Rigidity for the hyperbolic Monge-Ampère equation
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 3 (2004) no. 3, pp. 609-623.

Some properties of nonlinear partial differential equations are naturally associated with the geometry of sets in the space of matrices. In this paper we consider the model case when the compact set K is contained in the hyperboloid -1 , where -1 𝕄 sym 2×2 , the set of symmetric 2×2 matrices. The hyperboloid -1 is generated by two families of rank-one lines and related to the hyperbolic Monge-Ampère equation det 2 u=-1. For some compact subsets K -1 containing a rank-one connection, we show the rigidity property of K by imposing proper topology in the convergence of approximate solutions and affine boundary conditions.

Classification : 49J10, 74G65, 35L70
Lin, Chun-Chi 1

1 Department of Mathematics National Taiwan Normal University Taipei 116, Taiwan
@article{ASNSP_2004_5_3_3_609_0,
     author = {Lin, Chun-Chi},
     title = {Rigidity for the hyperbolic {Monge-Amp\`ere} equation},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {609--623},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 3},
     number = {3},
     year = {2004},
     mrnumber = {2099251},
     zbl = {1170.49304},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2004_5_3_3_609_0/}
}
TY  - JOUR
AU  - Lin, Chun-Chi
TI  - Rigidity for the hyperbolic Monge-Ampère equation
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2004
SP  - 609
EP  - 623
VL  - 3
IS  - 3
PB  - Scuola Normale Superiore, Pisa
UR  - http://www.numdam.org/item/ASNSP_2004_5_3_3_609_0/
LA  - en
ID  - ASNSP_2004_5_3_3_609_0
ER  - 
%0 Journal Article
%A Lin, Chun-Chi
%T Rigidity for the hyperbolic Monge-Ampère equation
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2004
%P 609-623
%V 3
%N 3
%I Scuola Normale Superiore, Pisa
%U http://www.numdam.org/item/ASNSP_2004_5_3_3_609_0/
%G en
%F ASNSP_2004_5_3_3_609_0
Lin, Chun-Chi. Rigidity for the hyperbolic Monge-Ampère equation. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 3 (2004) no. 3, pp. 609-623. http://www.numdam.org/item/ASNSP_2004_5_3_3_609_0/

[1] R. Aumann - S. Hart, Bi-convexity and bi-martingales, Israel J. Math. 54 (1986), 159-180. | MR | Zbl

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

[3] E. Casadio-Tarabusi, An algebraic characterization of quasi-convex functions, Ricerche Mat. 42 (1993), 11-24. | MR | Zbl

[4] N. Chaudhuri - S. Müller, Rank-one convexity implies quasiconvexity on certain hypersurfaces, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003), 1263-1272. | MR | Zbl

[5] M. Chlebík - B. Kirchheim, Rigidity for the four gradient problem, J. Reine Angew. Math. 551 (2002), 1-9. | MR | Zbl

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

[7] G. Dolzmann, “Variational Methods for Crystalline Microstructure-Analysis and Computation", Springer Lecture Notes in Mathematics 1803, 2003. | MR | Zbl

[8] L. C. Evans, “Partial differential equations", Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998. | MR | Zbl

[9] L. C. Evans - R. F. Gariepy, On the partial regularity of energy-minimizing, area-preserving maps, Calc. Var. Partial Differential Equations (4) 9 (1999), 357-372. | MR | Zbl

[10] H. Federer, “Geometric Measure Theory", Springer-verlag, New York, 1969. | MR | Zbl

[11] G. Friesecke - R. D. James - S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Comm. Pure Appl. Math. 55 (2002), 1461-1506. | MR | Zbl

[12] E. Heinz, Über die Lösungen der Minimalflä chengleichung, Nach. Akad. Wissensch. in Göttingen Math.-Phys. Kl. II (1952), 51-56. | MR | Zbl

[13] F. John, Bounds for deformations in terms of average strains, Inequalities, III (Proc. Third Sympos., Univ. California, Los Angeles, Calif., 1969; dedicated to the memory of Theodore S. Motzkin), pp. 129-144, Academic Press, New York, 1972. | MR | Zbl

[14] D. Kinderlehrer, Remarks about equilibrium configurations of crystals, in: “Material instabilities in continuum mechanics and related mathematical problems" J.M. Ball (eds.), Oxford University Press, 1988, pp. 217-242. | MR | Zbl

[15] B. Kirchheim, “Habilitation thesis", University of Leipzig, 2001.

[16] O. I. Mokhov - Y. Nutku, Bianchi transformation between the real hyperbolic Monge-Ampère equation and the Born-Infeld equation, Lett. Math. Phys. (2) 32 (1994), 121-123. | MR | Zbl

[17] S. Müller, Variational models for microstructure and phase transitions, in: “Calculus of variations and geometric evolution problems" (Cetraro, 1996 eds.), pp. 85-210, Lecture Notes in Math., 1713, Springer, Berlin, 1999. | MR | Zbl

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

[19] S. Müller - V. Šverák, Convex integration for Lipschitz mappings and counterexamples to regularity, Ann. of Math. (2-3) 157 (2003), 715-742. | MR | Zbl

[20] V. Nesi - G.W. Milton, Polycrystalline configurations that maximize electrical resistivity, J. Mech. Phys. Solids (4) 39 (1991), 525-542. | MR | Zbl

[21] V. Scheffer, “Regularity and irregularity of solutions to nonlinear second order elliptic systems of partial differential equations and inequalities", Dissertation, Princeton University, 1974. | MR

[22] R. Schoen - J. Wolfson, Minimizing volume among Lagrangian submanifolds, Differential equations: La Pietra 1996, Florence, pp. 181-199, Proc. Sympos. Pure Math., 65, Amer. Math. Soc., Providence, RI, 1999. | MR | Zbl

[23] F. Schulz, “Regularity theory for quasilinear elliptic systems and Monge-Ampeère equations in two dimensions", Lecture Notes in Mathematics, 1445, Springer-Verlag, Berlin, 1990. | MR | Zbl

[24] V. Šverák, “On regularity for the Monge-Ampère equations", preprint, Heriot-Watt University, 1991.

[25] 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, 1993, pp. 191-204. | MR | Zbl