1. Faceted Crystals Grown from Solution
A Stefan Type Problem with a Singular Interfacial Energy
Yoshikazu Giga
University of Tokyo and Hokkaido University COE
Joint work with Piotr Rybka
December , 2005 Lyon

2. A basic problem from pattern formation in the theory of crystal growth
A basic problem from pattern formation in the theory of crystal growth. In what situation a flat portion (a FACET) of crystal surface breaks or not ?
Goal : We shall prove :
‘All facets are stable near equilibrium for a cylindrical crystal by analysizing a Stefan type problem’

3. Contents 1 Model 2 Problem 3 Main mathematical results
4 Three ingredients
－ ODE analysis
－ Berg’s effect
－ Facet splitting criteria－
5 Open problems

4. 1 Model Crystals grown from vapor (snow crystal)
from solution (NaCl crystal)
<driving force : supersaturation>
(density of atoms outside crystal is small)

5. Stefan like Model

7. unnormalized version :

10. One phase Stefan problem with Gibbs-Thomson + kinetic effect
We shall consider
(1)－(5) for given
quasi-stationary
One phase Stefan problem with Gibbs-Thomson + kinetic effect

11. K. Deckelnik - C. Elliott ’99 ( Hele Shaw type )
Solvability (smooth )
K. Deckelnik - C. Elliott ’99
( Hele Shaw type )
No 　　… Friedman –Hu ’92
　　Liu – Yuan ’94

12. Kuroda-Irisawa-Ookawa ‘77 Stability of facets Experiment e.g.
Others (No )
Kuroda-Irisawa-Ookawa ‘77
Stability of facets
Experiment e.g.
Gonda-Gomi ’85
(No ) : Fingering :
Saffman-Taylor
R.Almgrem ’95

13. 2.Problem (specific to ours)

19. 3. Main Math Results

20. Th (Rybka-G ‘04) If
is close to the
Equilibrium
then the solution
solves
the original problem
(1),(2),(3),
(4),(5),(6),(7)
Near equilibrium
Facet does not break.

21. Reduction to ODE

27. Near equilibrium : close to zero / bounded away from zero

28. 5. Open problems Existence of solution of the Original
problem is widly open if
is not near
equilibrium
(Even if is given
M.-H. Giga – Y. Giga ’98
graphs)
( : constant
M.-H. Giga – Y. Giga ‘01
level set approach : unique
existence of generalized sol
(2-D))
Uniqueness of the solution of
the original problem
(Sol is unique for
Reduced problems)

30. All my preprints are in Hokkaido University Preprint Series on Math.