Optimal Crystal Shapes

Associated with any Borel function Ф defined on the unit sphere S n in R n+1 with values in R ⋃ {∞} (and, say, bounded from below) and any n-dimensional oriented rectifiable surface S in R n+1 is the integral $$\Phi(S) = \int_{x\,\epsilon\,S}\, \Phi(v_S\,(x))\,dH^nx;$$ here v S (·) denotes the unit normal vectorfield orienting S, and H n is Hausdorff n-dimensional surface measure. If, for example, S is composed of polygonal pieces S i with oriented unit normals v i , then $$\Phi(S) = \Sigma_{i}\,\Phi(v_i)$$ area(S i ) Perhaps the most important integrands Φ: S 2 → R arise as the surface free energy density functions for interfaces S between an ordered material A (hereafter called a crystal) and another phase or a crystal of another orientation. In this case v S (p) is the unit exterior normal to A at pєS and Ф(S) gives the surface free energy of S. Other interesting Ф’s need not be continuous or even bounded. See the sailboat example of [T1], in which Ф(v) is the time required to sail unit distance in direction v rotated by 90°.