The Wulff theorem revisited. (English) Zbl 0725.49017

Fonseca, Irene

Proc. R. Soc. Lond., Ser. A 432, No. 1884, 125-145 (1991).

The study of phase transitions leads in a natural way to the study of variational problems where the total energy functional involves bulk and surface energies terms. For solid crystals with sufficiently small grains the total energy reduces essentially to its interfacial component. Hence, assuming that interfaces are sharp, the surface free energy plays a definite role in determining the shape of the crystal approaching an equilibrium configuration of minimum energy. Considering a surface tension of the type \[ \int_{\partial E}\Gamma (n_ E(x))\quad dH_{N-1}(x), \] where E is a smooth subset of \({\mathbb{R}}^ N\), \(n_ E\) is the outward unit normal to its boundary and \(\Gamma\) denotes the anisotropic free energy density per unit area, the equilibrium problem \[ (P)\text{ minimize (1.1) subject to } meas(E)=text{constant}, \] is studied. The parametrized indicator measures and the Brunn-Minkowski inequality are used to prove that the Wulff set \(W_{\Gamma}:=\{x\in {\mathbb{R}}^ N|\) \(x.n<\Gamma (n)\), for all \(n\in S^{N-1}\}\) is a minimizer for (P) within the class \({\mathcal C}\) of all measurable sets (bounded or unbounded) with finite perimeter. The support of the indicator measures associated to minimizing sequences is characterized. This result allows one to show that if \(W_{\Gamma}\) is polyhedral then minimizing sequences cannot oscillate.

Reviewer: I.Fonseca

MSC:

49Q20

Variational problems in a geometric measure-theoretic setting

Keywords:

sets of finite perimeter; phase transitions; total energy functional; solid crystals; surface tension; parametrized indicator measures; Brunn- Minkowski inequality; Wulff set