"Convex geometric reasoning for crystalline energies" , Thaicia Stona ... 2016 , No . 1 , P. 51 - 62 , DOI : http://dx.doi.org/10.22039/cjcme.2016.05
Abstract :
The present work revisits the classical Wulff problem restricted to crystalline integrands, a class of surface energies that gives rise to finitely faceted crystals. The general proof of the Wulff theorem was given by J.E. Taylor (1978) by methods of Geometric Measure Theory. This work follows a simpler and direct way through Minkowski Theory by taking advantage of the convex properties of the considered Wulff shapes.
References :
1. Taylor, J.E. Crystalline variational problems, 1978
2. Burchard, A. A short course on rearrangement inequalities, 2009
3. Federer, H. Geometric Measure Theory, 1969
4. McCann, R. Equilibrium shapes for plannar crystals in an external field, 1998
5. Fu, J. A mathematical model for crystal growth and related problems,1976
6. Brazitikos, S., Giannopoulos, A., Valettas, P.,Vritsiou, B. Geometry of IsotropicConvex Bodies, 2014
7. Gibbs, J.W. Collected Works Vol.1, 1948
8. Wulff, G. Zeitschrift fur Krystallographie und Mineralogie, 1901
9. Taylor, J.E., Cahn, J.W., Handwerker, C.A. Evolving crystal forms: Frank's characteristics revisited, 1991
10. Wills, J.M. Wulff-Shape, Minimal Energy and Maximal Density, 2001
11. Micheletti, A., Patti, S., Villa, E. Crystal Growth Simulations: a new Mathematical Model based on the Minkowski Sum of Sets, 2005
12. Taylor, J.E. Crystalline Variational Methods, 2002
13. Cahn, J.W., Handwerker, C.A. Equilibrium geometries of anisotropic surfaces and interfaces, 1993