The equilibrium shape of a crystal

Calculus of variations involves important minimization problems encoding physical phenomena; for example, in the formation of bubbles, the molecules permute themselves to form the least energy-efficient structure, and the result is a ball. Mathematically, this formation is the isoperimetric inequality. Several extensions of this inequality exist via a free energy encoding a potential g. Hence, minimizing this free energy with a mass constraint can generate a convex crystal. Thus, the minimizer, in some contexts, encodes the property such that if one takes two points in the set and connects them with a line segment, the line segment will likewise be in the set. The relation between minimizers and crystals was encountered by Willard Gibbs and Pierre Curie in the 19^th century. In a more modern thermodynamics application, Almgren’s problem investigates the formation of a convex minimizer assuming convexity of the sub-level sets. Interestingly, even in two dimensions, this is complicated. The full physics application is in three dimensions, and few results exist for all masses. My talk will address this problem and will illuminate the importance of symmetry. The methods include the first variation, geometric measure theory, and PDEs. The results discuss the existence and non-existence of crystals, their shape geometry, and stability. A few conclusions will be discussed regarding extending the theory.