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Crystalline variational problems. (English) Zbl 0392.49022


MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
53C65 Integral geometry
49M99 Numerical methods in optimal control
82D25 Statistical mechanics of crystals
82D35 Statistical mechanics of metals
52A05 Convex sets without dimension restrictions (aspects of convex geometry)
52A40 Inequalities and extremum problems involving convexity in convex geometry
28A75 Length, area, volume, other geometric measure theory
49Q05 Minimal surfaces and optimization
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References:

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[3] 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. · Zbl 0327.49043
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[5] J. W. Cahn, personal communication.
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[8] Herbert Federer, Colloquium lectures on geometric measure theory, Bull. Amer. Math. Soc. 84 (1978), no. 3, 291 – 338. · Zbl 0392.49021
[9] J. H. G. Fu, A mathematical model for crystal growth and related problems, Junior paper, Princeton University.
[10] C. Herring, Some theorems on the free energy of crystal surface, Phys. Rev. 82 (1951), 87-93. · Zbl 0042.23201
[11] M. v. Laue, Der Wulffsche Satz für die Gleichgewichtsform von Kristallen, Zeitschrift für Kristallographie 105 (1943), 124-133. · Zbl 0060.46003
[12] H. Liebmann, Der Curie-Wulff’sche Satz über Combinationsformen von Krystallen, Zeitschrift für Krystallographie und Mineralogie 53 (1914), 171-177.
[13] Harold Parks, Explicit determination of area minimizing hypersurfacess, Duke Math. J. 44 (1977), no. 3, 519 – 534. · Zbl 0385.49026
[14] R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. · Zbl 0193.18401
[15] Jean E. Taylor, Existence and structure of solutions to a class of nonelliptic variational problems, Symposia Mathematica, Vol. XIV (Convegno di Teoria Geometrica dell’Integrazione e Varietà Minimali, INDAM, Roma, Maggio 1973), Academic Press, London, 1974, pp. 499 – 508.
[16] Jean E. Taylor, Unique structure of solutions to a class of nonelliptic variational problems, Differential geometry (Proc. Sympos. Pure. Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R.I., 1975, pp. 419 – 427.
[17] G. Wulff, Zur Frage der Geschwindigkeit des Wachsthums und der Auflösung der Krystallflachen, Zeitschrift für Krystallographie und Mineralogie 34 (1901), 449-530.
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