Abstract

We design an algorithm for computations of quasiconvex hulls of isotropic compact sets in in the space of 2×2 real matrices. Our approach uses a recent result by the first author [Adv. Calc. Var. 8 (2015) 43–53] on quasiconvex hulls of isotropic compact sets in the space of 2×2 real matrices. We show that our algorithm has the time complexity of O(Nlog⁡N)\mathcal{O}(N\log N) where N is the number of orbits of the set. Finally, we outline some applications of our results to relaxation of L∞ variational problems

Author information

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

Sebastian Heinz

Weierstraß-Institut für Analysis und Stochastik, Mohrenstr. 39, 10117 Berlin, Germany

heinz@wias-berlin.de

Martin Kružík

Institute of Information Theory and Automation, Academy of Sciences, Pod vodárenskou věží 4, 182 08 Praha 8, Czech Republic

kruzik@utia.cas.cz

References

  1. E. Acerbi, G. Buttazzo, F. Prinari: The class of functionals which can be represented by a supremum, J. Convex Analysis 9 (2002) 225–236.https://doi.org/10.68381/jca09010Google Scholar
  2. G. Aronsson, E. N. Barron: L∞ variational problems with running costs and constraints, Appl. Math. Optim. 65 (2012) 53–90.https://doi.org/10.1007/s00245-011-9151-zGoogle Scholar
  3. J. M. Ball: Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal. 63 (1977) 337–403.https://doi.org/10.1007/BF00279992Google Scholar
  4. J. M. Ball, R. D. James: Fine phase mixtures as minimizers of energy, Arch. Ration. Mech. Anal. 100 (1987) 15–52.https://doi.org/10.1007/BF00281246Google Scholar
  5. E. N. Barron, R. R. Jensen, C. Y. Wang: Lower semicontinuity of L∞ functionals, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 18 (2001) 495–517.https://doi.org/10.1016/s0294-1449(01)00070-1Google Scholar
  6. S. Bartels: Linear convergence in the approximation of rank-one convex envelopes, M2AN, Math. Model. Numer. Anal. 38 (2004) 811–820.https://doi.org/10.1051/m2an:2004040Google Scholar
  7. S. Bartels: Reliable and efficient approximation of polyconvex envelopes, SIAM J. Numer. Anal. 43 (2005) 363–385.https://doi.org/10.1137/S0036142903428840Google Scholar
  8. S. Bartels, M. Kružík: An efficient approach to the numerical solution of rate-independent problems with nonconvex energies, Multiscale Model. Simul. 9 (2011) 1276–1300.https://doi.org/10.1137/110821718Google Scholar
  9. B. Benešová: Global optimization numerical strategies for rate-independent processes, J. Glob. Optim. 50 (2011) 197–220.https://doi.org/10.1007/s10898-010-9560-6Google Scholar
  10. P. Cardaliaguet, R. Tahraoui: Equivalence between rank-one convexity and polyconvexity for isotropic sets of R2×2, Part I, Nonlinear. Anal. 50 (2002) 1179–1199.https://doi.org/10.1016/S0362-546X(01)00807-0Google Scholar
  11. P. Cardaliaguet, R. Tahraoui: Equivalence between rank-one convexity and polyconvexity for isotropic sets of R2×2, Part II, Nonlinear. Anal. 50 (2002) 1201–1239.https://doi.org/10.1016/S0362-546X(01)00808-2Google Scholar
  12. S. Conti, C. De Lellis, S. Müller, M. Romeo: Polyconvexity equals rank-one convexity for connected isotropic sets in M2×2, C. R., Math., Acad. Sci. Paris 337 (2003) 233–238.https://doi.org/10.1016/S1631-073X(03)00333-9Google Scholar
  13. B. Dacorogna: Direct Methods in the Calculus of Variations, 2nd Ed., Springer, Berlin (2008).https://doi.org/10.1142/p616Google Scholar
  14. G. Dolzmann: Numerical computation of rank-one convex envelopes, SIAM J. Numer. Anal. 36 (1999) 1621–1635.https://doi.org/10.1137/S0036142997325581Google Scholar
  15. G. Dolzmann, N. Walkington: Estimates for numerical approximations of rank one convex envelopes, Numer. Math. 85 (2000) 647–663.https://doi.org/10.1007/PL00005395Google Scholar
  16. R. D. Grigorieff: A note on von Neumann’s trace inequality, Math. Nachr. 151 (1991) 327–328.https://doi.org/10.1002/mana.19911510122Google Scholar
  17. S. Heinz: Quasiconvexity equals lamination convexity for isotropic sets of 2 × 2 matrices, Adv. Calc. Var. 8 (2015) 43–53.https://doi.org/10.1515/acv-2012-0008Google Scholar
  18. R. Kohn, G. Strang: Explicit relaxation of a variational problem in optimal design, Bull. Amer. Math. Soc. 9 (1983) 211–214.https://doi.org/10.1090/S0273-0979-1983-15158-3Google Scholar
  19. R. Kohn, G. Strang: Optimal design and relaxation of variational problems I, II, III, Commun. Pure Appl. Math. 39 (1986) 113–137, 139–182, 353–377.https://doi.org/10.1002/cpa.3160390305Google Scholar
  20. J. Kolář: Non-compact lamination convex hulls, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 20 (2003) 391–403.https://doi.org/10.1016/s0294-1449(02)00007-0Google Scholar
  21. M. Kružík, M. Luskin: The computation of martensitic microstructure with piecewise laminates, J. Sci. Comput. 19 (2003) 293–308.https://doi.org/10.1023/A:1025360126654Google Scholar
  22. J. Matoušek, P. Plecháč: On functional separately convex hulls, Discrete Comput. Geom. 19 (1998) 105–130.https://doi.org/10.1007/PL00009331Google Scholar
  23. A. Melkman: On-line construction of the convex hull of a simple polyline, Inf. Process. Lett. 25 (1987) 11–12.https://doi.org/10.1016/0020-0190(87)90086-XGoogle Scholar
  24. F. Prinari: Personal communication (March 2014).Google Scholar
  25. M. Šilhavý: Rank 1 convex hulls of isotropic functions in dimension 2 by 2, Math. Bohem. 126 (2001) 521–529.https://doi.org/10.21136/MB.2001.134029Google Scholar
  26. M. Šilhavý: Rotationally invariant rank 1 convex functions, Appl. Math. Optim. 44 (2001) 1–15.https://doi.org/10.1007/s00245-001-0012-zGoogle Scholar
  27. V. Šverák: Rank-one convexity does not imply quasiconvexity, Proc. R. Soc. Edinb., Sect. A 120 (1992) 185–189.https://doi.org/10.1017/S0308210500015080Google Scholar

Suggested citation

S. Heinz, M. Kružík. “Computations of Quasiconvex Hulls of Isotropic Sets.” Journal of Convex Analysis 24 (2017), No. 2, 477–492. https://doi.org/10.68381/jca24030

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2017. Rights now held by Banach Press LLC.