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  • Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Volume 119, Issue 1-2
  • January 1991, pp. 125-136

A uniqueness proof for the Wulff Theorem

  • Irene Fonseca (a1) and Stefan Müller (a2)
  • DOI:
  • Published online: 14 November 2011

The Wulff problem is a generalisation of the isoperimetric problem and is relevant for the equilibrium of (small) elastic crystals. It consists in minimising the (generally anisotropic) surface energy among sets of given volume. A solution of this problem is given by a geometric construction due to Wulff. In the class of sets of finite perimeter this was first shown by J. E. Taylor who, using methods of geometric measure theory, also proved uniqueness. Here a more analytic uniqueness proof is presented. The main ingredient is a sharpened version of the Brunn–Minkowski inequality.

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1W. Allard . On the first variation of a varifold. Ann. of Math. 95 (1972), 417491.

2F. J. Almgren , Jr. Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure. Ann. of Math. 87 (1968), 321391.

8I. Fonseca . The Wulff Theorem revisited Proc. Roy. Soc. London 432 (1991), 125145.

10E. Giusti . Minimal Surfaces and Functions of Bounded Variation (Basel: Birkhäuser, 1984).

11C. Herring . Some theorems on the free energies of crystal surfaces. Phys. Rev. 82 (1951), 8793.

12Yu. G. Reshetnyak . Weak convergence of completely additive vector functions on a set. Sib. Math. J. 9 (1968), 10391045 (translation of: Sibirsk. Mat. Z. 9 (1968), 1386–1394).

15J. Taylor . Crystalline variational problems. Bull. Amer. Math. Soc. 84 (1978), 568588.

17L. C. Young . Generalized surfaces in the calculus of variations I, II. Ann. of Math. 43 (1942), 84–103, 530544.

18W. P. Ziemer . Weakly differentiable functions (Berlin: Springer, 1989).

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Proceedings of the Royal Society of Edinburgh Section A: Mathematics
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