Moreover, we prove that this algorithm converges to the global solution of the equation.

math.AP↗
arXiv2025

On variational scheme modeling the anisotropic surface diffusion with elasticity in the plane

In this paper, we prove the existence of classical solutions for the anisotropic surface diffusion with elasticity in the plane using a minimizing movements scheme, provided that the initial set is sufficiently regular. This scheme is inspired by the one introduced by Cahn-Taylor [15] to modeling the surface diffusion. Moreover, we prove that this scheme converges to the global solution of the equation.

math.AP↗
arXiv2025

Consistency for the surface diffusion flat flow in three dimensions

We investigate the flat flow solution for the surface diffusion equation via the discrete minimizing movements scheme proposed by Cahn and Taylor. We prove that in dimension three the scheme converges to the unique smooth solution of the equation, provided that the initial set is sufficiently regular.

math.AP↗
arXiv2024

Discrete and Continuum Area-Preserving Mean-Curvature Flow of Rectangles

We investigate the area-preserving mean-curvature-type motion of a two-dimensional lattice crystal obtained by coupling constrained minimizing movements scheme introduced by Almgren, Taylor and Wang with a discrete-to-continuous analysis. We first examine the continuum counterpart of the model and establish the existence and uniqueness of the flat flow, originating from a rectangle. Additionally, we characterize the governing system of ordinary differential equations. Subsequently, in the atomistic setting, we identify geometric properties of the discrete-in-time flow and describe the governing system of finite-difference inclusions. Finally, in the limit where both spatial and time scales vanish at the same rate, we prove that a discrete-to-continuum evolution is expressed through a system of differential inclusions which does never reduce to a system of ODEs.

math.AP↗
arXiv2024

A notion of $s$-fractional mass for $1$-currents in higher codimension

In this paper we propose a notion of $s$-fractional mass for $1$-currents in $\R^d$. Such a notion generalizes the notion of $s$-fractional perimeters for sets in the plane to higher codimension one-dimensional singularities. Remarkably, the limit as $s\to 1$ of the $s$-fractional mass gives back the classical notion of length for regular enough curves in $\R^d$. We prove a lower semi-continuity and compactness result for sequences of $1$-currents with uniformly bounded fractional mass and support. Moreover, we prove the density of weighted polygonal, closed and compact oriented curves in the class of divergence-free 1-currents with compact support and finite fractional mass. Finally, we discuss some possible applications of our notion of fractional mass to build up purely geometrical approaches to the variational modeling of dislocation lines in crystals and to vortex filaments in superconductivity.

math.FA↗
arXiv2023

Variational analysis in one and two dimensions of a frustrated spin system: chirality transitions and magnetic anisotropic transitions

We study the energy of a ferromagnetic/antiferromagnetic frustrated spin system with values on two disjoint circumferences of the 3-dimensional unit sphere in a one-dimensional and two-dimensional domain. It consists on the sum of a term that depends on the nearest and next-to-nearest interactions and a penalizing term that counts the spin's magnetic anisotropy transitions. We analyze the asymptotic behaviour of the energy, that is when the system is close to the helimagnet/ferromagnet transition point as the number of particles diverges. In the one-dimensional setting we compute the $Γ$-limit of renormalizations of the energy at first and second order. As a result, it is shown how much energy the system spends for any magnetic anistropy transition and chirality transition. In the two-dimensional setting, by computing the $Γ$-limit of the renormalization of the energy at second order, we we prove the emergence and study the geometric rigidity of chirality transitions.

math-ph↗
arXiv2022

Asymptotic of the Discrete Volume-Preserving Fractional Mean Curvature Flow via a Nonlocal Quantitative Alexandrov Theorem

We characterize the long time behaviour of a discrete-in-time approximation of the volume preserving fractional mean curvature flow. In particular, we prove that the discrete flow starting from any bounded set of finite fractional perimeter converges exponentially fast to a single ball. As an intermediate result we establish a quantitative Alexandrov type estimate in the fractional setting for normal deformations of a ball. Finally, we provide existence for flat flows as limit points of the discrete flow when the time discretization parameter tends to zero.

math.AP↗
arXiv2021

The variational approach to $s$-fractional heat flows and the limit cases $s\to 0^+$ and $s\to 1^-$

This paper deals with the limit cases for $s$-fractional heat flows in a cylindrical domain, with homogeneous Dirichlet boundary conditions, as $s\to 0^+$ and $s\to 1^-$\,. To this purpose, we describe the fractional heat flows as minimizing movements of the corresponding Gagliardo seminorms, with respect to the $L^2$ metric. First, we provide an abstract stability result for minimizing movements in Hilbert spaces, with respect to a sequence of $Γ$-converging uniformly $λ$-convex energy functionals. Then, we provide the $Γ$-convergence analysis of the $s$-Gagliardo seminorms as $s\to 0^+$ and $s\to 1^-$\,, and apply the general stability result to such specific cases. As a consequence, we prove that $s$-fractional heat flows (suitably scaled in time) converge to the standard heat flow as $s\to 1^-$, and to a degenerate ODE type flow as $s\to 0^+$\,. Moreover, looking at the next order term in the asymptotic expansion of the $s$-fractional Gagliardo seminorm, we show that suitably forced $s$-fractional heat flows converge, as $s\to 0^+$\,, to the parabolic flow of an energy functional that can be seen as a sort of renormalized $0$-Gagliardo seminorm: the resulting parabolic equation involves the first variation of such an energy, that can be understood as a zero (or logarithmic) Laplacian.

math.AP↗
arXiv2021

Convergence of supercritical fractional flows to the mean curvature flow

We consider a core-radius approach to nonlocal perimeters governed by isotropic kernels having critical and supercritical exponents, extending the nowadays classical notion of $s$-fractional perimeter, defined for $0<s<1$, to the case $s\ge 1$\,. We show that, as the core-radius vanishes, such core-radius regularized $s$-fractional perimeters, suitably scaled, $Γ$-converge to the standard Euclidean perimeter. Under the same scaling, the first variation of such nonlocal perimeters gives back regularized $s$-fractional curvatures which, as the core radius vanishes, converge to the standard mean curvature; as a consequence, we show that the level set solutions to the corresponding nonlocal geometric flows, suitably reparametrized in time, converge to the standard mean curvature flow. Finally, we prove analogous results in the case of anisotropic kernels with applications to dislocation dynamics. Keywords: Fractional perimeters; $Γ$-convergence; Local and nonlocal geometric evolutions; Viscosity solutions; Level set formulation; Fractional mean curvature flow; Dislocation dynamics

math.AP↗
arXiv2020

Attractive Riesz potentials acting on hard spheres

In this paper we introduce a model for hard spheres interacting through attractive Riesz type potentials, and we study its thermodynamic limit. We show that the tail energy enforces optimal packing and round macroscopic shapes.

math.AP↗