×

zbMATH — the first resource for mathematics

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

MSC:
49Q20 Variational problems in a geometric measure-theoretic setting
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Dinghas, Zeitschrift für Kristallographie 105 pp 304– (1944)
[2] DOI: 10.1090/S0002-9904-1978-14499-1 · Zbl 0392.49022 · doi:10.1090/S0002-9904-1978-14499-1
[3] DOI: 10.2307/1970868 · Zbl 0252.49028 · doi:10.2307/1970868
[4] DOI: 10.1007/978-1-4612-1015-3 · Zbl 0692.46022 · doi:10.1007/978-1-4612-1015-3
[5] DOI: 10.2307/1968809 · Zbl 0063.08362 · doi:10.2307/1968809
[6] Wulff, Z. Krist. 34 pp 449– (1901)
[7] Taylor, Proc. Sympos. Pure Math. 27 pp 419– (1975) · doi:10.1090/pspum/027.1/0388225
[8] Taylor, Sympos. Math. 14 pp 499– (1974)
[9] Reshetnyak, Sib. Math. J. 9 pp 1039– (1968) · Zbl 0176.44402 · doi:10.1007/BF02196453
[10] Gel’fand, Generalized functions 5 (1966)
[11] DOI: 10.1103/PhysRev.82.87 · Zbl 0042.23201 · doi:10.1103/PhysRev.82.87
[12] DOI: 10.1098/rspa.1991.0009 · Zbl 0725.49017 · doi:10.1098/rspa.1991.0009
[13] Giusti, Minimal Surfaces and Functions of Bounded Variation (1984) · Zbl 0545.49018 · doi:10.1007/978-1-4684-9486-0
[14] Federer, Geometric Measure Theory (1969) · Zbl 0176.00801
[15] DOI: 10.2307/1970587 · Zbl 0162.24703 · doi:10.2307/1970587
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.