Abstract

We study the relationship between rank-one convexity and quasiconvexity in the space of 2×2 matrices. We show that a certain procedure for constructing homogeneous gradient Young measures from periodic deformations, that arises from V. Sverák's celebrated counterexample in higher dimensions, always yields laminates in the 2×2 case.

Author information

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

Gabriella Sebestyén

Dept. of Analysis, Faculty of Science, Eötvös Loránd University, 1117 Budapest, Hungary

Keywords

  • Gradients
  • quasiconvexity
  • rank-one convexity
  • laminates

Mathematics Subject Classification

49J10, 49K10

References

  1. I. Al Baernstein, S. Montgomery-Smith: Some conjectures about integral means of ∂f and ∂f ¯, Acta Univ. Upsaliensis C 64 (1999) 92–109.Google Scholar
  2. K. Astala, T. Iwaniec, I. Prause, E. Saksman: Burkholder integrals, Morrey’s problem and quasiconformal mappings, J. Amer. Math. Soc. 25(2) (2012) 507–531.https://doi.org/10.1090/S0894-0347-2011-00718-2Google Scholar
  3. L. Bandeira, A. Ornelas: On the characterization of a class of laminates for 2 × 2 symmetric gradients, J. Convex Analysis 18(1) (2011) 37–58.https://doi.org/10.68381/jca18002Google Scholar
  4. D. Faraco, L. Székelyhidi Jr.: Tartar’s conjecture and localization of the quasiconvex hull in R2×2, Acta Math. 200(2) (2008) 279–305.https://doi.org/10.1007/s11511-008-0028-1Google Scholar
  5. D. Kinderlehrer, P. Pedregal: Characterizations of Young measures generated by gradients, Arch. Rational Mech. Anal. 115(4) (1991) 329–365.https://doi.org/10.1007/BF00375279Google Scholar
  6. B. Kirchheim: Rigidity and Geometry of Microstructures, Habilitation Thesis, Univ. Leipzig (2003).Google Scholar
  7. J. Matoušek, P. Plechac: On functional separately convex hulls, Discrete Comput. Geom. 19(1) (1998) 105–130.https://doi.org/10.1007/PL00009331Google Scholar
  8. C. Morrey: Quasi-convexity and the lower semicontinuity of multiple integrals, Pacific J. Math. 2(1) (1952) 25–53.https://doi.org/10.2140/pjm.1952.2.25Google Scholar
  9. S. Müller: Variational models for microstructure and phase transitions, in: Calculus of Variations and Geometric Evolution Problems, Lectures given at the 2nd Session of the Centre Internazionale Matematico Estivo, Cetaro (1996).Google Scholar
  10. P. Pedregal: Laminates and microstructure, European J. Applied Mathematics 4(2) (1993) 121–149.https://doi.org/10.1017/S0956792500001030Google Scholar
  11. P. Pedregal: Some remarks on quasiconvexity and rank-one convexity, Proc. Royal Society of Edinburgh, Section A 126(5) (1996) 1055–1065.https://doi.org/10.1017/S0308210500023258Google Scholar
  12. P. Pedregal: Some evidence in favor of Morrey’s conjecture, arXiv (June 2014).Google Scholar
  13. P. Pedregal, V. Sverak: The quasiconvexification of a cubic polynomial at the origin, Nonlinear Analysis 30(8) (1997) 5093–5100.https://doi.org/10.1016/S0362-546X(97)00055-2Google Scholar
  14. P. Pedregal, V. Sverak: A note on quasiconvexity and rank-one convexity for 2 × 2 matrices, J. Convex Analysis 5(1) (1998) 107–117.https://doi.org/10.68381/jca05-8Google Scholar
  15. V. Sverak: New examples of quasiconvex functions, Arch. Rational Mech. Anal. 119(4) (1992) 293–300.https://doi.org/10.1007/BF01837111Google Scholar
  16. V. Sverak: Rank-one convexity does not imply quasiconvexity, Proc. Royal Society of Edinburgh, Section A 120(1-2) (1992) 185–189.https://doi.org/10.1017/S0308210500015080Google Scholar

Suggested citation

G. Sebestyén, L. Székelyhidi Jr.. “Laminates Supported on Cubes.” Journal of Convex Analysis 24 (2017), No. 4, 1217–1237. https://doi.org/10.68381/jca24070

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