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An approximation theorem for sequences of linear strains and its applications

Published online by Cambridge University Press:  15 March 2004

Kewei Zhang
Kewei Zhang*
Affiliation:
School of Mathematical Sciences, University of Sussex Brighton, BN1 9QH, UK; k.zhang@sussex.ac.uk.
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Abstract

We establish an approximation theorem for a sequence oflinear elastic strains approaching a compact set in L 1 by thesequence of linear strains of mapping bounded in Sobolev space W 1,p . We apply this result to establish equalities forsemiconvex envelopes for functions defined on linear strains via a construction of quasiconvex functions with linear growth.

Keywords

Linear strainsmaximal functionapproximate sequencesquasiconvex envelopequasiconvex hull.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 10 , Issue 2 , April 2004 , pp. 224 - 242
Copyright
© EDP Sciences, SMAI, 2004

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References

Acerbi, E. and Fusco, N., Semicontinuity problems in the calculus of variations. Arch. Ration. Mech. Anal. 86 (1984) 125-145. CrossRef
R.A. Adams, Sobolev Spaces. Academic Press (1975).
Ambrosio, L., Coscia, A. and Dal Maso, G., Fine properties of functions with bounded deformation. Arch. Ration. Mech. Anal. 139 (1997) 201-238. CrossRef
Ball, J.M., Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63 (1977) 337-403. CrossRef
J.M. Ball, A version of the fundamental theorem of Young measures, in Partial Differential Equations and Continuum Models of Phase Transitions, M. Rascle, D. Serre and M. Slemrod Eds., Springer-Verlag (1989) 207-215.
Ball, J.M. and James, R.D., Fine phase mixtures as minimizers of energy. Arch. Ration. Mech. Anal. 100 (1987) 13-52. CrossRef
Ball, J.M. and James, R.D., Proposed experimental tests of a theory of fine microstructures and the two-well problem. Phil. R. Soc. Lond. Sect. A 338 (1992) 389-450. CrossRef
Ball, J.M. and Zhang, K., Lower semicontinuity of multiple integrals and the biting lemma. Proc. R. Soc. Edinb. Sect. A 114 (1990) 367-379. CrossRef
Bhattacharya, K., Comparison of the geometrically nonlinear and linear theories of martensitic transformation. Continuum Mech. Thermodyn. 5 (1993) 205-242. CrossRef
Bhattacharya, K., Firoozy, N.B., James, R.D. and Kohn, R.V., Restrictions on Microstructures. Proc. R. Soc. Edinb. Sect. A 124 (1994) 843-878. CrossRef
B. Dacorogna, Direct Methods in the Calculus of Variations. Springer (1989).
Ebobisse, F.B., Luzin-type approximation of BD functions. Proc. R. Soc. Edin. Sect. A 129 (1999) 697-705. CrossRef
F.B. Ebobisse, On lower semicontinuity of integral functionals in LD(Ω) . Preprint Univ. Pisa.
I. Ekeland and R. Temam, Convex Analysis and Variational Problems. North-Holland (1976).
Fonseca, I. and Müller, S., A-quasiconvexity, lower semicontinuity and Young measures. SIAM J. Math. Anal. 30 (1999) 1355-1390. CrossRef
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order. Second edn, Academic Press (1983).
Z. Iqbal, Variational Methods in Solid Mechanics. Ph.D. thesis, University of Oxford (1999).
A.G. Khachaturyan, Theory of Structural Transformations in Solids. John Wiley and Sons (1983).
Kinderlehrer, D. and Pedregal, P., Characterizations of Young measures generated by gradients. Arch. Ration. Mech. Anal. 115 (1991) 329-365. CrossRef
Kondratev, V.A. and Oleinik, O.A., Boundary-value problems for the system of elasticity theory in unbounded domains. Russian Math. Survey 43 (1988) 65-119. CrossRef
R.V. Kohn, New estimates for deformations in terms of their strains. Ph.D. thesis, Princeton University (1979).
Kohn, R.V., The relaxation of a double-well energy. Cont. Mech. Therm. 3 (1991) 981-1000. CrossRef
Kristensen, J., Lower semicontinuity in spaces of weakly differentiable functions. J. Math. Ann. 313 (1999) 653-710. CrossRef
de Leeuw, K. and Mirkil, H., Majorations dans L ∞ des opérateurs différentiels à coefficients constants. C. R. Acad. Sci. Paris 254 (1962) 2286-2288.
Liu, F.C., Luzin, A type property of Sobolev functions. Ind. Univ. Math. J. 26 (1977) 645-651. CrossRef
C.B. Jr Morrey, Multiple integrals in the calculus of variations. Springer (1966).
Müller, S., A sharp version of Zhang's theorem on truncating sequences of gradients. Trans. AMS 351 (1999) 4585-4597. CrossRef
S. Müller and V. Šverák, Attainment results for the two-well problem by convex integration, in Geometric analysis and the calculus of variations, Internat. Press, Cambridge, MA (1996) 239-251.
R.T. Rockafellar, Convex Analysis. Princeton University Press (1970).
E.M. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970).
Šverák, V., Quasiconvex functions with subquadratic growth. Proc. R. Soc. Lond. Sect. A 433 (1991) 723-725. CrossRef
Šverák, V., Rank one convexity does not imply quasiconvexity. Proc. R. Soc. Edinb. Sect. A 120 (1992) 185-189. CrossRef
Šverák, V., On the problem of two wells in Microstructure and Phase Transition. IMA Vol. Math. Appl. 54 (1994) 183-189. CrossRef
L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear Analysis and Mechanics: Heriot-Watt Symp., R.J. Knops Ed., IV (1979) 136-212.
R. Temam, Problèmes Mathématiques en Plasticité. Gauthier-Villars (1983).
J.H. Wells and L.R. Williams, Embeddings and extensions in analysis. Springer-Verlag (1975).
B.-S. Yan, On W 1,p -quasiconvex hulls of set of matrices. Preprint.
K.-W. Zhang, A construction of quasiconvex functions with linear growth at infinity. Ann. Sc. Norm Sup. Pisa. Serie IV XIX (1992) 313-326.
Zhang, K.-W., Quasiconvex functions, SO(n) and two elastic wells. Anal. Nonlin. H. Poincaré 14 (1997) 759-785. CrossRef
Zhang, K.-W., On the structure of quasiconvex hulls. Anal. Nonlin. H. Poincaré 15 (1998) 663-686. CrossRef
Zhang, K.-W., On some quasiconvex functions with linear growth. J. Convex Anal. 5 (1988) 133-146.
Zhang, K.-W., Rank-one connections at infinity and quasiconvex hulls. J. Convex Anal. 7 (2000) 19-45.
Zhang, K.-W., On some semiconvex envelopes in the calculus of variations. NoDEA – Nonlinear Diff. Equ. Appl. 9 (2002) 37-44. CrossRef
Zhang, K.-W., On equality of relaxations for linear elastic strains. Commun. Pure Appl. Anal. 1 (2002) 565-573. CrossRef