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.

Keywords: linear strains, maximal function, approximate sequences, quasiconvex envelope, quasiconvex hull

@article{COCV_2004__10_2_224_0, author = {Zhang, Kewei}, title = {An approximation theorem for sequences of linear strains and its applications}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {224--242}, publisher = {EDP-Sciences}, volume = {10}, number = {2}, year = {2004}, doi = {10.1051/cocv:2004001}, mrnumber = {2083485}, zbl = {1085.49017}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv:2004001/} }

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 SP - 224 EP - 242 VL - 10 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2004001/ DO - 10.1051/cocv:2004001 LA - en ID - COCV_2004__10_2_224_0 ER -

%0 Journal Article %A Zhang, Kewei %T An approximation theorem for sequences of linear strains and its applications %J ESAIM: Control, Optimisation and Calculus of Variations %D 2004 %P 224-242 %V 10 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv:2004001/ %R 10.1051/cocv:2004001 %G en %F COCV_2004__10_2_224_0

Zhang, Kewei. An approximation theorem for sequences of linear strains and its applications. ESAIM: Control, Optimisation and Calculus of Variations, Volume 10 (2004) no. 2, pp. 224-242. doi : 10.1051/cocv:2004001. http://archive.numdam.org/articles/10.1051/cocv:2004001/

[1] Semicontinuity problems in the calculus of variations. Arch. Ration. Mech. Anal. 86 (1984) 125-145. | MR | Zbl

and ,[2] Sobolev Spaces. Academic Press (1975). | MR | Zbl

,[3] Fine properties of functions with bounded deformation. Arch. Ration. Mech. Anal. 139 (1997) 201-238. | MR | Zbl

, and ,[4] Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63 (1977) 337-403. | MR | Zbl

,[5] 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. | MR | Zbl

,[6] Fine phase mixtures as minimizers of energy. Arch. Ration. Mech. Anal. 100 (1987) 13-52. | MR | Zbl

and ,[7] Proposed experimental tests of a theory of fine microstructures and the two-well problem. Phil. R. Soc. Lond. Sect. A 338 (1992) 389-450. | Zbl

and ,[8] Lower semicontinuity of multiple integrals and the biting lemma. Proc. R. Soc. Edinb. Sect. A 114 (1990) 367-379. | MR | Zbl

and ,[9] Comparison of the geometrically nonlinear and linear theories of martensitic transformation. Continuum Mech. Thermodyn. 5 (1993) 205-242. | MR | Zbl

,[10] Restrictions on Microstructures. Proc. R. Soc. Edinb. Sect. A 124 (1994) 843-878. | MR | Zbl

, , and ,[11] Direct Methods in the Calculus of Variations. Springer (1989). | MR | Zbl

,[12] Luzin-type approximation of $BD$ functions. Proc. R. Soc. Edin. Sect. A 129 (1999) 697-705. | MR | Zbl

,[13] On lower semicontinuity of integral functionals in $LD\left(\Omega \right)$. Preprint Univ. Pisa. | MR | Zbl

,[14] Convex Analysis and Variational Problems. North-Holland (1976). | MR | Zbl

and ,[15] A-quasiconvexity, lower semicontinuity and Young measures. SIAM J. Math. Anal. 30 (1999) 1355-1390. | MR | Zbl

and ,[16] Elliptic Partial Differential Equations of Second Order. Second edn, Academic Press (1983). | MR | Zbl

and ,[17] Variational Methods in Solid Mechanics. Ph.D. thesis, University of Oxford (1999).

,[18] Theory of Structural Transformations in Solids. John Wiley and Sons (1983).

,[19] Characterizations of Young measures generated by gradients. Arch. Ration. Mech. Anal. 115 (1991) 329-365. | MR | Zbl

and ,[20] Boundary-value problems for the system of elasticity theory in unbounded domains. Russian Math. Survey 43 (1988) 65-119. | MR | Zbl

and ,[21] New estimates for deformations in terms of their strains. Ph.D. thesis, Princeton University (1979).

,[22] The relaxation of a double-well energy. Cont. Mech. Therm. 3 (1991) 981-1000. | MR | Zbl

,[23] Lower semicontinuity in spaces of weakly differentiable functions. J. Math. Ann. 313 (1999) 653-710. | MR | Zbl

,[24] Majorations dans ${L}_{\infty}$ des opérateurs différentiels à coefficients constants. C. R. Acad. Sci. Paris 254 (1962) 2286-2288. | MR | Zbl

and ,[25] A Luzin type property of Sobolev functions. Ind. Univ. Math. J. 26 (1977) 645-651. | MR | Zbl

,[26] Multiple integrals in the calculus of variations. Springer (1966). | MR | Zbl

,[27] A sharp version of Zhang's theorem on truncating sequences of gradients. Trans. AMS 351 (1999) 4585-4597. | Zbl

,[28] 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. | MR | Zbl

and ,[29] Convex Analysis. Princeton University Press (1970). | MR | Zbl

,[30] Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970). | MR | Zbl

,[31] Quasiconvex functions with subquadratic growth. Proc. R. Soc. Lond. Sect. A 433 (1991) 723-725. | MR | Zbl

,[32] Rank one convexity does not imply quasiconvexity. Proc. R. Soc. Edinb. Sect. A 120 (1992) 185-189. | MR | Zbl

,[33] On the problem of two wells in Microstructure and Phase Transition. IMA Vol. Math. Appl. 54 (1994) 183-189. | MR | Zbl

,[34] Compensated compactness and applications to partial differential equations, in Nonlinear Analysis and Mechanics: Heriot-Watt Symp., R.J. Knops Ed., IV (1979) 136-212. | MR | Zbl

,[35] Problèmes Mathématiques en Plasticité. Gauthier-Villars (1983). | MR | Zbl

,[36] Embeddings and extensions in analysis. Springer-Verlag (1975). | MR | Zbl

and ,[37] On ${W}^{1,p}$-quasiconvex hulls of set of matrices. Preprint.

,[38] A construction of quasiconvex functions with linear growth at infinity. Ann. Sc. Norm Sup. Pisa. Serie IV XIX (1992) 313-326. | EuDML | Numdam | MR | Zbl

,[39] Quasiconvex functions, $SO\left(n\right)$ and two elastic wells. Anal. Nonlin. H. Poincaré 14 (1997) 759-785. | EuDML | Numdam | MR | Zbl

,[40] On the structure of quasiconvex hulls. Anal. Nonlin. H. Poincaré 15 (1998) 663-686. | EuDML | Numdam | MR | Zbl

,[41] On some quasiconvex functions with linear growth. J. Convex Anal. 5 (1988) 133-146. | EuDML | MR | Zbl

,[42] Rank-one connections at infinity and quasiconvex hulls. J. Convex Anal. 7 (2000) 19-45. | EuDML | MR | Zbl

,[43] On some semiconvex envelopes in the calculus of variations. NoDEA - Nonlinear Diff. Equ. Appl. 9 (2002) 37-44. | Zbl

,[44] On equality of relaxations for linear elastic strains. Commun. Pure Appl. Anal. 1 (2002) 565-573. | MR | Zbl

,*Cited by Sources: *