Tartar's conjecture and localization of the quasiconvex hull in ℝ2x2

Acta Mathematica

Volumn 200, Issue 2, 2008, Pages 279-305

  Faraco, Daniel ^a   Székelyhidi Jr , László ^b  

^a UNIVERSIDAD AUTÓNOMA DE MADRID   (Spain)

^b ETH ZURICH   (Switzerland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 45449086650     PISSN: 00015962     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11511-008-0028-1     Document Type: Article

References (60)

1. Ahlfors, L. V., Lectures on Quasiconformal Mappings. The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole, Monterey, CA, 1987.
2. Astala, K., Analytic aspects of quasiconformality, in Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), pp. 617-626.
3. Astala, K. & Faraco, D., Quasiregular mappings and Young measures. Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 1045-1056.
4. Astala, K., Iwaniec, T. & Martin, G., Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane. Book in preparation.
5. Astala, K., Iwaniec, T. & Saksman, E., Beltrami operators in the plane. Duke Math. J., 107 (2001), 27-56.
6. Aumann, R. J. & Hart, S., Bi-convexity and bi-martingales. Israel J. Math., 54 (1986), 159-180.
7. Baernstein, A. II & Montgomery-Smith, S. J., Some conjectures about integral means of ∂f and ∂̄f, in Complex Analysis and Differential Equations (Uppsala, 1997), Acta Univ. Upsaliensis Skr. Uppsala Univ. C Organ. Hist., 64, pp. 92-109. Uppsala Univ., Uppsala, 1999.
8. Ball, J. M., Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal., 63 (1976/77), 337-403.
9. Ball, J. M. & James, R. D., Fine phase mixtures as minimizers of energy. Arch. Ration. Mech. Anal., 100 (1987), 13-52.
10. - Proposed experimental tests of a theory of fine microstructure and the two-well problem. Phil. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 338 (1992), 389-450.
11. - Incompatible sets of gradients and metastability. In preparation.
12. Bhattacharya, K., Firoozye, N. B., James, R. D. & Kohn, R. V., Restrictions on microstructure. Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 843-878.
13. Bojarski, B., Generalized solutions of a system of differential equations of first order and of elliptic type with discontinuous coefficients. Mat. Sb., 43(85) (1957), 451-503 (Russian).
14. Bojarski, B., D'Onofrio, L., Iwaniec, T. & Sbordone, C., G-closed classes of elliptic operators in the complex plane. Ricerche Mat., 54 (2005), 403-432 (2006).
15. Bojarski, B. & Iwaniec, T., Quasiconformal mappings and non-linear elliptic equations in two variables. I; II. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 22 (1974), 473-478; 479-484.
16. Casadio Tarabusi, E., An algebraic characterization of quasi-convex functions. Ricerche Mat., 42 (1993), 11-24.
17. Chipot, M. & Kinderlehrer, D., Equilibrium configurations of crystals. Arch. Ration. Mech. Anal., 103 (1988), 237-277.
18. Dacorogna, B., Direct Methods in the Calculus of Variations. Applied Mathematical Sciences, 78. Springer, Berlin-Heidelberg, 1989.
19. DiPerna, R. J., Compensated compactness and general systems of conservation laws. Trans. Amer. Math. Soc., 292 (1985), 383-420.
20. Ekeland, I. & Temam, R., Convex Analysis and Variational Problems. Studies in Mathematics and its Applications, 1. North-Holland, Amsterdam, 1976.
21. Faraco, D., Tartar conjecture and Beltrami operators. Michigan Math. J., 52 (2004), 83-104.
22. Gromov, M., Partial Differential Relations. Ergebnisse der Mathematik und ihrer Grenzgebiete, 9. Springer, Berlin-Heidelberg, 1986.
23. Iwaniec, T., Quasiconformal mapping problem for general nonlinear systems of partial differential equations, in Symposia Mathematica, Vol. XVIII (Convegno sulle Transformazioni Quasiconformi e Questioni Connesse, INdAM, Rome, 1974), pp. 501-517. Academic Press, London, 1976.
24. n. Trans. Amer. Math. Soc., 354:5 (2002), 1961-1995.
25. Iwaniec, T. & Martin, G., Geometric Function Theory and Non-Linear Analysis. Oxford Mathematical Monographs. Oxford University Press, New York, 2001.
26. Kinderlehrer, D. & Pedregal, P., Characterizations of Young measures generated by gradients. Arch. Ration. Mech. Anal., 115 (1991), 329-365.
27. Kirchheim, B., Rigidity and Geometry of Microstructures. Habilitation Thesis, University of Leipzig, 2003.
28. Kirchheim, B., Müller, S. & Šverák, V., Studying nonlinear PDE by geometry in matrix space, in Geometric Analysis and Nonlinear Partial Differential Equations, pp. 347-395. Springer, Berlin-Heidelberg, 2003.
29. Kristensen, J., On the non-locality of quasiconvexity. Ann. Inst. H. Poincaré Anal. Non Linéaire, 16 (1999), 1-13.
30. 1-isometric imbeddings. Nederl. Akad. Wetensch. Proc. Ser. A., 58 (1955), 545-556.
31. Lehto, O. & Virtanen, K. I., Quasiconformal Mappings in the Plane. Die Grundlehren der mathematischen Wissenschaften, 126. Springer, New York, 1973.
32. Leonetti, F. & Nesi, V., Quasiconformal solutions to certain first order systems and the proof of a conjecture of G. W. Milton. J. Math. Pures Appl., 76 (1997), 109-124.
33. Matoušek, J., On directional convexity. Discrete Comput. Geom., 25 (2001), 389-403.
34. Matoušek, J. & Plecháè, P., On functional separately convex hulls. Discrete Comput. Geom., 19 (1998), 105-130.
35. Milton, G. W. & Nesi, V., Polycrystalline configurations that maximize electrical resistivity. J. Mech. Phys. Solids, 39 (1991), 525-542.
36. Morrey, C. B., Jr., Quasi-convexity and the lower semicontinuity of multiple integrals. Pacific J. Math., 2 (1952), 25-53.
37. Müller, S., Rank-one convexity implies quasiconvexity on diagonal matrices. Internat. Math. Res. Notices, 1999:20 (1999), 1087-1095.
38. - Variational models for microstructure and phase transitions, in Calculus of Variations and Geometric Evolution Problems (Cetraro, 1996), Lecture Notes in Math., 1713, pp. 85-210. Springer, Berlin-Heidelberg, 1999.
39. Müller, S. & Šverák, V., Convex integration for Lipschitz mappings and counterexamples to regularity. Ann. of Math., 157 (2003), 715-742.
40. Murat, F., Compacité par compensation. Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5 (1978), 489-507.
41. - Compacité par compensation: condition nécessaire et suffisante de continuité faible sous une hypothèse de rang constant. Ann. Scuola Norm. Sup. Pisa Cl. Sci., 8 (1981), 69-102.
42. 1 isometric imbeddings. Ann. of Math., 60 (1954), 383-396.
43. Pedregal, P., Laminates and microstructure. European J. Appl. Math., 4 (1993), 121-149.
44. - Parametrized Measures and Variational Principles. Progress in Nonlinear Differential Equations and their Applications, 30. Birkhäuser, Basel, 1997.
45. Rudin, W., Real and Complex Analysis. McGraw-Hill, New York, 1987.
46. Scheffer, V., Regularity and Irregularity of Solutions to Nonlinear Second Order Elliptic Systems and Inequalities. Ph.D. Thesis, Princeton University, 1974.
47. Šverák, V., Rank-one convexity does not imply quasiconvexity. Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), 185-189.
48. - On Tartar's conjecture. Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 405-412.
49. - On the problem of two wells, in Microstructure and Phase Transition, IMA Vol. Math. Appl., 54, pp. 183-189. Springer, New York, 1993.
50. - Lower-semicontinuity of variational integrals and compensated compactness, in Proceedings of the International Congress of Mathematicians, Vol. 2 (Zürich, 1994), pp. 1153-1158. Birkhäuser, Basel, 1995.
51. Székelyhidi, L., Jr., The regularity of critical points of polyconvex functionals. Arch. Ration. Mech. Anal., 172 (2004), 133-152.
52. 2x2. Calc. Var. Partial Differential Equations, 22 (2005), 253-281.
53. - On quasiconvex hulls in symmetric 2∈×∈2 matrices. Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 865-876.
54. 2x2" [Calc. Var. Partial Differential Equations, 22 (2005), 253-281]. Calc. Var. Partial Differential Equations, 28 (2007), 545-546.
55. Tartar, L., Compensated compactness and applications to partial differential equations, in Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., 39, pp. 136-212. Pitman, Boston, MA, 1979.
56. - The compensated compactness method applied to systems of conservation laws, in Systems of Nonlinear Partial Differential Equations (Oxford, 1982), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 111, pp. 263-285. Reidel, Dordrecht, 1983.
57. - Some remarks on separately convex functions, in Microstructure and Phase Transition, IMA Vol. Math. Appl., 54, pp. 191-204. Springer, New York, 1993.
58. Zhang, K., A construction of quasiconvex functions with linear growth at infinity. Ann. Scuola Norm. Sup. Pisa Cl. Sci., 19 (1992), 313-326.
59. 2∈×∈2 without rank-one connections. Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 207-216.
60. - Quasiconvex functions, SO(n) and two elastic wells. Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 759-785.