# An approximation theorem for sequences of linear strains and its applications

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

- 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 (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

top- E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations. Arch. Ration. Mech. Anal.86 (1984) 125-145.
- R.A. Adams, Sobolev Spaces. Academic Press (1975).
- L. Ambrosio, A. Coscia and G. Dal Maso, Fine properties of functions with bounded deformation. Arch. Ration. Mech. Anal.139 (1997) 201-238.
- J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal.63 (1977) 337-403.
- 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.
- J.M. Ball and R.D. James, Fine phase mixtures as minimizers of energy. Arch. Ration. Mech. Anal.100 (1987) 13-52.
- 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.
- J.M. Ball and K. Zhang, Lower semicontinuity of multiple integrals and the biting lemma. Proc. R. Soc. Edinb. Sect. A114 (1990) 367-379.
- K. Bhattacharya, Comparison of the geometrically nonlinear and linear theories of martensitic transformation. Continuum Mech. Thermodyn.5 (1993) 205-242.
- K. Bhattacharya, N.B. Firoozy, R.D. James and R.V. Kohn, Restrictions on Microstructures. Proc. R. Soc. Edinb. Sect. A124 (1994) 843-878.
- B. Dacorogna, Direct Methods in the Calculus of Variations. Springer (1989).
- F.B. Ebobisse, Luzin-type approximation of BD functions. Proc. R. Soc. Edin. Sect. A129 (1999) 697-705.
- 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).
- I. Fonseca and S. Müller, A-quasiconvexity, lower semicontinuity and Young measures. SIAM J. Math. Anal.30 (1999) 1355-1390.
- 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).
- D. Kinderlehrer and P. Pedregal, Characterizations of Young measures generated by gradients. Arch. Ration. Mech. Anal.115 (1991) 329-365.
- 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.
- R.V. Kohn, New estimates for deformations in terms of their strains. Ph.D. thesis, Princeton University (1979).
- R.V. Kohn, The relaxation of a double-well energy. Cont. Mech. Therm.3 (1991) 981-1000.
- J. Kristensen, Lower semicontinuity in spaces of weakly differentiable functions. J. Math. Ann.313 (1999) 653-710.
- K. de Leeuw and H. Mirkil, Majorations dans L∞ des opérateurs différentiels à coefficients constants. C. R. Acad. Sci. Paris254 (1962) 2286-2288.
- F.C. Liu, A Luzin type property of Sobolev functions. Ind. Univ. Math. J.26 (1977) 645-651.
- C.B. Jr Morrey, Multiple integrals in the calculus of variations. Springer (1966).
- S. Müller, A sharp version of Zhang's theorem on truncating sequences of gradients. Trans. AMS351 (1999) 4585-4597.
- 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).
- V. Šverák, Quasiconvex functions with subquadratic growth. Proc. R. Soc. Lond. Sect. A433 (1991) 723-725.
- V. Šverák, Rank one convexity does not imply quasiconvexity. Proc. R. Soc. Edinb. Sect. A120 (1992) 185-189.
- V. Šverák, On the problem of two wells in Microstructure and Phase Transition. IMA Vol. Math. Appl.54 (1994) 183-189.
- 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 W1,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.
- K.-W. Zhang, Quasiconvex functions, SO(n) and two elastic wells. Anal. Nonlin. H. Poincaré14 (1997) 759-785.
- K.-W. Zhang, On the structure of quasiconvex hulls. Anal. Nonlin. H. Poincaré15 (1998) 663-686.
- K.-W. Zhang, On some quasiconvex functions with linear growth. J. Convex Anal.5 (1988) 133-146.
- K.-W. Zhang, Rank-one connections at infinity and quasiconvex hulls. J. Convex Anal.7 (2000) 19-45.
- K.-W. Zhang, On some semiconvex envelopes in the calculus of variations. NoDEA – Nonlinear Diff. Equ. Appl.9 (2002) 37-44.
- K.-W. Zhang, On equality of relaxations for linear elastic strains. Commun. Pure Appl. Anal.1 (2002) 565-573.

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