Related papers: Crystallization in two dimensions and a discrete G…
We investigate two-dimensional crystallization phenomena, i.e. minimality of a lattice's patch for interaction energies, with pair potentials of type $(x,y)\mapsto V(\|x-y\|)$ where $\|\cdot\|$ is an arbitrary norm on $\mathbb{R}^2$ and…
When flat or on a firm mechanical substrate, the atomic composition and atomistic structure of two-dimensional crystals dictate their chemical, electronic, optical, and mechanical properties. These properties change when the two-dimensional…
We investigate ground state configurations of atomic systems in two dimensions interacting via short range pair potentials. As the number of particles tends to infinity, we show that low-energy configurations converge to a macroscopic…
We prove finite crystallization for particles in the plane interacting through a soft disc potential, as originally shown by C. Radin \cite{Radin_soft}. We give an alternative proof that relies on the geometric decomposition of the energy…
We consider low energy configurations for the Heitmann-Radin sticky discs functional, in the limit of diverging number of discs. More precisely, we renormalize the Heitmann-Radin potential by subtracting the minimal energy per particle,…
We present new exact solutions for two-dimensional geometries generated by continuous distributions of topological defects within a conformal metric framework. By reformulating Einstein's equations in two dimensions as a Poisson equation…
We study static and dynamical properties that distinguish two dimensional crystals constrained to lie on a curved substrate from their flat space counterparts. A generic mechanism of dislocation unbinding in the presence of varying Gaussian…
We develop a rigorous framework for modelling the geometry equilibration of crystalline defects. We formulate the equilibration of crystal defects as a variational problems on a discrete energy space and establish qualitatively sharp…
A discrete-to-continuum analysis for free-boundary problems related to crystalline films deposited on substrates is performed by $\Gamma$-convergence. The discrete model here introduced is characterized by an energy with two contributions,…
We obtain improved local well-posedness results for the Lorentzian timelike minimal surface equation. In dimension $d=3$, for a surface of arbitrary co-dimension, we show a gain of $1/3$ derivative regularity compared to a generic equation…
We summarise some recent results on solution landscapes for two-dimensional (2D) problems in the Landau--de Gennes theory for nematic liquid crystals. We study energy-minimizing and non energy-minimizing solutions of the Euler--Lagrange…
We investigate the emergence of rigid polycrystalline structures from atomistic particle systems. The atomic interaction is governed by a suitably normalized pair interaction energy, where the `sticky disk' interaction potential models the…
We study the formulation of bond-orientational order in an arbitrary two dimensional geometry. We find that bond-orientational order is properly formulated within the framework of differential geometry with torsion. The torsion reflects the…
Herein, fundamentals of topology and symmetry breaking are used to understand crystallization and geometrical frustration in topologically close-packed structures. This frames solidification from a new perspective that is unique from…
We devise a new technique to prove two-dimensional crystallization results in the square lattice for finite particle systems. We apply this strategy to energy minimizers of configurational energies featuring two-body short-ranged particle…
Current quadratic smoothness energies for curved surfaces either exhibit distortions near the boundary due to zero Neumann boundary conditions, or they do not correctly account for intrinsic curvature, which leads to unnatural-looking…
The action for a class of three-dimensional dilaton-gravity theories, with an electromagnetic Maxwell field and a cosmological constant, can be recast in a Brans-Dicke-Maxwell type action, with its free $\omega$ parameter. For a negative…
We investigate one-dimensional periodic chains of alternate type of particles interacting through mirror symmetric potentials. The optimality of the equidistant configuration at fixed density -- also called crystallization -- is shown in…
For matrix analogues of embedded surfaces we define discrete curvatures and Euler characteristics, and a non-commutative Gauss--Bonnet theorem is shown to follow. We derive simple expressions for the discrete Gauss curvature in terms of…
Based on a theory of extra dimensional confinement of quantum particles [E. R. Hedin, Physics Essays 25, 2 (2012)], a simple model of a nucleon-nucleon (NN) central potential is derived which quantitatively reproduces the radial profile of…