Geometric Analysis & Crystalline Anisotropy

The anisoperimetric problem, also known as the crystalline variational problem, may not be as old as the isoperimetric problem, but it has certainly been explored by scientists of different knowledge domains for a while.

Minerals have been formally characterized for the first time by polyhedral shapes by Rene Hauy, in the beginning of the XIX century. Hauy suggested correlations between micro and macroscopic properties of materials, the hypothesis that the smallest units of matter - in his work, molecules - induce the macro behavior. This was an important step for Crystallography.

Under special conditions (equilibrium conditions) such very small scale properties can be used to determine the final shape of a growing crystal - its equilibrium shape. The Wulff construction, published in 1901, presents the steps to build this shape starting from the crystal surface energy, which is a function of the material crystallography.

The problem is not so obvious though for evolution of interfaces happening far from equilibrium, such as the one of a crystal growing from liquid or vapor. The most promising theoretical tool to interpret such conditions is Geometric Measure Theory (GMT).

This research project started as a childhood passion ignited in the Brazilian Central Plateau, where eroded lands expose crystalline strata and all types of crystals can be found: emeralds, diverse tourmalines, sapphires, quartz of all colors and varieties, including lots of amethyst, agate, onyx, rutile, citrine and more. Later, at the university, I soon thought that natural crystal shapes should be interesting objects, since they are the result of optimal processes and could be interpreted as optimal solutions or minimal surfaces. It took many years between my childhood and the day that I met Prof. Frank Morgan at the 16th Brazilian School of Differential Geometry, when he confirmed my intuition, letting me know that these are indeed minimal surfaces - in the sense of Geometric Measure Theory.


CONVEX GEOMETRIC REASONING FOR CRYSTALLINE ENERGIES
T. Stona de Almeida. Caspian Journal of Computational & Mathematical Engineering, 2016, 51-62 (arXiv:2102.12683)

cryst image 1 Abstract: The present work revisits the classical Wulff problem restricted to crystalline integrands, a class of surface energies that gives rise to finitely faceted crystals. The general proof of the Wulff theorem was given by J.E. Taylor (1978) by methods of Geometric Measure Theory. This work follows a simpler and direct way through Minkowski Theory by taking advantage of the convex properties of the considered Wulff shapes.

Keywords: Wulff Shape; Energy Minimization; Anisotropy; Surface Energy; Geometric Inequality.

MSC: 35A15-Variational methods; 49J40-Variational inequalities; 49K20-Problems involving PDEs; 49Q10-Shape optimization other than minimal surfaces; 52B60-Isoperimetric problems for polytopes; 82D25-Crystals.


THE MATHEMATICS OF EQUILIBRIUM SHAPES: FROM GEOMETRIC INEQUALITIES TO THE CRYSTALLINE VARIATIONAL PROBLEM
T. Stona de Almeida. Poster in Current Trends in Analysis and Partial Differential Equations, 2015, IMPA, Brazil.

cryst image 2 Abstract: This work explores the anisotropic version of the Isoperimetric Inequality for anisotropic surface energies γ, the Crystalline Variational Problem, also known as the Wulff Problem. Our starting point is Taylor’s geometric measure approach of the Wulff problem, Crystalline Variational Problems (1978). This project was developed at the Fields Institute thematic program on Variational Problems, Fall 2014.