Convex integration for Lipschitz mappings and counterexamples to regularity

Annals of Mathematics

Volumn 157, Issue 3, 2003, Pages 715-742

  Muller S ^a   Sverak V ^b  

^a MAX PLANCK INSTITUTE FOR MATHEMATICS IN THE SCIENCES   (Germany)

^b UNIVERSITY OF MINNESOTA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0038159405     PISSN: 0003486X     EISSN: None     Source Type: Journal    
DOI: 10.4007/annals.2003.157.715     Document Type: Article

References (29)

1. [AF 87] E. ACERBI and N. FUSCO, A regularity theorem for minimizers of quasiconvex integrals, Arch. Rational Mech. Anal. 99 (1987), 261-281.
2. [AH 86] R. AUMANN and S. HART, Bi-convexity and bi-martingales, Israel J. Math. 54 (1986), 159-180.
3. [Ba 80] J. M. BALL, Strict convexity, strong ellipticity and regularity in the calculus of variations, Math. Proc. Cambridge Philos. Soc. 87 (1980), 501-513.
4. [Ba 90] _, Sets of gradients with no rank-one connections, J. Math. Pures Appl. (9) 69 (1990), 241-259.
5. [BJ 87] J. M. BALL and R. D. JAMES, Fine phase mixtures as minimizers of energy, Arch. Rational Mech. Anal. 100 (1987), 13-52.
6. [BFJK 94] K. BHATTACHARYA, N. FIROOZYE, R. D. JAMES, and R. V. KOHN, Restrictions on microstructure, Proc. Roy. Soc. Edinburgh A 124 (1994), 843-878.
7. [CT 93] E. CASADIO-TARABUSI, An algebraic characterization of quasi-convex functions, Ricerche Mat. 42 (1993), 11-24.
8. [CK 88] M. CHIPOT and D. KINDERLEHRER, Equilibrium configurations of crystals, Arch. Rational Mech. Anal. 103 (1988), 237-277.
9. [DM 97] B. DACOROGNA and P. MARCELLINI, General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial cases, Acta Math. 178 (1997), 1-37.
10. [DG 68] E. DE GIORGI, Un esempio di estremali discontinue per un problema variazionale di tipo ellittico, Boll. Un. Mat. Ital. 4 (1968), 135-137.
11. [Ev 86] L. C. EVANS, Quasiconvexity and partial regularity in the calculus of variations, Arch. Rational Mech. Anal. 95 (1986), 227-252.
12. [Fe 69] H. FEDERER, Geometric Measure Theory, Springer-Verlag, New York, 1969.
13. [GS 85] M. GIAQUINTA and J. SOUǑEK, Caccioppoli's inequality and Legendre-Hadamard condition, Math. Ann. 270 (1985), 105-107.
14. [GM 68] E. Giusti and M. Miranda, Un esempio di soluzioni discontinue per un problema di minimo relativo ad un integrale regolare del calcolo delle variazioni, Boll. Un. Mat. Ital. 2 (1968), 219-226.
15. [Gr 86] M. Gromov, Partial Differential Relations, Springer-Verlag, New York, 1986.
16. [HLN 96] W. HAO, S. LEONARDI, and J. NEČAS, An example of irregular solution to a non-linear Euler-Lagrange elliptic system with real analytic coefficients. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 23 (1996), 57-67.
17. 1-isometric embeddings. I, Nederl. Akad. Wetensch. Proc. A 58 (1955), 545-556.
18. [Ma 85] P. MARCELLINI, Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals, Manuscripta Math. 51 (1985), 1-28.
19. [MP 98] J. MATOUŠEK and P. PLECHÁČ, On functional separately convex hulls, Discrete Comput. Geom. 19 (1998), 105-130.
20. [Mo 66] C. B. MORREY, Multiple Integrals in the Calculus of Variations, Springer-Verlag, New York, 1966.
21. [MS 96] S. MÜLLER and V. ŠVERÁK, Attainment results for the two-well problem by convex integration, in Geometric Analysis and the Calculus of Variations (J. Jost, ed.), Internat. Press, Cambridge, MA, 1996, 239-251.
22. 1 isometric imbeddings, Ann. of Math. 60 (1954), 383-396.
23. [NM 91] V. NESI and G. W. MILTON, Polycrystalline configurations that maximize electrical resistivity, J. Mech. Phys. Solids 39 (1991), 525-542.
24. [Pe 93] P. PEDREGAL, Laminates and microstructure, European J. Appl. Math. 4 (1993), 121-149.
25. [Sch 74] V. SCHEFFER, Regularity and irregularity of solutions to nonlinear second order elliptic systems of partial differential equations and inequalities, Dissertation, Princeton University, 1974 (unpublished).
26. [Sv 90] V. ŠVERÁK, Examples of rank-one convex functions, Proc. Roy. Soc. Edinburgh A 114 (1990), 237-242.
27. [Sv 92a] _, Rank-one convexity does not imply quasiconvexity, Proc. Roy. Soc. Edinburgh A 120 (1992), 185-189.
28. [Sv 92b] _, New examples of quasiconvex functions, Arch. Rational Mech. Anal. 119 (1992), 293-300.
29. [Ta 93] L. TARTAR, Some remarks on separately convex functions, in: Microstructure and Phase Transitions, IMA Vol. Math. Appl. 54 (D. Kinderlehrer, R. D. James, M. Luskin and J. L. Ericksen, eds.), Springer-Verlag, New York (1993), 191-204.