An approximation theorem for sequences of linear strains and its applications

Kewei Zhang

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 10, Issue: 2, page 224-242
  • ISSN: 1292-8119

Abstract

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We establish an approximation theorem for a sequence of linear elastic strains approaching a compact set in L1 by the sequence of linear strains of mapping bounded in Sobolev space W1,p . We apply this result to establish equalities for semiconvex envelopes for functions defined on linear strains via a construction of quasiconvex functions with linear growth.

How to cite

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Zhang, Kewei. "An approximation theorem for sequences of linear strains and its applications." ESAIM: Control, Optimisation and Calculus of Variations 10.2 (2010): 224-242. <http://eudml.org/doc/90727>.

@article{Zhang2010,
abstract = { We establish an approximation theorem for a sequence of linear elastic strains approaching a compact set in L1 by the sequence of linear strains of mapping bounded in Sobolev space W1,p . We apply this result to establish equalities for semiconvex envelopes for functions defined on linear strains via a construction of quasiconvex functions with linear growth. },
author = {Zhang, Kewei},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Linear strains; maximal function; approximate sequences; quasiconvex envelope; quasiconvex hull.; linear strains; quasiconvex hull},
language = {eng},
month = {3},
number = {2},
pages = {224-242},
publisher = {EDP Sciences},
title = {An approximation theorem for sequences of linear strains and its applications},
url = {http://eudml.org/doc/90727},
volume = {10},
year = {2010},
}

TY - JOUR
AU - Zhang, Kewei
TI - An approximation theorem for sequences of linear strains and its applications
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 10
IS - 2
SP - 224
EP - 242
AB - We establish an approximation theorem for a sequence of linear elastic strains approaching a compact set in L1 by the sequence of linear strains of mapping bounded in Sobolev space W1,p . We apply this result to establish equalities for semiconvex envelopes for functions defined on linear strains via a construction of quasiconvex functions with linear growth.
LA - eng
KW - Linear strains; maximal function; approximate sequences; quasiconvex envelope; quasiconvex hull.; linear strains; quasiconvex hull
UR - http://eudml.org/doc/90727
ER -

References

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  1. E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations. Arch. Ration. Mech. Anal.86 (1984) 125-145.  
  2. R.A. Adams, Sobolev Spaces. Academic Press (1975).  
  3. L. Ambrosio, A. Coscia and G. Dal Maso, Fine properties of functions with bounded deformation. Arch. Ration. Mech. Anal.139 (1997) 201-238.  
  4. J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal.63 (1977) 337-403.  
  5. 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.  
  6. J.M. Ball and R.D. James, Fine phase mixtures as minimizers of energy. Arch. Ration. Mech. Anal.100 (1987) 13-52.  
  7. J.M. Ball and R.D. James, Proposed experimental tests of a theory of fine microstructures and the two-well problem. Phil. R. Soc. Lond. Sect. A338 (1992) 389-450.  
  8. J.M. Ball and K. Zhang, Lower semicontinuity of multiple integrals and the biting lemma. Proc. R. Soc. Edinb. Sect. A114 (1990) 367-379.  
  9. K. Bhattacharya, Comparison of the geometrically nonlinear and linear theories of martensitic transformation. Continuum Mech. Thermodyn.5 (1993) 205-242.  
  10. K. Bhattacharya, N.B. Firoozy, R.D. James and R.V. Kohn, Restrictions on Microstructures. Proc. R. Soc. Edinb. Sect. A124 (1994) 843-878.  
  11. B. Dacorogna, Direct Methods in the Calculus of Variations. Springer (1989).  
  12. F.B. Ebobisse, Luzin-type approximation of BD functions. Proc. R. Soc. Edin. Sect. A129 (1999) 697-705.  
  13. F.B. Ebobisse, On lower semicontinuity of integral functionals in LD(Ω) . Preprint Univ. Pisa.  
  14. I. Ekeland and R. Temam, Convex Analysis and Variational Problems. North-Holland (1976).  
  15. I. Fonseca and S. Müller, A-quasiconvexity, lower semicontinuity and Young measures. SIAM J. Math. Anal.30 (1999) 1355-1390.  
  16. D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order. Second edn, Academic Press (1983).  
  17. Z. Iqbal, Variational Methods in Solid Mechanics. Ph.D. thesis, University of Oxford (1999).  
  18. A.G. Khachaturyan, Theory of Structural Transformations in Solids. John Wiley and Sons (1983).  
  19. D. Kinderlehrer and P. Pedregal, Characterizations of Young measures generated by gradients. Arch. Ration. Mech. Anal.115 (1991) 329-365.  
  20. V.A. Kondratev and O.A. Oleinik, Boundary-value problems for the system of elasticity theory in unbounded domains. Russian Math. Survey43 (1988) 65-119.  
  21. R.V. Kohn, New estimates for deformations in terms of their strains. Ph.D. thesis, Princeton University (1979).  
  22. R.V. Kohn, The relaxation of a double-well energy. Cont. Mech. Therm.3 (1991) 981-1000.  
  23. J. Kristensen, Lower semicontinuity in spaces of weakly differentiable functions. J. Math. Ann.313 (1999) 653-710.  
  24. K. de Leeuw and H. Mirkil, Majorations dans L∞ des opérateurs différentiels à coefficients constants. C. R. Acad. Sci. Paris254 (1962) 2286-2288.  
  25. F.C. Liu, A Luzin type property of Sobolev functions. Ind. Univ. Math. J.26 (1977) 645-651.  
  26. C.B. Jr Morrey, Multiple integrals in the calculus of variations. Springer (1966).  
  27. S. Müller, A sharp version of Zhang's theorem on truncating sequences of gradients. Trans. AMS351 (1999) 4585-4597.  
  28. 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.  
  29. R.T. Rockafellar, Convex Analysis. Princeton University Press (1970).  
  30. E.M. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970).  
  31. V. Šverák, Quasiconvex functions with subquadratic growth. Proc. R. Soc. Lond. Sect. A433 (1991) 723-725.  
  32. V. Šverák, Rank one convexity does not imply quasiconvexity. Proc. R. Soc. Edinb. Sect. A120 (1992) 185-189.  
  33. V. Šverák, On the problem of two wells in Microstructure and Phase Transition. IMA Vol. Math. Appl.54 (1994) 183-189.  
  34. 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.  
  35. R. Temam, Problèmes Mathématiques en Plasticité. Gauthier-Villars (1983).  
  36. J.H. Wells and L.R. Williams, Embeddings and extensions in analysis. Springer-Verlag (1975).  
  37. B.-S. Yan, On W1,p-quasiconvex hulls of set of matrices. Preprint.  
  38. K.-W. Zhang, A construction of quasiconvex functions with linear growth at infinity. Ann. Sc. Norm Sup. Pisa. Serie IV XIX (1992) 313-326.  
  39. K.-W. Zhang, Quasiconvex functions, SO(n) and two elastic wells. Anal. Nonlin. H. Poincaré14 (1997) 759-785.  
  40. K.-W. Zhang, On the structure of quasiconvex hulls. Anal. Nonlin. H. Poincaré15 (1998) 663-686.  
  41. K.-W. Zhang, On some quasiconvex functions with linear growth. J. Convex Anal.5 (1988) 133-146.  
  42. K.-W. Zhang, Rank-one connections at infinity and quasiconvex hulls. J. Convex Anal.7 (2000) 19-45.  
  43. K.-W. Zhang, On some semiconvex envelopes in the calculus of variations. NoDEA – Nonlinear Diff. Equ. Appl.9 (2002) 37-44.  
  44. K.-W. Zhang, On equality of relaxations for linear elastic strains. Commun. Pure Appl. Anal.1 (2002) 565-573.  

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