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A necessary and sufficient condition for nonattainment and formation of microstructure almost everywhere in scalar variational problems

Published online by Cambridge University Press:  14 November 2011

Gero Friesecke
Gero Friesecke
Affiliation:
Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, Scotland, U.K.
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Abstract

For scalar variational problems

subject to linear boundary values, we determine completely those integrands W: ℝn → ℝ for which the minimum is not attained, thereby completing previous efforts such as a recent nonexistence theorem of Chipot [9] and unifying a large number of examples and counterexamples in the literature.

As a corollary, we show that in case of nonattainment (and provided W grows superlinearly at infinity), every minimising sequence converges weakly but not strongly in W1,1(Ω) to a unique limit, namely the linear deformation prescribed at the boundary, and develops fine structure everywhere in Ω, that is to say every Young measure associated with the sequence of its gradients is almost-nowhere a Dirac mass.

Connections with solid–solid phase transformations are indicated.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 3 , 1994 , pp. 437 - 471
Copyright
Copyright © Royal Society of Edinburgh 1994

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References

1Ball, J. M.. A version of the fundamental theorem for Young measures. In Partial differential equations and continuum models of phase transitions, eds Rascle, M., Serre, D. and Slemrod, M.. Springer Lecture Notes in Physics 344, 207–215 (Berlin: Springer, 1989).Google Scholar
2Ball, J. M., Holmes, P. J., James, R. D., Pego, R. L. and Swart, P. J.. On the dynamics of fine structure. J. Nonlinear Sci. 1 (1991), 17–70.Google Scholar
3Ball, J. M. and James, R. D.. Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal. 100 (1987), 13–52.CrossRefGoogle Scholar
4Ball, J. M. and James, R. D.. Proposed experimental tests of a theory of fine microstructure and the two-well problem. Philos. Trans. Roy. Soc. London Ser. A 338 (1992), 389–450.Google Scholar
5Ball, J. M. and Mizel, V.. One-dimensional variational problems whose minimizers do not satisfy the Euler–Lagrange equation. Arch. Rational Mech. Anal. 90 (1985), 325–388.Google Scholar
6Bauman, P. and Phillips, D.. A nonconvex variational problem related to change of phase. Appl. Math. Optim. 20 (1990), 113–138.Google Scholar
7Cellina, A.. On minima of a functional of the gradient: necessary conditions. Nonlinear Anal. Theory Methods Appl. 20 (1993), 337–341.CrossRefGoogle Scholar
8Cellina, A.. On minima of a functional of the gradient: sufficient conditions. Nonlinear Anal. Theory Methods Appl. 20 (1993) 343–347.CrossRefGoogle Scholar
9Chipot, M.. Numerical analysis of oscillations in nonconvex problems. Numer. Math. 59 (1991), 747–767.Google Scholar
10Chipot, M. and Collins, C.. Numerical approximations in variational problems with potential wells (IMA preprint #696, University of Minnesota, Minneapolis, 1990).Google Scholar
11Chipot, M. and Kinderlehrer, D.. Equilibrium configurations of crystals. Arch. Rational Mech. Anal. 103(1988), 237–277.Google Scholar
12Dacorogna, B.. Direct Methods in the Calculus of Variations (Berlin: Springer, 1988).Google Scholar
13Ekeland, I. and Temam, R.. Analyse Convexe et Problèmes Variationnels (Paris: Dunod, 1974).Google Scholar
14Ericksen, J. L.. Equilibrium of bars. J. Elasticity 5 (1975), 191–201.Google Scholar
15Ferguson, T. S.. Mathematical Statistics. A decision theoretic approach (New York: Academic Press, 1967).Google Scholar
16Fonseca, I.. The lower quasiconvex envelope of the stored energy function for an elastic crystal. J. Math. Pures Appl. 67 (1988), 175–195.Google Scholar
17Khatchaturyan, A.. Some questions concerning the theory of phase transformation in solids. Soviet Phys – Solid State 8 (9) (1967), 2163–2168.Google Scholar
18Khatchaturyan, A.. Theory of Structural Transformations in Solids (New York: J. Wiley & Sons, 1983).Google Scholar
19Khatchaturyan, A. and Shatalov, G.. Theory of macroscopic periodicity for a phase transition in the solid state. Soviet Phys. JETP 29 (3) (1969), 557–561.Google Scholar
20Kohn, R. V. and Müller, S.. Branching of twins near an austenite/twinned martensite interface. Philos. Mag. Ser. A 66 (1992) 697–715.Google Scholar
21Mania, B.. Sopra un esempio di Lavrentieff. Boll. Vn. Mat. 13 (1934), 147–153.Google Scholar
22Marcellini, P.. Alcune osservazioni sull'esistenza del minimo di integrali del calcolo delle variazioni senza ipotesi di convessità. Rend. Mat. 13 (1980), 271–281.Google Scholar
23Marcellini, P.. A relation between existence of minima for nonconvex integrals and uniqueness for non strictly convex integrals of the calculus of variations. In Mathematical Theories of Optimization, eds Cecconi, J. P. and Zolezzi, T.. Springer Lecture Notes in Mathematics 979, 216–231 (Berlin: Springer, 1983).Google Scholar
24Mascolo, E. and Schianchi, R.. Existence theorems for non convex problems. J. Math. Pures Appl. 62 (1983), 349–359.Google Scholar
25Maxwell, J. C.. On the dynamical evidence of the molecular constitution of bodies. Nature XI; reprinted in Niven, W. D. (ed.) The scientific papers of Maxwell, J. C. (New York: Dover Publications, 1890).Google Scholar
26Morrey, C. B.. Multiple integrals in the calculus of variations (Berlin: Springer, 1966).CrossRefGoogle Scholar
27Pedregal, P.. Jensen's inequality in the calculus of variations Diff. and Int. Equations 7 (1994), 57–72.Google Scholar
28Raymond, J. P.. Théorème d'existence pour des problèmes variationnels non convexes. Proc. Roy. Soc. Edinburgh Sect. A 107 (1987), 43–64.Google Scholar
29Roitburd, A.. The domain structure of crystals formed in the solid phase. Soviet Phys. Solid State 10(12) (1969), 2870–2876.Google Scholar
30Roitburd, A.. Martensitic transformation as a typical phase transformation in solids. Solid State Physics 33, 317–390 (New York: Academic Press, 1978).Google Scholar
31Rybka, P.. Dynamical modelling of phase transitions by means of viscoelasticity in many dimensions. Proc. Roy. Soc. Edinburgh Sect. A 121 (1992), 101–138.Google Scholar
32De Simone, A.. Energy minimizers for large ferromagnetic bodies. Arch. Rat. Mech. Anal, 125 (1993), 99–143.CrossRefGoogle Scholar
33Stein, E.. Singular Integrals and Differentiability Properties of Functions (Princeton: Princeton Univeristy Press, 1970).Google Scholar
34Swart, P. J.. The dynamical creation of microstructure in material phase transitions (PhD thesis, Cornell University, 1991).Google Scholar
35Swart, P. J. and Holmes, P. J.. Energy minimization and the formation of microstructure in dynamic anti-plane shear. Arch. Rational Mech. Anal. 121 (1992), 37–85.Google Scholar
36Tartar, L.. Compensated compactness and applications to partial differential equations. In Nonlinear Analysis and Mechanics: Heriot-Watt symposium Vol. IV, ed. J. Knops, R., Pitman Research Notes in Mathematics, 136–192 (London: Pitman, 1979).Google Scholar
37Tartar, L.. The compensated compactness method applied to systems of conservation laws. In NATO Advanced Study Institute Series, Vol. C111, ed. Ball, J. M., 263–285 (Amsterdam: Reidel, 1983).Google Scholar
38Tartar, L.. H-measures, a new approach for studying homogenization, oscillations and concentration effects in partial differential equations. Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), 193–230.Google Scholar