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Riccardo Cristoferi

Publications and source records attributed to Riccardo Cristoferi.

28 records · Page 2Linked to original sources
arXiv2020

A homogenization result in the gradient theory of phase transitions

A variational model in the context of the gradient theory for fluid-fluid phase transitions with small scale heterogeneities is studied. In particular, the case where the scale $\varepsilon$ of the small homogeneities is of the same order of the scale governing the phase transition is considered. The interaction between homogenization and the phase transitions process will lead, in the limit as $\varepsilon\to0$, to an anisotropic interfacial energy.

math.AP↗
arXiv2020

On the Gamma convergence of functionals defined over pairs of measures and energy-measures

A novel general framework for the study of $Γ$-convergence of functionals defined over pairs of measures and energy-measures is introduced. This theory allows us to identify the $Γ$-limit of these kind of functionals by knowing the $Γ$-limit of the underlining energies. In particular, the interaction between the functionals and the underlining energies results, in the case these latter converge to a non continuous energy, in an additional effect in the relaxation process. This study was motivated by a question in the context of epitaxial growth evolution with adatoms. Interesting cases of application of the general theory are also presented.

math.AP↗
arXiv2020

Sharp interface limit of a multi-phase transitions model under nonisothermal conditions

A vectorial Modica--Mortola functional is considered and the convergence to a sharp interface model is studied. The novelty of the paper is that the wells of the potential are not constant, but depend on the spatial position in the domain $Ω$. The mass constrained minimization problem and the case of Dirichlet boundary conditions are also treated. The proofs rely on the precise understanding of minimizing geodesics for the degenerate metric induced by the potential.

math.AP↗
arXiv2018

Large data limit for a phase transition model with the p-Laplacian on point clouds

The consistency of a nonlocal anisotropic Ginzburg-Landau type functional for data classification and clustering is studied. The Ginzburg-Landau objective functional combines a double well potential, that favours indicator valued function, and the $p$-Laplacian, that enforces regularity. Under appropriate scaling between the two terms minimisers exhibit a phase transition on the order of $ε=ε_n$ where $n$ is the number of data points. We study the large data asymptotics, i.e. as $n\to \infty$, in the regime where $ε_n\to 0$. The mathematical tool used to address this question is $Γ$-convergence. In particular, it is proved that the discrete model converges to a weighted anisotropic perimeter.

math.AP↗
arXiv2018

Equilibria configurations for epitaxial crystal growth with adatoms

The behavior of a surface energy $\mathcal F(E,u)$, where $E$ is a set of finite perimeter and $u\in L^1(\partial^* E, \mathbb R_+)$ is studied. These energies have been recently considered in the context of materials science to derive a new model in crystal growth that takes into account the effect of atoms freely diffusing on the surface (called adatoms), which are responsible for morphological evolution through an attachment and detachment process. Regular critical points, existence and uniqueness of minimizers are discussed and the relaxation of $\mathcal F$ in a general setting under the $L^1$ convergence of sets and the vague convergence of measures is characterized. This is part of an ongoing project aimed at an analytical study of diffuse interface approximations of the associated evolution equations.

math.AP↗
arXiv2018

Piecewise constant reconstruction of damaged color images

A variational model for reconstruction of damaged color images is studied, in particular in the case where only finitely many colors are admissible for the reconstructed image. An existence result and regularity properties of minimizers are presented.

math.OC↗
arXiv2017

On periodic critical points and local minimizers of the Ohta-Kawasaki functional

In this paper we collect some new observations about periodic critical points and local minimizers of a nonlocal isoperimetric problem, arising in the modeling of diblock copolymers. In the main result, by means of a purely variational procedure, we show that it is possible to construct (locally minimizing) periodic critical points whose shape resemble that of any given strictly stable constant mean curvature (periodic) hypersurface. Along the way, we establish several auxiliary results of independent interest.

math.AP↗
arXiv2013

Local and global minimality results for a nonlocal isoperimetric problem on R^N

We consider a nonlocal isoperimetric problem defined in the whole space $\R^N$, whose nonlocal part is given by a Riesz potential with exponent $α\in(0, N-1)$. We show that critical configurations with positive second variation are local minimizers and satisfy a quantitative inequality with respect to the $L^1$-norm. This criterion provides the existence of a (explicitly determined) critical threshold determining the interval of volumes for which the ball is a local minimizer, and allows to address several global minimality issues.

math.AP↗