메뉴 건너뛰기




Volumn 195, Issue 3, 1998, Pages 699-723

Equilibrium shapes for planar crystals in an external field

Author keywords

[No Author keywords available]

Indexed keywords


EID: 0032368101     PISSN: 00103616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s002200050409     Document Type: Article
Times cited : (25)

References (67)
  • 1
    • 0027559276 scopus 로고
    • Curvature-driven flows: A variational approach
    • Almgren, F., Taylor, J.E. and Wang, L.: Curvature-driven flows: A variational approach. SIAM J. Control Optim. 31, 387-438 (1993)
    • (1993) SIAM J. Control Optim. , vol.31 , pp. 387-438
    • Almgren, F.1    Taylor, J.E.2    Wang, L.3
  • 2
    • 33847303953 scopus 로고
    • Multiphase thermomechanics with interfacial structure 2. Evolution of an isothermal interface
    • Angenent, S. and Gurtin, M.E.: Multiphase thermomechanics with interfacial structure 2. Evolution of an isothermal interface. Arch. Rational Mech. Anal. 108, 323-391 (1989)
    • (1989) Arch. Rational Mech. Anal. , vol.108 , pp. 323-391
    • Angenent, S.1    Gurtin, M.E.2
  • 3
    • 84961475060 scopus 로고
    • Anisotropic motion of a phase interface: Well-posedness of the initial value problem and qualitative properties of the interface
    • Angenent, S. and Gurtin, M.E.: Anisotropic motion of a phase interface: Well-posedness of the initial value problem and qualitative properties of the interface. J. Reine Angew. Math. 446, 1-47 (1994)
    • (1994) J. Reine Angew. Math. , vol.446 , pp. 1-47
    • Angenent, S.1    Gurtin, M.E.2
  • 4
    • 0037697944 scopus 로고
    • Equilibrium shapes of crystals in a gravitational field: Crystals on a table
    • Avron, J.E., Taylor, J.E. and Zia, R.K.P.: Equilibrium shapes of crystals in a gravitational field: Crystals on a table. J. Stat. Phys. 33, 493-522 (1983)
    • (1983) J. Stat. Phys. , vol.33 , pp. 493-522
    • Avron, J.E.1    Taylor, J.E.2    Zia, R.K.P.3
  • 6
    • 0001016311 scopus 로고
    • Décomposition polaire et réarrangement monotone des champs de vecteurs
    • Brenier, Y.: Décomposition polaire et réarrangement monotone des champs de vecteurs. C.R. Acad. Sci. Paris Sér. I Math. 305, 805-808 (1987)
    • (1987) C.R. Acad. Sci. Paris Sér. I Math. , vol.305 , pp. 805-808
    • Brenier, Y.1
  • 7
    • 84990623199 scopus 로고
    • Polar factorization and monotone rearrangement of vector-valued functions
    • Brenier, Y.: Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math. 44, 375-417 (1991)
    • (1991) Comm. Pure Appl. Math. 44, 375-417
    • Brenier, Y.1
  • 8
    • 84968482174 scopus 로고
    • The regularity of mappings with a convex potential
    • Caffarelli, L.A.: The regularity of mappings with a convex potential. J. Am. Math. Soc. 5, 99-104 (1992)
    • (1992) J. Am. Math. Soc. , vol.5 , pp. 99-104
    • Caffarelli, L.A.1
  • 9
    • 84990563964 scopus 로고
    • Boundary regularity of maps with convex potentials
    • Caffarelli, L.A.: Boundary regularity of maps with convex potentials. Comm. Pure Appl. Math. 45, 1141-1151 (1992)
    • (1992) Comm. Pure Appl. Math. , vol.45 , pp. 1141-1151
    • Caffarelli, L.A.1
  • 10
    • 0030366438 scopus 로고    scopus 로고
    • Boundary regularity of maps with convex potentials-II
    • Caffarelli, L. A.: Boundary regularity of maps with convex potentials-II. Ann. of Math. (2) 144, 453-496 (1996)
    • (1996) Ann. of Math. , vol.144 , Issue.2 , pp. 453-496
    • Caffarelli, L.A.1
  • 11
    • 0016118199 scopus 로고
    • A vector thermodynamics for anisotropic surfaces - II. Curved and faceted surfaces
    • Cahn, J.W. and Hoffman, D.W.: A vector thermodynamics for anisotropic surfaces - II. Curved and faceted surfaces. Acta Metallurgica 22, 1205-1215 (1974)
    • (1974) Acta Metallurgica , vol.22 , pp. 1205-1215
    • Cahn, J.W.1    Hoffman, D.W.2
  • 12
    • 84972499852 scopus 로고
    • Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations
    • Chen, Y.-G., Giga, Y. and Goto, S.: Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Differ. Geom. 33, 749-786 (1991)
    • (1991) J. Differ. Geom. , vol.33 , pp. 749-786
    • Chen, Y.-G.1    Giga, Y.2    Goto, S.3
  • 14
    • 0039481273 scopus 로고
    • Wulff theorem and best constant in Sobolev inequality
    • Dacorogna, B. and Pfister, C.E. Wulff theorem and best constant in Sobolev inequality. J. Math. Pures Appl. 71, 97-118 (1992)
    • (1992) J. Math. Pures Appl. , vol.71 , pp. 97-118
    • Dacorogna, B.1    Pfister, C.E.2
  • 15
    • 0001539326 scopus 로고
    • Über einen geometrischen satz von wulff für die gleichgewichtsform von kristallen
    • Dinghas, A.: Über einen geometrischen Satz von Wulff für die Gleichgewichtsform von Kristallen. Z. Krist. 105, 304-314 (1944)
    • (1944) Z. Krist. , vol.105 , pp. 304-314
    • Dinghas, A.1
  • 18
    • 84972518016 scopus 로고
    • Motion of level sets by mean curvature
    • Evans, L.C. and Spruck, J.: Motion of level sets by mean curvature. I. J. Differ. Geom. 33, 635-681 (1991)
    • (1991) I. J. Differ. Geom. , vol.33 , pp. 635-681
    • Evans, L.C.1    Spruck, J.2
  • 20
    • 84972544319 scopus 로고
    • The sessile liquid drop, 1. Symmetric case
    • Finn, R.: The sessile liquid drop, 1. Symmetric case. Pacific J. Math. 88, 549-587 (1980)
    • (1980) Pacific J. Math. , vol.88 , pp. 549-587
    • Finn, R.1
  • 24
    • 0000500198 scopus 로고
    • Curve shortening makes convex curves circular
    • Gage, M.E.: Curve shortening makes convex curves circular. Invent. Math. 76, 357-364 (1984)
    • (1984) Invent. Math. , vol.76 , pp. 357-364
    • Gage, M.E.1
  • 25
    • 85033889076 scopus 로고
    • Minkowski plane geometry and anisotropic curvature flow of curves
    • A. Damlamian, J. Spruck and A. Visintin, editors, Curvature Flows and Related Topics, Tokyo: GAKUTO
    • Gage, M.E.: Minkowski plane geometry and anisotropic curvature flow of curves. In: A. Damlamian, J. Spruck and A. Visintin, editors, Curvature Flows and Related Topics, volume 5 of Internat. Ser. Math. Sci. Appl., Tokyo: GAKUTO, 1995, pp. 83-97
    • (1995) Internat. Ser. Math. Sci. Appl. , vol.5 , pp. 83-97
    • Gage, M.E.1
  • 26
    • 84974005139 scopus 로고
    • Evolving plane curves by curvature in relative geometries
    • Gage, M.E.: Evolving plane curves by curvature in relative geometries. Duke Math. J. 72, 441-466 (1993)
    • (1993) Duke Math. J. , vol.72 , pp. 441-466
    • Gage, M.E.1
  • 27
    • 84963023423 scopus 로고
    • The heat equation shrinking convex plane curves
    • Gage, M.E. and Hamilton, R.S.: The heat equation shrinking convex plane curves. J. Differential Geom. 23, 69-96 (1986)
    • (1986) J. Differential Geom. , vol.23 , pp. 69-96
    • Gage, M.E.1    Hamilton, R.S.2
  • 28
    • 0345788532 scopus 로고
    • Evolving plane curves by curvature in relative geometries. Ii
    • Gage, M.E. and Li. Y.: Evolving plane curves by curvature in relative geometries. II. Duke Math. J. 75, 79-98 (1994)
    • (1994) Duke Math. J. , vol.75 , pp. 79-98
    • Gage, M.E.1    Li, Y.2
  • 29
    • 21844484336 scopus 로고
    • An elementary proof of the polar factorization of vector-valued functions
    • Gangbo, W.: An elementary proof of the polar factorization of vector-valued functions. Arch. Rational Mech. Anal. 128, 381-399 (1994)
    • (1994) Arch. Rational Mech. Anal. , vol.128 , pp. 381-399
    • Gangbo, W.1
  • 30
    • 0040219531 scopus 로고
    • On the equilibrium of heterogeneous substances. 1878
    • esp. footnote p. 325. New York: Longmans, Green and Company
    • Gibbs, J.W.: On the equilibrium of heterogeneous substances. 1878. In: Collected Works, Volume 1, esp. footnote p. 325. New York: Longmans, Green and Company, 1898, pp. 55-353
    • (1898) Collected Works , vol.1 , pp. 55-353
    • Gibbs, J.W.1
  • 32
    • 84972528479 scopus 로고
    • The heat equation shrinks embedded plane curves to round points
    • Grayson. M.A.: The heat equation shrinks embedded plane curves to round points. J. Differ. Geom. 26, 285-314 (1987)
    • (1987) J. Differ. Geom. , vol.26 , pp. 285-314
    • Grayson, M.A.1
  • 33
    • 0040219530 scopus 로고
    • Anisotropic motion of an interface relaxed by the formation of infinitesimal wrinkles
    • Gurtin, M.E., Soner, H.M. and Souganidas, P.E.: Anisotropic motion of an interface relaxed by the formation of infinitesimal wrinkles. J. Differ. Eqs. 119, 54-108 (1995)
    • (1995) J. Differ. Eqs. , vol.119 , pp. 54-108
    • Gurtin, M.E.1    Soner, H.M.2    Souganidas, P.E.3
  • 34
    • 0002318515 scopus 로고
    • Brunn-Minkowskischer satz und isoperimetrie
    • Hadwiger, H. and Ohmann, D.: Brunn-Minkowskischer Satz und Isoperimetrie. Math. Z. 66, 1-8 (1956)
    • (1956) Math. Z. , vol.66 , pp. 1-8
    • Hadwiger, H.1    Ohmann, D.2
  • 35
    • 36149008975 scopus 로고
    • Some theorems on the free energy of crystal surfaces
    • Herring, C.: Some theorems on the free energy of crystal surfaces. Phys. Rev. 82, 87-93 (1951)
    • (1951) Phys. Rev. , vol.82 , pp. 87-93
    • Herring, C.1
  • 36
    • 0002358415 scopus 로고
    • The use of classical macroscopic concepts in surface-energy problems
    • R. Gomer and C.S. Smith, editors, Chicago II: University of Chicago Press
    • Herring, C.: The use of classical macroscopic concepts in surface-energy problems. In: R. Gomer and C.S. Smith, editors, Structure and Properties of Solid Surfaces, Chicago II: University of Chicago Press 1953, pp. 5-72
    • (1953) Structure and Properties of Solid Surfaces , pp. 5-72
    • Herring, C.1
  • 37
    • 33745017533 scopus 로고
    • Establishment of the equilibrium shape of metal crystallites on a foreign substrate: Gold on graphite
    • Heyraud, J.C. and Métois, J.J.: Establishment of the equilibrium shape of metal crystallites on a foreign substrate: Gold on graphite. J. Crystal Growth 50, 571-574 (1980)
    • (1980) J. Crystal Growth , vol.50 , pp. 571-574
    • Heyraud, J.C.1    Métois, J.J.2
  • 38
    • 33744588032 scopus 로고
    • Equilibrium shape and temperature: Lead on graphite
    • Heyraud, J.C. and Métois, J.J.: Equilibrium shape and temperature: Lead on graphite. Surface Science 128, 334-342 (1983)
    • (1983) Surface Science , vol.128 , pp. 334-342
    • Heyraud, J.C.1    Métois, J.J.2
  • 39
  • 40
    • 84972507688 scopus 로고
    • Flow by mean curvature of convex surfaces into spheres
    • Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20, 237-266 (1984)
    • (1984) J. Differ. Geom. , vol.20 , pp. 237-266
    • Huisken, G.1
  • 42
    • 0040814074 scopus 로고
    • Der curie-wulff'sehe satz über combinationsformen von krystallen
    • Liebmann, H.: Der Curie-Wulff'sehe Satz über Combinationsformen von Krystallen. Z. Krist. 53, 171-177 (1914)
    • (1914) Z. Krist. , vol.53 , pp. 171-177
    • Liebmann, H.1
  • 43
    • 0000850415 scopus 로고
    • Implicit time discretization for the mean curvature flow equation
    • Luckhaus, S. and Sturzenhecker, T.: Implicit time discretization for the mean curvature flow equation. Calc. Var. Partial Differ. Eqs. 3, 253-271 (1995)
    • (1995) Calc. Var. Partial Differ. Eqs. , vol.3 , pp. 253-271
    • Luckhaus, S.1    Sturzenhecker, T.2
  • 44
    • 0040814075 scopus 로고
    • The surface of crystalline helium-4
    • February
    • Maris, H.J. and Andreev, A.F.: The surface of crystalline helium-4. Physics Today, 25-30, February (1987)
    • (1987) Physics Today , pp. 25-30
    • Maris, H.J.1    Andreev, A.F.2
  • 46
    • 84974004014 scopus 로고
    • Existence and uniqueness of monotone measure-preserving maps
    • McCann, R.J. Existence and uniqueness of monotone measure-preserving maps. Duke Math. J. 80, 309-323 (1995)
    • (1995) Duke Math. J. , vol.80 , pp. 309-323
    • McCann, R.J.1
  • 47
    • 0031161793 scopus 로고    scopus 로고
    • A convexity principle for interacting gases
    • McCann, R.J.: A convexity principle for interacting gases. Adv. Math. 128, 153-179 (1997)
    • (1997) Adv. Math. , vol.128 , pp. 153-179
    • McCann, R.J.1
  • 48
    • 85033873578 scopus 로고    scopus 로고
    • Personal communication
    • Okikiolu, K.: Personal communication
    • Okikiolu, K.1
  • 49
    • 0007191718 scopus 로고
    • Experimental investigation of the surface melting of equilibrium form faces of diphenyl
    • Pavlovska, A. and Nenow, D.: Experimental investigation of the surface melting of equilibrium form faces of diphenyl. Surface Science 27, 211-217 (1971)
    • (1971) Surface Science , vol.27 , pp. 211-217
    • Pavlovska, A.1    Nenow, D.2
  • 50
    • 0040814073 scopus 로고
    • Experimental study of the equilibrium form of negative crystals in diphenyl
    • Pavlovska, A. and Nenow, D.: Experimental study of the equilibrium form of negative crystals in diphenyl. J. Crystal Growth 8, 209-212 (1971)
    • (1971) J. Crystal Growth , vol.8 , pp. 209-212
    • Pavlovska, A.1    Nenow, D.2
  • 51
    • 0040219529 scopus 로고
    • Les surfaces non-singulières sur la forme d'équilibre du naphtalène
    • Pavlovska, A. and Nenow, D. Les surfaces non-singulières sur la forme d'équilibre du naphtalène. J. Crystal Growth 12, 9-12 (1971)
    • (1971) J. Crystal Growth , vol.12 , pp. 9-12
    • Pavlovska, A.1    Nenow, D.2
  • 52
    • 0001503348 scopus 로고
    • Weak convergence of completely additive vector functions on a set
    • Reshetnyak, Yu.G.: Weak convergence of completely additive vector functions on a set. Siberian Math. J. 9, 1039-1045 (1968)
    • (1968) Siberian Math. J. , vol.9 , pp. 1039-1045
    • Reshetnyak, Yu.G.1
  • 54
    • 0030543461 scopus 로고    scopus 로고
    • Constrained variational problem with applications to the ising model
    • Schonmann, R.H. and Shlosman, S.B.: Constrained variational problem with applications to the Ising model. J. Stat. Phys. 83, 867-905 (1996)
    • (1996) J. Stat. Phys. , vol.83 , pp. 867-905
    • Schonmann, R.H.1    Shlosman, S.B.2
  • 55
    • 0000935065 scopus 로고
    • Motion of a set by the mean curvature of its boundary
    • Soner, H.M.: Motion of a set by the mean curvature of its boundary. J. Differ. Eqs. 101, 313-372 (1993)
    • (1993) J. Differ. Eqs. , vol.101 , pp. 313-372
    • Soner, H.M.1
  • 56
    • 0001123141 scopus 로고
    • Existence and structure of solutions to a class of nonelliptic variational problems
    • London: Academic Press
    • Taylor, J.E. Existence and structure of solutions to a class of nonelliptic variational problems. Volume 14 of Symposia Mathematica, London: Academic Press, 1974, pp. 499-508
    • (1974) Symposia Mathematica , vol.14 , pp. 499-508
    • Taylor, J.E.1
  • 57
    • 0000500627 scopus 로고
    • Unique structure of solutions to a class of nonelliptic variational problems
    • Differential geometry. Providence, RI: American Mathematical Society
    • Taylor, J.E.: Unique structure of solutions to a class of nonelliptic variational problems. In: Differential geometry. Volume 27 of Proc. Sympos. Pure. Math., Providence, RI: American Mathematical Society, 1975, pp. 419-427
    • (1975) Proc. Sympos. Pure. Math. , vol.27 , pp. 419-427
    • Taylor, J.E.1
  • 58
    • 84966261517 scopus 로고
    • Crystalline variational problems
    • Taylor, J.E.: Crystalline variational problems. Bull. Am. Math. Soc. 84, 568-588 (1978)
    • (1978) Bull. Am. Math. Soc. , vol.84 , pp. 568-588
    • Taylor, J.E.1
  • 59
    • 0026899706 scopus 로고
    • II-mean curvature and weighted mean curvature
    • Taylor, J.E.: II-Mean curvature and weighted mean curvature. Acta Metallurgica et Materiala 40, 1475-1485 (1992)
    • (1992) Acta Metallurgica et Materiala , vol.40 , pp. 1475-1485
    • Taylor, J.E.1
  • 60
  • 62
    • 0001095414 scopus 로고
    • Unbounded parametric surfaces of prescribed mean curvature
    • Vogel, T.I.: Unbounded parametric surfaces of prescribed mean curvature. Indiana Univ. Math. J. 31, 281-288 (1982)
    • (1982) Indiana Univ. Math. J. , vol.31 , pp. 281-288
    • Vogel, T.I.1
  • 63
    • 0010212156 scopus 로고
    • Der wulffsche satz für die gleichgewichtsform von kristallen
    • von Laue, M.: Der Wulffsche Satz für die Gleichgewichtsform von Kristallen. Z. Krist. 105, 124-133 (1943)
    • (1943) Z. Krist. , vol.105 , pp. 124-133
    • Von Laue M1
  • 64
    • 84972492577 scopus 로고
    • The symmetry of sessile and pendant drops
    • Wente, H.C.: The symmetry of sessile and pendant drops. Pacific J. Math. 88, 307-397 (1980)
    • (1980) Pacific J. Math. , vol.88 , pp. 307-397
    • Wente, H.C.1
  • 65
    • 18844410961 scopus 로고
    • Equilibrium shape of a small particle in contact with a foreign substrate
    • Winterbottom, W.L.: Equilibrium shape of a small particle in contact with a foreign substrate. Acta Metallurgica 15, 303-310 (1967)
    • (1967) Acta Metallurgica , vol.15 , pp. 303-310
    • Winterbottom, W.L.1
  • 66
    • 0001421454 scopus 로고
    • Zur frage der geschwindigkeit des wachsthums und der auflösung der krystallflächen
    • Wulff, G.: Zur Frage der Geschwindigkeit des Wachsthums und der Auflösung der Krystallflächen. Z. Krist. 34, 449-530 (1901)
    • (1901) Z. Krist. , vol.34 , pp. 449-530
    • Wulff, G.1
  • 67
    • 0000198102 scopus 로고
    • Crystals in corners: The summertop construction
    • Zia, R.K.P., Avron, J.E. and Taylor, J.E.: Crystals in corners: the summertop construction. J. Stat. Phys. 50, 727-736 (1988)
    • (1988) J. Stat. Phys. , vol.50 , pp. 727-736
    • Zia, R.K.P.1    Avron, J.E.2    Taylor, J.E.3


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.