Some mathematical challenges in materials science
Author:
Jean E. Taylor
Journal:
Bull. Amer. Math. Soc. 40 (2003), 6987
MSC (2000):
Primary 74N20; Secondary 49Q20, 49Q15, 35R99, 53A17, 65K10
Published electronically:
October 15, 2002
MathSciNet review:
1943134
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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 . The figures are selected copies of transparencies presented at the lecture on which this paper is based.
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 W. K. Allard, On the first variation of a varifold, Annals of Math. 95 (1972), 417491. MR 46:6136
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 Frederick J. Almgren, Jr, Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Memoirs Amer. Math. Soc. 4 165 (1976) viii + 199 pages. MR 54:8420
 [Alm99]
 Selected Works of Frederick J. Almgren, Jr., American Mathematical Society, 1999. MR 2001f:01053
 [Alm00]
 Frederick J. Almgren, Jr., Almgren's Big Regularity Paper: 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
 [AT]
 Fred Almgren and Jean E. Taylor, Flat flow is motion by crystalline curvature for curves with crystalline energies, J. Differential Geometry 42 (1995), 122. MR 96h:58034
 [ATW]
 Fred Almgren, Jean Taylor and Lihe Wang, Curvature Driven Flows: A Variational Approach, SIAM Journal of Control and Optimization 31 (1993), 387438. MR 94h:58067
 [AG]
 S. Angenent and M. Gurtin, Multiphase thermomechanics with interfacial structure. 2. Evolution of an isothermal interface. Arch Rat. Mech. Anal 108 (1989) 323391. MR 91d:73004
 [BNP]
 G. Bellettini, M. Novaga, and M. Paolini, Facetbreaking for three dimensional crystals evolving by mean curvature, Interfaces and Free Boundaries, 1 (1999), 3955.
 [B]
 Kenneth A. Brakke, The surface evolver, Experiment. Math. 1 (1992), no. 2, 141165. MR 93k:53006
 [CH]
 J. W. Cahn and D. W. Hoffman, A vector thermodynamics for anisotropic surfaces II. Curved and faceted surfaces, Acta Met. 22 (1974) 12051214.
 [Car]
 David Caraballo, thesis, Princeton University, 1996.
 [CRCT]
 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),43094323.
 [Cha]
 Sheldon Chang, Two dimensional area minimizing integral currents are classical minimal surfaces, J. Amer. Math. Soc. 1 (1988), 699778. MR 89i:49028
 [Chu]
 Kin Yan Chung, Ph.D. thesis, Princeton University, 1997.
 [CCGW]
 Computational Crystal Growers Workshop (Jean E. Taylor, ed.), Selected Lectures in Mathematics, American Mathematical Society (1992). MR 94f:58007
 [COG]
 Computing Optimal Geometries, J. E. Taylor, ed., Selected Lectures in Mathematics, Amer. Math. Soc. 1991. MR 93a:65021
 [ES]
 Lawrence C. Evans and Joel Spruck, Motion of Level Sets by Mean Curvature I, II, III, and IV, in (respectively) J. Differential Geometry 33(1991), 635681; Trans. Amer. Math. Soc. 330 (1992), no. 1, 321332 J. Geom. Anal. 2 (1992), no. 2, 121150; J. Geom. Anal. 5 (1995), no. 1, 77114. MR 92h:35097; MR 92f:58050; MR 93d:58044; MR 96a:35077
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 H. Federer, Geometric Measure Theory, SpringerVerlag, New York, 1969. MR 41:1976
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 M.H. Giga and Y. Giga, Evolving graphs by singular weighted curvature, Arch. Rational Mech. Anal. 141 (1998), 117198. (The paper by Yoshikazu Giga, Morton E. Gurtin, and José Matias, On the dynamics of crystalline motion, Japan Journal of Industrial and Applied Mathematics 15 (1998), 750, includes a comprehensive definition, but incorrectly ``proves'' that no stepping occurs.) MR 99j:35118; MR 99h:73008
 [GN]
 Harald Garcke and Amy NovickCohen, A singular limit for a system of degenerate CahnHilliard equations, Adv. Differential Equations 5 (2000), no. 46, 401434. MR 2001e:35097
 [GNS]
 Harald Garcke, Britta Nestler, and Barbara Stoth, On anisotropic order parameter models for multiphase systems and their sharp interface limits. Phys. D 115 (1998), no. 12, 87108. MR 99g:82023
 [Hab]
 R. Haberman, Mathematical Models, PrenticeHall, Inc., Engelwood Cliffs, NJ, 1977. Pages 257394 deal with traffic flow, through the use of characteristics with shocks and fans. MR 55:14066
 [Hal]
 Thomas C. Hales, The honeycomb conjecture. Discrete Comput. Geom. 25 (2001), no. 1, 122. MR 2002a:52020
 [HC]
 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) 368388.
 [HKS]
 Lucas Hsu, Rob Kusner, and John Sullivan, Minimizing the squared mean curvature integral for surfaces in space forms, Experiment. Math. 1 (1992), no. 3, 191207. MR 94a:53015
 [HMRS]
 Michael Hutchings, Frank Morgan, Manuel Ritoré, and Antonio Ros, Proof of the double bubble conjecture, Annals of Math. 155 (2002), 459489.
 [Max]
 James Clerk Maxwell, Capillary Action, in Encyclopaedia Britannica, 11th Edition, 5, 256275.
 [Mor00]
 Frank Morgan, Geometric Measure Theory: A Beginner's Guide, Third Edition, 2000. Academic Press, Inc., San Diego, CA, 2000. MR 2001j:49001
 [Mor94]
 Frank Morgan, Clusters minimizing area plus length of singular curves. Math. Ann. 299 (1994), no. 4, 69771. MR 95g:49083
 [NCE]
 Amy NovickCohen, J. W. Cahn, and C. M. Elliott, "The CahnHilliard equation: motion by the Laplacian of the mean curvature," Euro. J. Appl. Math., 7, 287301 (1996). MR 97g:80010
 [OS]
 Stan Osher and James A. Sethian, Fronts propagating with curvaturedependent speed: algorithms based on Hamilton Jacobi formulations. J.Comput.Phys. 79 (1988), no. 1, 1249. MR 89h:80012
 [RS]
 Fernando Reitich and H. Mete Soner, Threephase boundary motions under constant velocities. I. The vanishing surface tension limit, Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), no. 4, 837865. MR 97h:73012
 [SC]
 S.G. Srinivasan and J.W. Cahn, Challenging some freeenergy reduction criteria for grain growth, preprint.
 [T76]
 Jean E. Taylor, The structure of singularities in soapbubblelike and soapfilmlike minimal surfaces, Annals of Math. 103 (1976), 489539. MR 55:1208a
 [T78]
 Jean E. Taylor, Crystalline variational problems, Bull. AMS 84 (1978), 568588; Mean curvature and weighted mean curvature, Acta Metall. Mater. 40 (1992), 14751485. MR 58:12649
 [T89]
 Jean E. Taylor, Crystals, in equilibrium and otherwise, video tape of AMSMAA joint lecture, Boulder, CO 1989, American Mathematical Society, Providence, RI. MR 92c:58019
 [T91]
 Jean E. Taylor, Constructions and Conjectures in Crystalline Nondifferential Geometry, in Differential Geometry, B. Lawson and K. Tenenblat, eds, Pitman Monographs and Surveys in Pure and Applied Math 52 (1991), 321336. MR 93e:49004
 [T93]
 Jean E. Taylor, Motion of curves by crystalline curvature, including triple junctions and boundary points, Differential Geometry, Proceedings of Symposia in Pure Math. 51 (part 1) (1993), 417438. MR 94c:53012
 [T95]
 Jean E. Taylor, The motion of multiplephase junctions under prescribed phaseboundary velocities, J. Diff. Eq. 119 (1995), 109136. MR 96e:73013
 [T99m]
 Jean E. Taylor, Mathematical Models of Triple Junctions, Interface Science 7 (1999), 243250.
 [T99v]
 Jean E. Taylor, A Variational Approach to Crystalline Triple Junction Motion, J. Stat. Phys. 95 (1999), 12211244. MR 2000i:74073
 [TC1]
 Jean E. Taylor and J. W. Cahn, Linking Anisotropic Sharp and Diffuse Surface Motion Laws via Gradient Flows, J. Stat. Phys. 77 (1994), 183197. MR 95j:58029
 [TC2]
 Jean E. Taylor and John W. Cahn, On motioninduced rotation of embedded crystals, preprint.
 [Teg]
 James Tegart, ThreeDimensional Fluid Interfaces in a Cylindrical Container, in Computing Optimal Geometries, J. E. Taylor, ed., Selected Lectures in Mathematics, Amer. Math. Soc. 1991. MR 93a:65021
 [V]
 Arthur F. Voter, Hyperdynamics: Accelerated molecular dynamics of infrequent events, Phys. Rev. Lett. 78 (1997), 39083911.
 [W]
 Brian White, Existence of leastarea mappings of dimensional domains. Ann. of Math. (2) 118 (1983), no. 1, 179185; Mappings that minimize area in their homotopy classes. J. Differential Geom. 20 (1984), no. 2, 433446. MR 85e:49063; MR 86f:49107
 [Yip]
 NungKwan Yip, Ph.D. thesis, Princeton University, 1996. Also see: Stochastic motion by mean curvature. Arch. Rational Mech. Anal. 144 (1998), no. 4, 313355. MR 99m:60100
 [Yun]
 Jason Yunger, Ph.D. thesis, Rutgers University, 1998.
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Additional Information
Jean E. Taylor
Affiliation:
Department of Mathematics, Rutgers University, Piscataway, New Jersey 08855
Email:
taylor@math.rutgers.edu
DOI:
http://dx.doi.org/10.1090/S0273097902009679
PII:
S 02730979(02)009679
Received by editor(s):
November 6, 2000
Received by editor(s) in revised form:
February 21, 2002
Published electronically:
October 15, 2002
Article copyright:
© Copyright 2002
American Mathematical Society
