Abstract

We establish (i) that the quasiconvexification of the distance function to any closed (possibly unbounded) subset of the space of conformal matrices E∂E_\partial in M²×² is bounded from below by the distance function itself, that is, Q dist(·, K) ≥ c dist(·, K), where c > 0 is a constant independent of K; (ii) some estimates of quasiconvexifications of the distance function to a closed subset of M²×² which is ‘supported’ by E∂E_\partial; (iii) Q distᵖ(·, K) = Q distᵖ(·, Qₚ(K)) for any p ≥ 1 and any closed K ⊂ Mᴺ×ⁿ; (iv) for some nonconvex K ⊂ M²×², Q dist(·, K) is homogeneous of degree one, conjugate invariant and convex, and Q₁(K) = C(K).

Author information

Contact details are reproduced from the original publication and may be historical.

Kewei Zhang

Department of Mathematics, Macquarie University, North Ryde, Sydney 2109, Australia

References

  1. J. M. Ball: Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1977) 337-403.https://doi.org/10.1007/BF00279992Google Scholar
  2. J. M. Ball, R. D. James: Proposed experimental tests of a theory of fine microstructures and the two-well problem, Phil. Royal Soc. Lon. 338A (1992) 389-450.https://doi.org/10.1098/rsta.1992.0013Google Scholar
  3. M. Chipot, D. Kinderlehrer: Equilibrium configurations of crystals, Arch. Rational Mech. Anal. 103 (1988) 237-277.https://doi.org/10.1007/BF00251759Google Scholar
  4. B. Dacorogna: Direct Methods in the Calculus of Variations, Springer-Verlag, Berlin et al., 1989.https://doi.org/10.1007/978-3-642-51440-1Google Scholar
  5. B. Dacorogna: On rank one convex functions which are homogeneous of degree one (1994), preprint.Google Scholar
  6. I. Ekeland, R. Temam: Convex Analysis and Variational Problems, North-Holland, 1976.Google Scholar
  7. I. Fonseca: The lower quasiconvex envelope of the stored energy function for an elastic crystal, J. Math. Pures et Appl. 67 (1988) 175-195.Google Scholar
  8. Z. Iqbal, K.-W. Zhang: Quasiconvex functions, subspaces without rank-one connections and linear elliptic systems, in preparation.Google Scholar
  9. C. B. Jr Morrey: Multiple Integrals in the Calculus of Variations, Springer-Verlag, Berlin et al., 1966.https://doi.org/10.1007/978-3-540-69952-1Google Scholar
  10. S. Müller: On quasiconvex functions which are homogeneous of degree one, Indiana Math. J. 41 (1992) 295-300.https://doi.org/10.1512/iumj.1992.41.41017Google Scholar
  11. E. M. Stein: Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970.https://doi.org/10.1515/9781400883882Google Scholar
  12. V. Šverák: Quasiconvex functions with subquadratic growth, Proc. Royal Soc. Lond. 433A (1991) 733-752.https://doi.org/10.1098/rspa.1991.0073Google Scholar
  13. V. Šverák: Rank one convexity does not imply quasiconvexity, Proc. Royal Soc. Edin. 120A (1992) 185-189.https://doi.org/10.1017/S0308210500015080Google Scholar
  14. B.-S. Yan: Remarks on the set of quasi-conformal matrices in higher dimensions (1994), preprint.Google Scholar
  15. K.-W. Zhang: On non-negative quasiconvex functions with unbounded zero sets, Proc. Royal Soc. Edin. 127A (1997) 411-422.https://doi.org/10.1017/S0308210500023726Google Scholar
  16. K.-W. Zhang: Quasiconvex functions, SO(n) and two elastic wells, Anal. Nonlin. H. Poincaré Inst. 14 (1997) 759-785.https://doi.org/10.1016/s0294-1449(97)80132-1Google Scholar

Suggested citation

K. Zhang. “On Some Quasiconvex Functions with Linear Growth.” Journal of Convex Analysis 5 (1998), No. 1, 133–146. https://doi.org/10.68381/jca05-10

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 1998. Rights now held by Banach Press LLC.