References on variational models and weak convergence
methods for crystalline solids (for AMSC 698V) - B. Li
AMSC 698V: Advanced Topics in Applied Mathematics, Fall, 2003
MATHEMATICAL AND COMPUTATIONAL PROBLEMS
IN MATERIALS SCIENCE
Instructor: Bo Li
References on Variational Models and Weak Convergence
Methods for Crystalline solids
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E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of
variations, Arch. Rat. Mech. Anal., 86:125-145, 1986.
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E. Acerbi and N. Fusco, A regularity theorem for minimizers of
quasiconvex integrals, Arch. Rat. Mech. Anal., 99:261-281, 1987.
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J. M. Ball, Convexity conditions and existence theorems in nonlinear
elasticity}, Arch. Rat. Mech. Anal., 63:337-403, 1977.
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J. M. Ball, A version of the fundamental theorem for Young measures,
in Partial Differential Equations and Continuum Models of Phase Transition,
M. Rascle, D. Serre, and M. Slemrod, eds.,
Lecture Notes in Physics 344, Springer-Verlag, Berlin, 207-215, 1989.
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J. M. Ball, J. C. Currie, and P. J. Olver, Null Lagrangians,
weak continuity and variational problems of arbitrary order,
J. Funct. Anal., 41:135-174, 1981.
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J. Ball and F. Murat, W^{1,p}-Quasiconvexity and variational
problems for multiple integrals, J. Funct. Anal., 58:225-253, 1984.
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J. M. Ball and R. D. James, Fine phase mixtures as minimizers of
energy, Arch. Rat. Mech. Anal., 100:13-52, 1987.
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J. M. Ball and R. D. James, Proposed experimental tests of a theory
of fine microstructure and the two-well problem, Phil. Trans. Roy. Soc.
Lond. A, 338:389-450, 1992.
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K. Bhattacharya, N. Firoozye, R. D. James, and R. V. Kohn,
Restrictions on microstructure, Proc. Roy. Soc. Edinburgh,
124A:843-878, 1994.
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M. Chipot and D. Kinderlehrer, Equilibrium configurations of
crystals}, Arch. Rat. Mech. Anal., 103:237-277, 1988.
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B. Dacorogna, Direct Methods in the Calculus of Variations,
Springer-Verlag, Berlin, 1989.
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L. C. Evans, Quasi-convexity and partial regularity in the
calculus of variations, Arch. Rat. Mech. Anal., 95:227-252, 1986.
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L. C. Evans, Weak Convergence Methods for Nonlinear Partial
Differential Equations, CBMS Regional Conference Series in Mathematics
74, Amer. Math. Soc., Providence, RI, 1990.
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R. D. James and D. Kinderlehrer, Theory of diffusionless phase
transitions, in Partial Differential Equations and Continuum Models of
Phase Transition, M. Rascle, D. Serre, and M. Slemrod, eds.,
Lecture Notes in Physics 344, Springer-Verlag, Berlin, 51-84, 1989.
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D. Kinderlehrer and P. Pedregal, Characterizations of gradient
Young measures, Arch. Rat. Mech. Anal., 115:329-365, 1991.
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R. V. Kohn, Relaxation of a double-well energy, Cont. Mech. Thermodyn.,
3:193-236, 1991.
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N. G. Meyers, Quasi-convexity and lower semicontinuity in the
multiple variational integrals of any order, Trans. Amer. Math. Soc.,
119:125-149, 1965.
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C. B. Morrey, Jr., Quasiconvexity and the lower semicontinuity of
multiple integrals, Pacific J. Math., 2:25-53, 1952.
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C. B. Morrey, Jr., Multiple Integrals in the Calculus of Variations,
Springer-Verlag, 1966.
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L. Tartar, Compensated compactness and applications to partial
differential equations, in Nonlinear Analysis and Mechanics,
Heriot-Watt Symposium IV, R. Knops, ed.,
Pitman Research Notes in Mathematics 39, Pitman, Boston, 136-212, 1979.
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V. Sverák, Rank-one convexity does not imply
quasiconvexity, Proc. Roy. Soc. Edinburgh, 120A:185-189, 1992.
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V. Sverák, On the problem of two wells, in Microstructure
and Phase Transition, J. Ericksen, R. James, D. Kinderlehrer, and
M. Luskin, eds., IMA Volumes in Mathematics and Its Applications 54,
Springer-Verlag, 183-190, 1993.
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V. Sverák,
Lower-semicontinuity of variational integrals
and compensated compactness, in Proceedings ICM 94, Birkhäuser,
ZÜrich, 1995.