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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
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##### 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
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