An approximation theorem for sequences of linear strains and its applications
ESAIM: Control, Optimisation and Calculus of Variations (2004)
- Volume: 10, Issue: 2, page 224-242
- ISSN: 1292-8119
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topZhang, Kewei. "An approximation theorem for sequences of linear strains and its applications." ESAIM: Control, Optimisation and Calculus of Variations 10.2 (2004): 224-242. <http://eudml.org/doc/245938>.
@article{Zhang2004,
abstract = {We establish an approximation theorem for a sequence of linear elastic strains approaching a compact set in $L^1$ by the sequence of linear strains of mapping bounded in Sobolev space $W^\{1,p\}$. 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},
language = {eng},
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/245938},
volume = {10},
year = {2004},
}
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
PY - 2004
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 $L^1$ by the sequence of linear strains of mapping bounded in Sobolev space $W^{1,p}$. 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
UR - http://eudml.org/doc/245938
ER -
References
top- [1] E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations. Arch. Ration. Mech. Anal. 86 (1984) 125-145. Zbl0565.49010MR751305
- [2] R.A. Adams, Sobolev Spaces. Academic Press (1975). Zbl0314.46030MR450957
- [3] L. Ambrosio, A. Coscia and G. Dal Maso, Fine properties of functions with bounded deformation. Arch. Ration. Mech. Anal. 139 (1997) 201-238. Zbl0890.49019MR1480240
- [4] J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63 (1977) 337-403. Zbl0368.73040MR475169
- [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. Zbl0991.49500MR1036070
- [6] J.M. Ball and R.D. James, Fine phase mixtures as minimizers of energy. Arch. Ration. Mech. Anal. 100 (1987) 13-52. Zbl0629.49020MR906132
- [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. A 338 (1992) 389-450. Zbl0758.73009
- [8] J.M. Ball and K. Zhang, Lower semicontinuity of multiple integrals and the biting lemma. Proc. R. Soc. Edinb. Sect. A 114 (1990) 367-379. Zbl0716.49011MR1055554
- [9] K. Bhattacharya, Comparison of the geometrically nonlinear and linear theories of martensitic transformation. Continuum Mech. Thermodyn. 5 (1993) 205-242. Zbl0780.73005MR1236099
- [10] K. Bhattacharya, N.B. Firoozy, R.D. James and R.V. Kohn, Restrictions on Microstructures. Proc. R. Soc. Edinb. Sect. A 124 (1994) 843-878. Zbl0808.73063MR1303758
- [11] B. Dacorogna, Direct Methods in the Calculus of Variations. Springer (1989). Zbl0703.49001MR990890
- [12] F.B. Ebobisse, Luzin-type approximation of functions. Proc. R. Soc. Edin. Sect. A 129 (1999) 697-705. Zbl0939.49012MR1718471
- [13] F.B. Ebobisse, On lower semicontinuity of integral functionals in . Preprint Univ. Pisa. Zbl1027.49014MR1795030
- [14] I. Ekeland and R. Temam, Convex Analysis and Variational Problems. North-Holland (1976). Zbl0322.90046MR463994
- [15] I. Fonseca and S. Müller, A-quasiconvexity, lower semicontinuity and Young measures. SIAM J. Math. Anal. 30 (1999) 1355-1390. Zbl0940.49014MR1718306
- [16] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order. Second edn, Academic Press (1983). Zbl0562.35001MR737190
- [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. Zbl0754.49020MR1120852
- [20] V.A. Kondratev and O.A. Oleinik, Boundary-value problems for the system of elasticity theory in unbounded domains. Russian Math. Survey 43 (1988) 65-119. Zbl0669.73005MR971465
- [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. Zbl0825.73029MR1122017
- [23] J. Kristensen, Lower semicontinuity in spaces of weakly differentiable functions. J. Math. Ann. 313 (1999) 653-710. Zbl0924.49012MR1686943
- [24] K. de Leeuw and H. Mirkil, Majorations dans des opérateurs différentiels à coefficients constants. C. R. Acad. Sci. Paris 254 (1962) 2286-2288. Zbl0112.33002MR139964
- [25] F.C. Liu, A Luzin type property of Sobolev functions. Ind. Univ. Math. J. 26 (1977) 645-651. Zbl0368.46036MR450488
- [26] C.B. Jr Morrey, Multiple integrals in the calculus of variations. Springer (1966). Zbl0142.38701MR202511
- [27] S. Müller, A sharp version of Zhang’s theorem on truncating sequences of gradients. Trans. AMS 351 (1999) 4585-4597. Zbl0942.49013
- [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. Zbl0930.35038MR1449410
- [29] R.T. Rockafellar, Convex Analysis. Princeton University Press (1970). Zbl0193.18401MR274683
- [30] E.M. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970). Zbl0207.13501MR290095
- [31] V. Šverák, Quasiconvex functions with subquadratic growth. Proc. R. Soc. Lond. Sect. A 433 (1991) 723-725. Zbl0741.49016MR1116970
- [32] V. Šverák, Rank one convexity does not imply quasiconvexity. Proc. R. Soc. Edinb. Sect. A 120 (1992) 185-189. Zbl0777.49015MR1149994
- [33] V. Šverák, On the problem of two wells in Microstructure and Phase Transition. IMA Vol. Math. Appl. 54 (1994) 183-189. Zbl0797.73079MR1320537
- [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. Zbl0437.35004MR584398
- [35] R. Temam, Problèmes Mathématiques en Plasticité. Gauthier-Villars (1983). Zbl0547.73026MR711964
- [36] J.H. Wells and L.R. Williams, Embeddings and extensions in analysis. Springer-Verlag (1975). Zbl0324.46034MR461107
- [37] B.-S. Yan, On -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. Zbl0778.49015MR1205403
- [39] K.-W. Zhang, Quasiconvex functions, and two elastic wells. Anal. Nonlin. H. Poincaré 14 (1997) 759-785. Zbl0918.49014MR1482901
- [40] K.-W. Zhang, On the structure of quasiconvex hulls. Anal. Nonlin. H. Poincaré 15 (1998) 663-686. Zbl0917.49014MR1650974
- [41] K.-W. Zhang, On some quasiconvex functions with linear growth. J. Convex Anal. 5 (1988) 133-146. Zbl0915.49008MR1649465
- [42] K.-W. Zhang, Rank-one connections at infinity and quasiconvex hulls. J. Convex Anal. 7 (2000) 19-45. Zbl0976.49009MR1773175
- [43] K.-W. Zhang, On some semiconvex envelopes in the calculus of variations. NoDEA – Nonlinear Diff. Equ. Appl. 9 (2002) 37-44. Zbl1012.49012
- [44] K.-W. Zhang, On equality of relaxations for linear elastic strains. Commun. Pure Appl. Anal. 1 (2002) 565-573. Zbl1032.49019MR1942285
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