Skip to main content
SpringerLink
Log in

A New Approach to Counterexamples to L1 Estimates: Korn’s Inequality, Geometric Rigidity, and Regularity for Gradients of Separately Convex Functions

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract.

The derivation of counterexamples to L1 estimates can be reduced to a geometric decomposition procedure along rank-one lines in matrix space. We illustrate this concept in two concrete applications. Firstly, we recover a celebrated, and rather complex, counterexample by Ornstein, proving the failure of Korn’s inequality, and of the corresponding geometrically nonlinear rigidity result, in L1. Secondly, we construct a function f:ℝ2→ℝ which is separately convex but whose gradient is not in BVloc, in the sense that the mixed derivative ∂2f/∂x1∂x2 is not a bounded measure.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Alberti, G., Ambrosio, L.: A geometrical approach to monotone functions in ℝn. Math. Z. 230, 259–316 (1999)

    MATH  Google Scholar 

  2. Astala, K., Faraco, D., Székelyhidi, L.: Convex integration and Lp theory of elliptic equations. MPI-MIS Preprint 70/2004

  3. Aumann, R.J., Hart, S.: Bi-convexity and bi-martingales. Israel J. Math. 54, 159–180 (1986)

    MATH  Google Scholar 

  4. Bourgain, J., Brezis, H.: On the equation divy=f and application to control of phases. J. Am. Math. Soc. 16, 393–426 (2002)

    MATH  Google Scholar 

  5. Burago, D., Kleiner, B.: Separated nets in Euclidean space and Jacobians of bilipschitz maps. GAFA, Geom. funct. anal. 8, 273–282 (1998)

    Google Scholar 

  6. Burkholder, D.L.: Martingales and Fourier analysis in Banach spaces. Lecture Notes in Mathematics, vol. 1206, pp. 61–108, Springer-Verlag, 1986

  7. Conti, S., Faraco, D., Maggi, F., Müller, S.: Rank-one convex functions on 2×2 symmetric matrices and laminates on rank-three lines. MPI-MIS Preprint 50/2004

  8. Conti, S., Ortiz, M.: Dislocation microstructures and the effective behavior of single crystals. MPI-MIS Preprint 5/2004, to appear in Arch. Rational. Mech. Anal.

  9. Conti, S., Schweizer, B.: Rigidity and Gamma convergence for solid-solid phase transitions with SO(2)-invariance. MPI-MIS Preprint 69/2004

  10. Dacorogna, B., Fusco, N., Tartar, L.: On the solvability of the equation div u=f in L1 and in C0. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 14, 239–245 (2003)

  11. Faraco, D.: Milton’s conjecture on the regularity of solutions to isotropic equations. Ann. I. H. Poincaré 20, 889–909 (2003)

    MATH  Google Scholar 

  12. Faraco, D.: Tartar conjecture and Beltrami operators. Michigan Math. Journal 52, 83–104 (2004)

    MATH  Google Scholar 

  13. Friesecke, G., James, R., Müller, S.: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity. Comm. Pure Appl. Math 55, 1461–1506 (2002)

    MATH  Google Scholar 

  14. Gehring, F.W.: Rings and quasiconformal mappings in space. Proc. Nat. Acad. Sci. U.S.A. 47, 98–105 (1961)

    MATH  Google Scholar 

  15. John, F.: Rotation and strain. Comm. Pure Appl. Math. 14, 391–413 (1961)

    MATH  Google Scholar 

  16. Kirchheim, B.: Rigidity and geometry of microstructures. MPI-MIS Lecture notes n. 16, 2002

  17. Kirchheim B., Lorent, A.: In preparation

  18. Kirchheim, B., Müller, S., Šverák, V.: Studying nonlinear pde by geometry in matrix space. Geometric analysis and nonlinear partial differential equations (Hildebrandt, S., Karcher, H. eds.), Springer, 2003, pp. 347–395

  19. Kohn, R.V.: New estimates for deformations in terms of their strains. Ph.D. thesis, Princeton University, 1979

  20. Kohn, R.V.: New integral estimates for deformations in terms of their nonlinear strains. Arch. Rat. Mech. Anal. 78, 131–172 (1982)

    MATH  Google Scholar 

  21. McMullen, C.T.: Lipschitz maps and nets in Euclidean space. GAFA, Geom. funct. anal. 8, 304–314 (1998)

    Google Scholar 

  22. Müller, S.: Rank-one convexity implies quasiconvexity on diagonal matrices. Int. Math. Res. Not. 20, 1087–1095 (1999)

    Google Scholar 

  23. Müller, S.: Variational models for microstructure and phase transitions. In: Calculus of variations and geometric evolution problems (F. Bethuel et al., eds.), Springer Lecture Notes in Math. 1713, Springer, Berlin, 1999, pp. 85–210

  24. Müller, S., Šverák, V.: Convex integration for Lipschitz mappings and counterexamples to regularity. Ann. Math. 157, 715–742 (2003)

    MathSciNet  Google Scholar 

  25. Ornstein, D.A.: Non-inequality for differential operators in the L1-norm. Arch. Rational. Mech. Anal. 11, 40–49 (1962)

    MATH  Google Scholar 

  26. Preiss, D.: Unpublished, the results are quoted in [30]

  27. Reshetnyak, Y.G.: Liouville’s theory on conformal mappings under minimal regularity assumptions. Sibirskii Math. J. 8, 69–85 (1967)

    MATH  Google Scholar 

  28. Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton University Press, 1970

  29. Strauss, M.J.: Variations of Korn’s and Sobolev’s inequalities. In: Partilal Differential Equation, Amer. Math. Soc. 207–214 (1973)

  30. Tartar, L.: Some remarks on separately convex functions. Microstructure and phase transition (D. Kinderlehrer et al., eds.), IMA Vol. Mat. Appl., vol. 54, Springer, 1993, pp. 191–204

  31. Temam, R.: Mathematical problems in plasticity. Bordas, Paris, 1985

  32. Šverák, V.: Unpublished, the results are quoted in [30]

Download references

Author information

Authors and Affiliations

  1. Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22–26, 04103, Leipzig, Germany

    Sergio Conti & Daniel Faraco

  2. Dipartimento di Matematica “U. Dini”, Università di Firenze Viale Morgagni 67/A, 50134, Firenze, Italy

    Francesco Maggi

Authors
  1. Sergio Conti
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daniel Faraco
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Francesco Maggi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sergio Conti.

Additional information

Communicated by V. Šverák

Acknowledgement We thank BERND KIRCHHEIM for bringing the question of regularity of separately convex functions to our attention. This work was partially supported by the EU Research Training Network Hyperbolic and Kinetic Equations, contract HPRN-CT-2002-00282, and by the Deutsche Forschungsgemeinschaft through the Schwerpunktprogramm 1095 Analysis, Modeling and Simulation of Multiscale Problems.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Conti, S., Faraco, D. & Maggi, F. A New Approach to Counterexamples to L1 Estimates: Korn’s Inequality, Geometric Rigidity, and Regularity for Gradients of Separately Convex Functions. Arch. Rational Mech. Anal. 175, 287–300 (2005). https://doi.org/10.1007/s00205-004-0350-5

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-004-0350-5

Keywords

Search

Navigation