# Anisotropic functions: a genericity result with crystallographic implications

Victor J. Mizel; Alexander J. Zaslavski

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 10, Issue: 4, page 624-633
- ISSN: 1292-8119

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topMizel, Victor J., and Zaslavski, Alexander J.. "Anisotropic functions: a genericity result with crystallographic implications." ESAIM: Control, Optimisation and Calculus of Variations 10.4 (2010): 624-633. <http://eudml.org/doc/90747>.

@article{Mizel2010,

abstract = {
In the 1950's and 1960's surface physicists/metallurgists such as
Herring and Mullins applied ingenious thermodynamic arguments to explain a
number of experimentally observed surface phenomena in crystals. These insights permitted
the successful engineering of a large number of alloys, where the
major mathematical novelty was that the surface response to external stress was anisotropic.
By examining step/terrace (vicinal) surface defects it was discovered through
lengthy and tedious experiments that the stored energy density (surface tension)
along a step edge was a smooth symmetric function β of the azimuthal angle θ to the
step, and that the positive function β attains its minimum value at $\theta = \pi/2$ and its maximum value at $\theta = 0$. The function β provided the crucial thermodynamic
parameters needed for the engineering of these materials. Moreover the minimal energy
configuration of the step is determined by the values of the stiffness function$\beta'' + \beta$ which ultimately leads to the magnitude and direction of surface mass flow for
these materials. In the 1990's there was a dramatic improvement in electron microscopy
which permitted real time observation of the meanderings of a step edge under
Brownian heat oscillations. These observations provided much more rapid
determination of the relevant thermodynamic parameters for the step edge, even for crystals at
temperatures below their roughening temperature. Use of these tools led J.
Hannon and his coexperimenters to discover that some crystals behave in a highly anti-intuitive
manner as their temperature is varied. The present article is devoted to
a model described by a class of variational problems. The main result of the paper
describes the solutions of the corresponding problem for a generic integrand.
},

author = {Mizel, Victor J., Zaslavski, Alexander J.},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Complete metric space; generic property;
variational problem.; complete metric space; variational problem},

language = {eng},

month = {3},

number = {4},

pages = {624-633},

publisher = {EDP Sciences},

title = {Anisotropic functions: a genericity result with crystallographic implications},

url = {http://eudml.org/doc/90747},

volume = {10},

year = {2010},

}

TY - JOUR

AU - Mizel, Victor J.

AU - Zaslavski, Alexander J.

TI - Anisotropic functions: a genericity result with crystallographic implications

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 10

IS - 4

SP - 624

EP - 633

AB -
In the 1950's and 1960's surface physicists/metallurgists such as
Herring and Mullins applied ingenious thermodynamic arguments to explain a
number of experimentally observed surface phenomena in crystals. These insights permitted
the successful engineering of a large number of alloys, where the
major mathematical novelty was that the surface response to external stress was anisotropic.
By examining step/terrace (vicinal) surface defects it was discovered through
lengthy and tedious experiments that the stored energy density (surface tension)
along a step edge was a smooth symmetric function β of the azimuthal angle θ to the
step, and that the positive function β attains its minimum value at $\theta = \pi/2$ and its maximum value at $\theta = 0$. The function β provided the crucial thermodynamic
parameters needed for the engineering of these materials. Moreover the minimal energy
configuration of the step is determined by the values of the stiffness function$\beta'' + \beta$ which ultimately leads to the magnitude and direction of surface mass flow for
these materials. In the 1990's there was a dramatic improvement in electron microscopy
which permitted real time observation of the meanderings of a step edge under
Brownian heat oscillations. These observations provided much more rapid
determination of the relevant thermodynamic parameters for the step edge, even for crystals at
temperatures below their roughening temperature. Use of these tools led J.
Hannon and his coexperimenters to discover that some crystals behave in a highly anti-intuitive
manner as their temperature is varied. The present article is devoted to
a model described by a class of variational problems. The main result of the paper
describes the solutions of the corresponding problem for a generic integrand.

LA - eng

KW - Complete metric space; generic property;
variational problem.; complete metric space; variational problem

UR - http://eudml.org/doc/90747

ER -

## References

top- B. Dacorogna and C.E. Pfister, Wulff theorem and best constant in Sobolev inequality. J. Math. Pures Appl.71 (1992) 97-118. Zbl0676.46031
- I. Fonseca, The Wulff theorem revisited. Proc. R. Soc. Lond. A432 (1991) 125-145. Zbl0725.49017
- J. Hannonet al., Step faceting at the (001) surface of boron doped silicon. Phys. Rev. Lett.79 (1997) 4226-4229.
- J. Hannon, M. Marcus and V.J. Mizel, A variational problem modelling behavior of unorthodox silicon crystals. ESAIM: COCV9 (2003) 145-149. Zbl1066.49011
- H.C. Jeng and E.D. Williams, Steps on surfaces: experiment and theory. Surface Science Reports34 (1999) 175-294.

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