## Some mathematical challenges in materials science

HTML articles powered by AMS MathViewer

- by Jean E. Taylor PDF
- Bull. Amer. Math. Soc.
**40**(2003), 69-87 Request permission

## Abstract:

Four challenges to mathematics research posed by the field of materials science are given, plus an additional challenge purely from the field of geometric measure theory. The problems all concern the effects of surface and grain boundary free energy: motion by weighted mean curvature and/or surface diffusion and/or other kinetics, proofs of minimality of soap bubble clusters and their anisotropic analogs, shapes of crystals in a gravitational or other field, incorporating data from simulations into mathematics, and understanding flat chains modulo $\nu$. The figures are selected copies of transparencies presented at the lecture on which this paper is based.## References

- William K. Allard,
*On the first variation of a varifold*, Ann. of Math. (2)**95**(1972), 417–491. MR**307015**, DOI 10.2307/1970868 - F. J. Almgren Jr.,
*Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints*, Mem. Amer. Math. Soc.**4**(1976), no. 165, viii+199. MR**420406**, DOI 10.1090/memo/0165 - Frederick J. Almgren Jr.,
*Selected works of Frederick J. Almgren, Jr*, Collected Works, vol. 13, American Mathematical Society, Providence, RI, 1999. Edited by Jean E. Taylor. MR**1747253** - Frederick J. Almgren, Jr.,
*Almgren’s Big Regularity Paper: $Q$-valued functions minimizing Dirichlet’s integral and the regularity of area minimizing rectifiable currents up to codimension two,*Vladimir Scheffer and Jean E. Taylor, editors, World Scientific Press, 2000 - Fred Almgren and Jean E. Taylor,
*Flat flow is motion by crystalline curvature for curves with crystalline energies*, J. Differential Geom.**42**(1995), no. 1, 1–22. MR**1350693** - Fred Almgren, Jean E. Taylor, and Lihe Wang,
*Curvature-driven flows: a variational approach*, SIAM J. Control Optim.**31**(1993), no. 2, 387–438. MR**1205983**, DOI 10.1137/0331020 - Sigurd Angenent and Morton E. Gurtin,
*Multiphase thermomechanics with interfacial structure. II. Evolution of an isothermal interface*, Arch. Rational Mech. Anal.**108**(1989), no. 4, 323–391. MR**1013461**, DOI 10.1007/BF01041068 - G. Bellettini, M. Novaga, and M. Paolini, Facet-breaking for three dimensional crystals evolving by mean curvature, Interfaces and Free Boundaries, 1 (1999), 39–55.
- Kenneth A. Brakke,
*The surface evolver*, Experiment. Math.**1**(1992), no. 2, 141–165. MR**1203871**, DOI 10.1080/10586458.1992.10504253 - J. W. Cahn and D. W. Hoffman, A vector thermodynamics for anisotropic surfaces II. Curved and faceted surfaces, Acta Met.
**22**(1974) 1205–1214. - David Caraballo, thesis, Princeton University, 1996.
- Craig Carter, Andrew Roosen, John Cahn, and Jean E. Taylor, Shape evolution by surface diffusion and surface attachment limited kinetics on completely faceted surfaces, Acta Metal. Mater.
**43**(1995),4309-4323. - Sheldon Xu-Dong Chang,
*Two-dimensional area minimizing integral currents are classical minimal surfaces*, J. Amer. Math. Soc.**1**(1988), no. 4, 699–778. MR**946554**, DOI 10.1090/S0894-0347-1988-0946554-0 - Kin Yan Chung, Ph.D. thesis, Princeton University, 1997.
- Jean E. Taylor (ed.),
*Computational Crystal Growers Workshop*, Selected Lectures in Mathematics, American Mathematical Society, Providence, RI, 1992. MR**1224451** - Jean E. Taylor (ed.),
*Computing optimal geometries*, Selected Lectures in Mathematics, American Mathematical Society, Providence, RI, 1991. Lectures presented at the AMS Special Session held in San Francisco, California, January 16–19, 1991. MR**1164472** - W. J. Trjitzinsky,
*General theory of singular integral equations with real kernels*, Trans. Amer. Math. Soc.**46**(1939), 202–279. MR**92**, DOI 10.1090/S0002-9947-1939-0000092-6 - Herbert Federer,
*Geometric measure theory*, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR**0257325** - Lawrence M. Graves,
*The Weierstrass condition for multiple integral variation problems*, Duke Math. J.**5**(1939), 656–660. MR**99** - Harald Garcke and Amy Novick-Cohen,
*A singular limit for a system of degenerate Cahn-Hilliard equations*, Adv. Differential Equations**5**(2000), no. 4-6, 401–434. MR**1750107** - Harald Garcke, Britta Nestler, and Barbara Stoth,
*On anisotropic order parameter models for multi-phase systems and their sharp interface limits*, Phys. D**115**(1998), no. 1-2, 87–108. MR**1616772**, DOI 10.1016/S0167-2789(97)00227-3 - Richard Haberman,
*Mathematical models*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1977. Mechanical vibrations, population dynamics, and traffic flow; An introduction to applied mathematics. MR**0441202** - T. C. Hales,
*The honeycomb conjecture*, Discrete Comput. Geom.**25**(2001), no. 1, 1–22. MR**1797293**, DOI 10.1007/s004540010071 - D. W. Hoffman and J. W. Cahn, A vector thermodynamics for anisotropic surfaces I. Fundamentals and application to plane surface junctions, Surf. Sci.
**31**(1972) 368–388. - Lucas Hsu, Rob Kusner, and John Sullivan,
*Minimizing the squared mean curvature integral for surfaces in space forms*, Experiment. Math.**1**(1992), no. 3, 191–207. MR**1203874**, DOI 10.1080/10586458.1992.10504258 - Michael Hutchings, Frank Morgan, Manuel Ritoré, and Antonio Ros, Proof of the double bubble conjecture, Annals of Math.
**155**(2002), 459-489. - James Clerk Maxwell, Capillary Action, in Encyclopaedia Britannica, 11th Edition,
**5**, 256-275. - Frank Morgan,
*Geometric measure theory*, 3rd ed., Academic Press, Inc., San Diego, CA, 2000. A beginner’s guide. MR**1775760** - Frank Morgan,
*Clusters minimizing area plus length of singular curves*, Math. Ann.**299**(1994), no. 4, 697–714. MR**1286892**, DOI 10.1007/BF01459806 - J. W. Cahn, C. M. Elliott, and A. Novick-Cohen,
*The Cahn-Hilliard equation with a concentration dependent mobility: motion by minus the Laplacian of the mean curvature*, European J. Appl. Math.**7**(1996), no. 3, 287–301. MR**1401172**, DOI 10.1017/S0956792500002369 - Stanley Osher and James A. Sethian,
*Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations*, J. Comput. Phys.**79**(1988), no. 1, 12–49. MR**965860**, DOI 10.1016/0021-9991(88)90002-2 - Fernando Reitich and H. Mete Soner,
*Three-phase boundary motions under constant velocities. I. The vanishing surface tension limit*, Proc. Roy. Soc. Edinburgh Sect. A**126**(1996), no. 4, 837–865. MR**1405760**, DOI 10.1017/S0308210500023106 - S.G. Srinivasan and J.W. Cahn, Challenging some free-energy reduction criteria for grain growth, preprint.
- Jean E. Taylor,
*The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces*, Ann. of Math. (2)**103**(1976), no. 3, 489–539. MR**428181**, DOI 10.2307/1970949 - Jean E. Taylor,
*Crystalline variational problems*, Bull. Amer. Math. Soc.**84**(1978), no. 4, 568–588. MR**493671**, DOI 10.1090/S0002-9904-1978-14499-1 - Jean Taylor,
*Crystals, in equilibrium and otherwise*, AMS-MAA Joint Lecture Series, American Mathematical Society, Providence, RI, 1990. A joint AMS-MAA lecture presented in Boulder, Colorado, August 1989. MR**1109716** - Jean E. Taylor,
*Constructions and conjectures in crystalline nondifferential geometry*, Differential geometry, Pitman Monogr. Surveys Pure Appl. Math., vol. 52, Longman Sci. Tech., Harlow, 1991, pp. 321–336. MR**1173051**, DOI 10.1111/j.1439-0388.1991.tb00191.x - Jean E. Taylor,
*Motion of curves by crystalline curvature, including triple junctions and boundary points*, Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990) Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 417–438. MR**1216599**, DOI 10.1090/pspum/054.1/1216599 - Jean E. Taylor,
*The motion of multiple-phase junctions under prescribed phase-boundary velocities*, J. Differential Equations**119**(1995), no. 1, 109–136. MR**1334489**, DOI 10.1006/jdeq.1995.1085 - Jean E. Taylor, Mathematical Models of Triple Junctions, Interface Science
**7**(1999), 243-250. - Jean E. Taylor,
*A variational approach to crystalline triple-junction motion*, J. Statist. Phys.**95**(1999), no. 5-6, 1221–1244. MR**1712448**, DOI 10.1023/A:1004523005442 - Jean E. Taylor and John W. Cahn,
*Linking anisotropic sharp and diffuse surface motion laws via gradient flows*, J. Statist. Phys.**77**(1994), no. 1-2, 183–197. MR**1300532**, DOI 10.1007/BF02186838 - Jean E. Taylor and John W. Cahn, On motion-induced rotation of embedded crystals, preprint.
- Jean E. Taylor (ed.),
*Computing optimal geometries*, Selected Lectures in Mathematics, American Mathematical Society, Providence, RI, 1991. Lectures presented at the AMS Special Session held in San Francisco, California, January 16–19, 1991. MR**1164472** - Arthur F. Voter,
*Hyperdynamics: Accelerated molecular dynamics of infrequent events*, Phys. Rev. Lett.**78**(1997), 3908–3911. - Alfred Rosenblatt,
*Sur les points singuliers des équations différentielles*, C. R. Acad. Sci. Paris**209**(1939), 10–11 (French). MR**85** - Nung Kwan Yip,
*Stochastic motion by mean curvature*, Arch. Rational Mech. Anal.**144**(1998), no. 4, 313–355. MR**1656479**, DOI 10.1007/s002050050120 - Jason Yunger, Ph.D. thesis, Rutgers University, 1998.

## Additional Information

**Jean E. Taylor**- Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08855
- MR Author ID: 171220
- Email: taylor@math.rutgers.edu
- Received by editor(s): November 6, 2000
- Received by editor(s) in revised form: February 21, 2002
- Published electronically: October 15, 2002
- © Copyright 2002 American Mathematical Society
- Journal: Bull. Amer. Math. Soc.
**40**(2003), 69-87 - MSC (2000): Primary 74N20; Secondary 49Q20, 49Q15, 35R99, 53A17, 65K10
- DOI: https://doi.org/10.1090/S0273-0979-02-00967-9
- MathSciNet review: 1943134