Convex geometric reasoning for crystalline energies

“Convex geometric reasoning for crystalline energies” ,  Thaicia Stona …  2016  ,  No . 1 ,  P. 51 – 62   , DOI :

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

14. Cahn, J.W., Hoffman, D.W. A vector thermodynamics for anisotropic  surfaces – II. curved and faceted surfaces, 1974

15. Palmer, B. Stable closed equilibria for anisotropic surface energies:  Surfaces with edges, 2011

16. Koiso, M., Palmer, B. Stable surfaces with constant anisotropic mean  curvature and circular boundary, 2013

17. Craig Carter, W., Taylor, J.E., Cahn, J.W. Variational Methods for  Microstructural Evolution, 1997

18. Herring, C. Some theorems on the free energies of crystal surfaces, 1951

19. Almgren, F., Taylor, J.E., Wang, L. Curvature driven ows: a variational   approach, 1993

20. Eggleston, H.G. Convexity, 1958

21. Schneider, R. Convex Bodies: The Brunn{Minkowski Theory, 2014

22. Peng, D., Osher, S., Merriman, B., Zhao, H. The geometry of Wulf Crystals Shapes and its relations with Riemann problems, 1998

23. Micheletti, A., Burger, M. Stochastic and deterministic simulation of nonisothermal crystallization of polymers, 2001

24.Burchard, A. How to achieve radial symmetry through simple rearangements, 2012