Hostname: page-component-76d6cb85b7-6jg5l Total loading time: 0 Render date: 2026-07-13T16:22:12.268Z Has data issue: false hasContentIssue false
  • English
  • Français

A uniqueness proof for the Wulff Theorem

Published online by Cambridge University Press:  14 November 2011

Irene Fonseca  and
Stefan Müller
Irene Fonseca
Affiliation:
Department of Mathematics, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A.
Stefan Müller
Affiliation:
Institut für Angewandte Mathematik, Universität Bonn, Beringstr. 4, W-5300 Bonn 1, Germany
Get access Rights & Permissions [Opens in a new window]

Synopsis

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.

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 119 , Issue 1-2 , 1991 , pp. 125 - 136
Copyright
Copyright © Royal Society of Edinburgh 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1Allard, W.. On the first variation of a varifold. Ann. of Math. 95 (1972), 417–491.CrossRefGoogle Scholar
2Almgren, F. J., 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), 321–391.CrossRefGoogle Scholar
3Dacorogna, B. and Pfister, C. E.. Wulff theorem and best constant in Sobolev inequality (to appear).Google Scholar
4Dinghas, A.. Über einen geometrischen Satz von Wulff für die Gleichgewichtsform von Kristallen. Zeitschrift für Kristallographie 105 (1944), 304–314.CrossRefGoogle Scholar
5Evans, L. C. and Gariepy, R. F.. Lecture Notes on Measure Theory and Fine Properties of Functions (Kentucky EPSCoR Preprint Series).Google Scholar
6Federer, H.. Geometric Measure Theory (Berlin: Springer, 1969).Google Scholar
7Fonseca, I.. Lower semicontinuity of surface energies (to appear).Google Scholar
8Fonseca, I.. The Wulff Theorem revisited Proc. Roy. Soc. London 432 (1991), 125–145.Google Scholar
9Gel'fand, I. M., Graev, M. I. and Vilenkin, N.. Generalized functions, Vol. 5. (New York: Academic Press, 1966).Google Scholar
10Giusti, E.. Minimal Surfaces and Functions of Bounded Variation (Basel: Birkhäuser, 1984).CrossRefGoogle Scholar
11Herring, C.. Some theorems on the free energies of crystal surfaces. Phys. Rev. 82 (1951), 87–93.CrossRefGoogle Scholar
12Reshetnyak, Yu. G.. Weak convergence of completely additive vector functions on a set. Sib. Math. J. 9 (1968), 1039–1045 (translation of: Sibirsk. Mat. Z. 9 (1968), 1386–1394).CrossRefGoogle Scholar
13Taylor, J.. Existence and structure of solutions to a class of nonelliptic variational problems. Sympos. Math. 14 (1974), 499–508.Google Scholar
14Taylor, J.. Unique structure of solutions to a class of nonellipic variational problems. Proc. Sympos. Pure Math. 27 (1975), 419–427.CrossRefGoogle Scholar
15Taylor, J.. Crystalline variational problems. Bull. Amer. Math. Soc. 84 (1978), 568–588.CrossRefGoogle Scholar
16Wulff, G.. Zur Frage der Geschwindigkeit des Wachstums und der Auflösung der Kristallflächen. Z. Krist. 34 (1901), 449–530.Google Scholar
17Young, L. C.. Generalized surfaces in the calculus of variations I, II. Ann. of Math. 43 (1942), 84–103, 530–544.CrossRefGoogle Scholar
18Ziemer, W. P.. Weakly differentiable functions (Berlin: Springer, 1989).CrossRefGoogle Scholar