Summary
We study numerically the pattern of the minimizing sequences of nonconvex problems which do not admit a minimizer.
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References
[B.] Ball, J.M. (1989): A version of the fundamental theorem for Young measures. In: M. Rascle, D. Serre, M. Slemrod (eds.). Partial Differential Equations and Continuum Models of Phase Transitions, Lecture Notes in Physics #344, Springer, Berlin Heidelberg New York, pp. 207–215
[B.J.1] Ball, J.M., James, R.D. (1987): Fine phase mixtures as minimizers of energy. Arch. Rat. Mech. Anal.100, 13–52
[B.J.2] Ball, J.M., James, R.D. (1988): Experimental tests of a theory of fine microstructures. Preprint.
[B.M.] Ball, J.M., Murat, F. (1984):W 1,p quasiconvexity and variational problems for multiple integrals. J. Funct. Anal.58, 225–253
[C.] Chipot, M. (1990): Hyperelasticity for crystals. Eur. J. Appl. Math.1, 113–129
[C.C.] Chipot, M., Collins, C. (1990): Numerical approximation in variational problems with potential wells. Preprint (to appear)
[C.C.K.] Chipot, M., Collins, C., Kinderlehrer, D.: In preparation
[C.K.] Chipot, M., Kinderlehrer, D. (1988): Equilibrium configurations of crystals. Arch. Rational Mech. Anal.103, 237–277
[Co.] Collins, C. (1990): Thesis, Univeristy of Minnesota
[C.K.L.] Collins, C., Kinderlehrer, D., Luskin, M. (1990): Numerical approximation of the solution of a variational problem with a double well potential. IMA preprint #605. SIAM J. Numer. Anal. (to appear)
[C.L.1] Collins, C., Luskin, M. (1989): The computation of the austenitic-martensitic phase transition. In: M. Rascle, D. Serre, M. Slemrod (eds.), Partial Differential Equations and Continuum Models of Phase Transitions, Lecture Notes in Physic #344, Springer, Berlin Heidelberg New York, pp 34–50
[C.L.2] Collins, C., Luskin, M. (1989): Computational results for phase transitions in shape memory materials. In: C. Rogers (ed.), Smart Materials, Structures, and Mathematical Issues. Technomic Publishing, Lancaster, Pennsylvania, pp. 198–215
[C.L.3] Collins, C., Luskin, M. (1991): Numerical modeling of the microstructure of crystals with symmetry-related variants. In: Proceedings of the ARO US-Japan Workshop on Smart/Intelligent Materials. Lancaster, Pennsylvania, Technomic Publishing (to appear).
[C.L.4] Collins, C., Luskin, M. (1990): Optimal order error estimates for the finite element approximation of the solution of a nonconvex variational problem. Preprint
[D.] Dacorogna, B. (1982): Weak continuity and weak lower semicontinuity of nonlinear functionals. Springer Lecture Notes #922. Springer, Berlin Heidelberg New York
[Er.] Ericksen, J.L. (1987): Some constrained elastic crystals. In: J.M. Ball (ed.), Material Instabilities in Continuum Mechanics and Related Problems. Oxford University Press, Oxford, pp. 119–137
[Ev.] Evans, L.C. (1989): Weak Convergence Methods for Nonlinear Partial Differential Equations. A.M.S. Regional Conference Series in Mathematics #74
[F.1] Fonseca, I. (1985): Variational methods for elastic crystals. Arch. Rat. Mech. Anal.97, 189–220
[F.2] Fonseca, I. (1988): The lower quasiconvex envelope of the stored energy function for an elastic crystal. J. Math. Pures Appl.67, 175–195
[Fr.] French, D.A. (1990): On the convergence of finite element approximations of a relaxed variational problem. SIAM J. Numer. Anal.26, 3, 419–436
[G.T.] Gilbard, D., Trudinger, N.S. (1985): Elliptic Partial Differential Equations of Second Order. Springer, Berlin Heidelberg New York
[J.1] James, R. (1989): Basic principles for the improvement of shape-memory and related materials. In: C. Rogers (ed.), Smart Materials, Structures, and Mathematical Issues. Technomic Publishing, Lancaster, Pennsylvania
[J.2] James, R.D. (1987): Microstructure and weak convergence. In: J.M. Ball (ed.), Material Instabilities in Continuum Mechanics and Related Problems. Oxford University Press, Oxford, pp. 175–196
[J.K.] James, R.D., Kinderlehrer, D. (1989): Theory of diffusionless phase transitions. In: M. Rascle, D. Serre, M. Slemrod (eds.), Partial Differential Equations and Continuum Models of Phase Transitions, Lecture Notes in Physics #344. Springer, Berlin Heidelberg New York, pp. 51–84
[K.] Kinderlehrer, D. (1987): Remarks about equilibrium configurations of crystals. In: J.M. Ball (ed.), Material Instabilities in Continuum Mechanics and Related Problems. Oxford University Press, Oxford, pp. 217–242
[K.P.1] Kinderlehrer, D., Pedregal, P. (1991): Analysis of young measures. Preprint
[K.P.2] Kinderlehrer, D., Pedregal, P. (1991): Weak convergence of integrands and the young measure representation. Preprint
[Ko.] Kohn, R. (1989): The relationship between linear and nonlinear variational models of coherent phase transitions. In: Proceedings of the Seventh Army Conference on Applied Mathematics and Computing, West Point
[P.] Pedregal, P. (1989): Thesis, University of Minnesota
[S.] Silling, S.A. (1988): Phase changes induced by deformation in isothermal elastic crystals. Preprint
[T.] Tartar, L. (1979): Compensated compactness and application to partial differential equations. In: R.J. Knops (ed.) Nonlinear analysis and mechanics: Heriot-Watt Symp IV. Pitman, London, pp. 136–212
[V.] Valadier, M.: Young measures. CIME Course, Preprint (1989)