A uniqueness proof for the Wulff theorem. (English) Zbl 0752.49019

Fonseca, Irene; Müller, Stefan

Proc. R. Soc. Edinb., Sect. A 119, No. 1-2, 125-136 (1991).

The authors treat the following variational problem: \[ \text{Minimise}\int_{\partial E}\Gamma(n_ E(x))dH_{n-1}(x)\quad\text{ s.t. \quad meas}(E)=\text{constant}, (P) \] where \(E\) is a smooth subset of \(R^ N\), \(n_ E\) is the outward unit normal to its boundary, \(\Gamma:S^{N-1}\to (0,\infty)\) is a continuous function (called the surface free energy) and \(H_{N-1}\) denotes the \(N-1\) dimensional Hausdorff measure in \(R^ N\).
As it is known, the Wulff set \(W_ \Gamma:=\{x\in R^ N\mid\langle x,n\rangle\leq\Gamma(n)\) for all \(x\in S^{N-1}\}\) is a solution of problem (P) with the constraint \(\text{meas}(E)=\text{meas}(W_ \Gamma)\) [see A. Dinghas, Z. Kristallogr. 105, 304-314 (1944; Zbl 0028.43001); J. E. Taylor, in: Symp. math. 14, Geom. simplett. Fis. mat., Teor. geom. Integr. Var. minim., Convegni 1973, 499-508 (1974; Zbl 0317.49053); and in: Differ. Geom., Proc. Symp. Pure Math. 27, Part 1, Stanford 1973, 419-427 (1975; Zbl 0317.49054); and Bull. Am. Math. Soc. 84, 568-588 (1978; Zbl 0392.49022)]. The authors prove that the Wulff set is the unique solution of problem (P) up to translation in \({\mathcal C}\), where \({\mathcal C}\) is a certain class of measurable sets with finite perimeter.
The proof of the uniqueness is based on a sharpened version of the Brunn-Minkowski inequality, which is also stated and proved in this paper.

Reviewer: Zs.Páles (Debrecen)

MSC:

49Q20

Variational problems in a geometric measure-theoretic setting

Keywords:

isoperimetric problem; surface free energy; Wulff set; uniqueness; Brunn-Minkowski inequality

References:

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