On quasiconvex hulls in symmetric $2\times 2$ matrices

Székelyhidi, László

Annales de l'I.H.P. Analyse non linéaire, Tome 23 (2006) no. 6, pp. 865-876.

[1] Astala K., Faraco D., Quasiregular mappings and Young measures, Proc. Roy. Soc. Edinburgh Sect. A 132 (5) (2002) 1045-1056. | MR | Zbl

[2] K. Astala, T. Iwaniec, G. Martin, Pucci's conjecture and the Alexandrov inequality for elliptic PDEs in the plane, J. Reine Angew. Math., in press. | Zbl

[3] Ball J.M., A version of the fundamental theorem for Young measures, in: PDEs and Continuum Models of Phase Transitions (Nice, 1988), Lecture Notes in Phys., vol. 344, Springer, Berlin, 1989, pp. 207-215. | MR | Zbl

[4] J.M. Ball, R.D. James, Incompatible sets of gradients and metastability, in preparation.

[5] Ball J.M., James R.D., Fine phase mixtures as minimizers of energy, Arch. Rational Mech. Anal. 100 (1987) 13-52. | MR | Zbl

[6] B. Bojarski, L. D'Onofrio, T. Iwaniec, C. Sbordone, G-closed classes of elliptic operators in the complex plane, in press.

[7] Caffarelli L.A., Cabré X., Fully Nonlinear Elliptic Equations, Amer. Math. Soc. Colloq. Publ., vol. 43, American Mathematical Society, Providence, RI, 1995. | MR | Zbl

[8] Cardaliaguet P., Tahraoui R., Sur l’équivalence de la 1-rang convexité et de la polyconvexité des ensembles isotropiques de $R^{2\times 2}$, C. R. Acad. Sci. Paris Sér. I Math. 331 (11) (2000) 851-856. | MR | Zbl

[9] Chipot M., Kinderlehrer D., Equilibrium configurations of crystals, Arch. Rational Mech. Anal. 103 (1988) 237-277. | MR | Zbl

[10] Conti S., De Lellis C., Müller S., Romeo M., Polyconvexity equals rank-one convexity for connected isotropic sets in $M^{2\times 2}$, C. R. Math. Acad. Sci. Paris, Ser. I 337 (4) (2003) 233-238. | MR | Zbl

[11] Dacorogna B., Tanteri C., Implicit partial differential equations and the constraints of nonlinear elasticity, J. Math. Pures Appl. (9) 81 (4) (2002) 311-341. | MR | Zbl

[12] Dolzmann G., Variational Methods for Crystalline Microstructure - Analysis and Computation, Lecture Notes in Math., vol. 1803, Springer-Verlag, Berlin, 2003. | Zbl

[13] Faraco D., Zhong X., Quasiconvex functions and Hessian equations, Arch. Rational Mech. Anal. 168 (3) (2003) 245-252. | MR | Zbl

[14] Gilbarg D., Trudinger N.S., Elliptic Partial Differential Equations of Second Order, Classics Math., Springer-Verlag, Berlin, 2001, (Reprint of the 1998 edition). | MR | Zbl

[15] Kinderlehrer D., Pedregal P., Characterizations of Young measures generated by gradients, Arch. Rational Mech. Anal. 115 (1991) 329-365. | MR | Zbl

[16] Matos J.P., Young measures and the absence of fine microstructures in a class of phase transitions, Eur. J. Appl. Math. 3 (1) (1992) 31-54. | MR | Zbl

[17] Müller S., Rank-one convexity implies quasiconvexity on diagonal matrices, Int. Math. Res. Notices 20 (1999) 1087-1095. | MR | Zbl

[18] Müller S., Variational models for microstructure and phase transitions, in: Calculus of Variations and Geometric Evolution Problems (Cetraro, 1996), Lecture Notes in Math., vol. 1713, Springer, Berlin, 1999, pp. 85-210. | MR | Zbl

[19] V. Šverák, On regularity for the Monge-Ampère equations, Preprint, Heriot-Watt University, 1991.

[20] Šverák V., New examples of quasiconvex functions, Arch. Rational Mech. Anal. 119 (4) (1992) 293-300. | MR | Zbl

[21] Székelyhidi L., Rank-one convex hulls in $R^{2\times 2}$, Calc. Var. Partial Differential Equations 22 (3) (2005) 253-281. | MR | Zbl

[22] Tartar L., Compensated compactness and applications to partial differential equations, in: Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, vol. IV, Res. Notes in Math., vol. 39, Pitman, Boston, MA, 1979, pp. 136-212. | MR | Zbl

[23] Zhang K., On separation of gradient Young measures, Calc. Var. Partial Differential Equations 17 (1) (2003) 85-103. | MR | Zbl