Crystalline variational problems

Abstract

Surface tension is commonly thought of as a fluid phenomenon; the mere mention of the term brings to mind bugs skimming over water, liquids rising or falling in capillary tubes-and soap films and soap bubbles. But there is in fact a notion of surface tension (which is surface energy per unit surface area) for the interface between any two substances, or even between one substance and a vacuum. This surface energy arises from the fact that atoms (or molecules, or ions) of a given substance have a different environment at the interface between that substance and another than those in the bulk of the substance. (Sometimes even the composition of the surface is different from the bulk; this occurs for instance in soapy water having an interface with air.)In this article we will deal with "surface tension functions" which are an outgrowth of the surface tensions of solids having their atoms arranged in some regular way. If one fixes the orientation of the lattice in R 3 9 then the environment of an atom on a planar interface between something else and such a regular structure can be different for different plane directions. Thus the surface tension between one substance and another can be a function F:G 0 (3 9 2)->R +where G 0 (3, 2) is the Grassmannian of oriented 2-planes through the origin in R 3 and R + is the set of positive real numbers. For the interface between, say, soapy water and air, the function JF would be identically a constant; but for that between, say, a single crystal of ordinary salt (NaQ) and air, F would depend quite strongly on direction. If one wishes to allow the regular structure to vary over relatively long distances, one can even obtain a function(This notion of surface tension for solid bodies is not at all esoteric, by the way: it is commonly used by metallurgists and others who deal with the surface properties of materials.)Article based on an invited hour address to the Eastern Sectional Meeting of the American

Motion of spirals by crystalline curvature

“…The origin is kept fixed during the evolution. The unit normal vector ν i to L i (t) will be oriented outwardly and independent of t. The set N of possible orientation ν i is finite, which appears in the so-called Wulff crystal [23]; that is, the unique crystal which minimizes the surface energy at the fixed enclosed area. We remark that L i (t) is contained in {x ∈ R 2 |x·ν i = d i (t)}, where d i (t) represents the distance from the origin to the line containing L i (t).…”

Aubry sets and the differentiability of the minimal average action in codimension one

“…This problem is motivated by an observation of Gibbs', recalled in [2] and [22], which says that 1 √ 1+|ρ| 2 β(ρ) can be interpreted as the energy per unit of area of the face of a (n + 1)-dimensional crystal which is orthogonal to (−ρ, 1). This energy is called a Wulff functional by crystalline people (see for instance [23]); we want to study what kind of corners are possible for Wulff functionals which arise from a microscopic theory like that of Gibbs'.…”

Maximum area with Minkowski measures of perimeter

“…The existence of a minimizing S and its uniqueness and form have been established for increasingly general domains and integrands [1,9,21,41,53]. Morgan outlines a history of this effort [40, see also 37,44] that is fascinatingly independent of convex geometry.…”