Edoardo Mainini

Finite crystallization in the square lattice

This paper addresses two-dimensional crystallization in the square lattice. A suitable configurational potential featuring both two- and three-body short-ranged particle interactions is considered. We prove that every ground state is a connected subset of the square lattice. Moreover, we discuss the global geometry of ground states and their optimality in terms of discrete isoperimetric inequalities on the square graph. Eventually, we study the aspect ratio of ground states and quantitatively prove the emergence of a square macroscopic Wulff shape as the number of particles grows.

A gradient flow approach to the porous medium equation with fractional pressure

We consider a family of porous media equations with fractional pressure, recently studied by Caffarelli and Vázquez. We show the construction of a weak solution as the Wasserstein gradient flow of a square fractional Sobolev norm. The energy dissipation inequality, regularizing effect and decay estimates for the Lp norms are established. Moreover, we show that a classical porous medium equation can be obtained as a limit case.

Ground states in the diffusion-dominated regime

We consider macroscopic descriptions of particles where repulsion is modelled by non-linear power-law diffusion and attraction by a homogeneous singular kernel leading to variants of the Keller–Segel model of chemotaxis. We analyse the regime in which diffusive forces are stronger than attraction between particles, known as the diffusion-dominated regime, and show that all stationary states of the system are radially symmetric non-increasing and compactly supported. The model can be formulated as a gradient flow of a free energy functional for which the overall convexity properties are not known. We show that global minimisers of the free energy always exist. Further, they are radially symmetric, compactly supported, uniformly bounded and

EXPONENTIAL AND POLYNOMIAL DECAY FOR FIRST ORDER LINEAR VOLTERRA EVOLUTION EQUATIONS

C^\infty

On the signed porous medium flow

C∞ inside their support. Global minimisers enjoy certain regularity properties if the diffusion is not too slow, and in this case, provide stationary states of the system. In one dimension, stationary states are characterised as optimisers of a functional inequality which establishes equivalence between global minimisers and stationary states, and allows to deduce uniqueness.

Gradient flow of the Chapman–Rubinstein–Schatzman model for signed vortices

The fine geometry of carbon nanotubes is investigated from the viewpoint of molecular mechanics. Actual nanotube configurations are characterized as locally minimizers of a given configurational energy, including both two- and three-body contributions. By focusing on so-called zigzag and armchair topologies, we prove that the configurational energy is strictly minimized within specific, one-parameter families of periodic configurations. Such optimal configurations are checked to be stable with respect to a large class of small nonperiodic perturbations and do not coincide with classical rolled-up nor polyhedral geometries.

Crystallization in Carbon Nanostructures

We consider, in an abstract setting, an instance of the Coleman-Gurtin model for heat conduction with memory, that is, the Volterra integro-differential equation ∂tu(t) − β∆u(t) − ∫ t 0 k(s)∆u(t − s)ds = 0. We establish new results for the exponential and polynomial decay of solutions, by means of conditions on the convolution kernel which are weaker than the classical differential inequalities.

Stability of abstract linear semigroups arising from heat conduction with memory

Carbon nanotubes are modeled as point configurations and investigated by minimizing configurational energies including two- and three-body interactions. Optimal configurations are identified with local minima and their fine geometry is fully characterized in terms of lower-dimensional problems. Under moderate tension, we prove the existence of periodic local minimizers, which indeed validates the so-called Cauchy–Born rule in this setting.