Existence of Lipschitz minimizers for the three-well problem in solid-solid phase transitions
Annales de l'I.H.P. Analyse non linéaire, Volume 24 (2007) no. 6, p. 953-962
@article{AIHPC_2007__24_6_953_0,
     author = {Conti, Sergio and Dolzmann, Georg and Kirchheim, Bernd},
     title = {Existence of Lipschitz minimizers for the three-well problem in solid-solid phase transitions},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {24},
     number = {6},
     year = {2007},
     pages = {953-962},
     doi = {10.1016/j.anihpc.2006.10.002},
     zbl = {1131.74037},
     mrnumber = {2371114},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2007__24_6_953_0}
}
Conti, Sergio; Dolzmann, Georg; Kirchheim, Bernd. Existence of Lipschitz minimizers for the three-well problem in solid-solid phase transitions. Annales de l'I.H.P. Analyse non linéaire, Volume 24 (2007) no. 6, pp. 953-962. doi : 10.1016/j.anihpc.2006.10.002. http://www.numdam.org/item/AIHPC_2007__24_6_953_0/

[1] Acerbi E., Fusco N., Semicontinuity problems in the calculus of variations, Arch. Ration. Mech. Anal. 86 (1984) 125-145. | MR 751305 | Zbl 0565.49010

[2] Adams J., Conti S., Desimone A., Soft elasticity and microstructure in smectic C elastomers, Cont. Mech. Thermodyn. 18 (2007) 319-334. | MR 2270449 | Zbl 1170.76303

[3] Ball J.M., Some open problems in elasticity, in: Newton P., Holmes P., Weinstein A. (Eds.), Geometry, Mechanics, and Dynamics, Springer, New York, 2002, pp. 3-59. | MR 1919825 | Zbl 1054.74008

[4] Ball J.M., James R.D., Fine phase mixtures as minimizers of the energy, Arch. Ration. Mech. Anal. 100 (1987) 13-52. | MR 906132 | Zbl 0629.49020

[5] Ball J.M., James R.D., Proposed experimental tests of a theory of fine microstructure and the two-well problem, Philos. Trans. R. Soc. Lond. A 338 (1992) 389-450. | Zbl 0758.73009

[6] Bhattacharya K., Self-accommodation in martensite, Arch. Ration. Mech. Anal. 120 (1992) 201-244. | MR 1183551 | Zbl 0771.73007

[7] Chipot M., Kinderlehrer D., Equilibrium configurations of crystals, Arch. Ration. Mech. Anal. 103 (1988) 237-277. | MR 955934 | Zbl 0673.73012

[8] Conti S., Desimone A., Dolzmann G., Müller S., Otto F., Multiscale modeling of materials - the role of analysis, in: Kirkilionis M., Krömker S., Rannacher R., Tomi F. (Eds.), Trends in Nonlinear Analysis (Heidelberg), Springer, 2002, pp. 375-408. | Zbl 1065.74056

[9] Dacorogna B., Marcellini P., Sur le problème de Cauchy-Dirichlet pour les systèmes d'équations non linéaires du premier ordre, C. R. Acad. Sci. Paris Sér. I Math. 323 (1996) 599-602. | Zbl 0860.35020

[10] Dacorogna B., Marcellini P., General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial cases, Acta Math. 178 (1997) 1-37. | Zbl 0901.49027

[11] Dacorogna B., Marcellini P., Implicit Partial Differential Equations, Progress in Nonlinear Differential Equations and their Applications, vol. 37, Birkhäuser, 1999. | MR 1702252 | Zbl 0938.35002

[12] Desimone A., Dolzmann G., Macroscopic response of nematic elastomers via relaxation of a class of SO 3-invariant energies, Arch. Ration. Mech. Anal. 161 (2002) 181-204. | MR 1894590 | Zbl 1017.74049

[13] Dolzmann G., Kirchheim B., Liquid-like behavior of shape memory alloys, C. R. Math. Acad. Sci. Paris, Ser. I 336 (2003) 441-446. | MR 1979361 | Zbl 1113.74411

[14] Dolzmann G., Müller S., Microstructures with finite surface energy: the two-well problem, Arch. Ration. Mech. Anal. 132 (1995) 101-141. | MR 1365827 | Zbl 0846.73054

[15] Gromov M., Partial Differential Relations, Springer-Verlag, 1986. | MR 864505 | Zbl 0651.53001

[16] B. Kirchheim, Lipschitz minimizers of the 3-well problem having gradients of bounded variation, Preprint 12, Max Planck Institute for Mathematics in the Sciences, Leipzig, 1998.

[17] Kirchheim B., Deformations with finitely many gradients and stability of quasiconvex hulls, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001) 289-294. | MR 1817378 | Zbl 0989.49013

[18] B. Kirchheim, Rigidity and geometry of microstructures, MPI-MIS Lecture Notes 16, 2002.

[19] Kirchheim B., Müller S., Šverák V., Studying nonlinear pde by geometry in matrix space, in: Hildebrandt S., Karcher H. (Eds.), Geometric Analysis and Nonlinear Partial Differential Equations, Springer-Verlag, 2003, pp. 347-395. | MR 2008346 | Zbl pre01944370

[20] Marcellini P., Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals, Manuscripta Math. 51 (1985) 1-28. | MR 788671 | Zbl 0573.49010

[21] Morrey C.B., Quasi-convexity and the lower semicontinuity of multiple integrals, Pacific J. Math. 2 (1952) 25-53. | MR 54865 | Zbl 0046.10803

[22] Müller S., Šverák V., Attainment results for the two-well problem by convex integration, in: Geometric Analysis and the Calculus of Variations, Internat. Press, Cambridge, MA, 1996, pp. 239-251. | MR 1449410 | Zbl 0930.35038

[23] Müller S., Šverák V., Convex integration with constraints and applications to phase transitions and partial differential equations, J. Eur. Math. Soc. (JEMS) 1 (1999) 393-442. | MR 1728376 | Zbl 0953.35042

[24] Müller S., Sychev M.A., Optimal existence theorems for nonhomogeneous differential inclusions, J. Funct. Anal. 181 (2001) 447-475. | MR 1821703 | Zbl 0989.49012

[25] Šverák V., On the problem of two wells, in: Microstructure and Phase Transition, IMA Vol. Math. Appl., vol. 54, Springer, New York, 1993, pp. 183-189. | MR 1320537 | Zbl 0797.73079

[26] Sychev M.A., Comparing two methods of resolving homogeneous differential inclusions, Calc. Var. Partial Differential Equations 13 (2001) 213-229. | MR 1861098 | Zbl 0994.35038