We investigate ground state configurations for atomic potentials including both two- and three-body nearest-neighbor interaction terms. The aim is to prove that such potentials may describe crystallization in carbon nanostructures such as graphene, nanotubes, and fullerenes. We give conditions in order to prove that planar energy minimizers are necessarily honeycomb, namely graphene patches. Moreover, we provide an explicit formula for the ground state energy which exactly quantifies the lower-order surface energy contribution. This allows us to give some description of the geometry of ground states. By recasting the minimization problem in three-space dimensions, we prove that ground states are necessarily nonplanar and, in particular, rolled-up structures like nanotubes are energetically favorable. Eventually, we check that the C20 and C60 fullerenes are strict local minimizers, hence stable

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 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.

This paper addresses two-dimensional crystallization in the square lattice. A suitable configurational potential featuring both two-and three-body shortranged 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.

We continue the study of Ambrosio and Serfaty (2008) [4] on the Chapman-Rubinstein-Schatzman-E evolution model for superconductivity, viewed as a gradient flow on the space of measures equipped with the quadratic Wasserstein structure. In Ambrosio and Serfaty (2008) [4] we considered the case of positive (probability) measures, while here we consider general real measures, as in the physical model. Understanding the evolution as a gradient flow in this context gives rise to several new questions, in particular how to define a "Wasserstein" distance for signed measures. We generalize the minimizing movement scheme of Ambrosio et al. (2005) [3] in this context, we show the entropy argument of Ambrosio and Serfaty (2008) [4] still carries through, and derive an evolution equation for the measure which contains an error term compared to the Chapman-Rubinstein-Schatzman-E model. Moreover, we also show the same applies to a very similar dissipative model on the whole plane.