Rigidity for the hyperbolic Monge-Ampère equation

Chun-Chi Lin[1]

  • [1] Department of Mathematics National Taiwan Normal University Taipei 116, Taiwan

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2004)

  • Volume: 3, Issue: 3, page 609-623
  • ISSN: 0391-173X

Abstract

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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 ℋ - 1 , where ℋ - 1 ⊂ 𝕄 sym 2 × 2 , the set of symmetric 2 × 2 matrices. The hyperboloid ℋ - 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 ⊂ ℋ - 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.

How to cite

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Lin, Chun-Chi. "Rigidity for the hyperbolic Monge-Ampère equation." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 3.3 (2004): 609-623. <http://eudml.org/doc/84542>.

@article{Lin2004,
abstract = {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 $\{\mathcal \{H\}\}_\{-1\}$, where $\{\mathcal \{H\}\}_\{-1\}\subset \{\mathbb \{M\}\}_\{\rm sym\}^\{2\times 2\}$, the set of symmetric $2\times 2$ matrices. The hyperboloid $\{\mathcal \{H\}\}_\{-1\}$ is generated by two families of rank-one lines and related to the hyperbolic Monge-Ampère equation $\det \nabla ^\{2\}u=-1$. For some compact subsets $K\subset \{\mathcal \{H\}\}_\{-1\}$ 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.},
affiliation = {Department of Mathematics National Taiwan Normal University Taipei 116, Taiwan},
author = {Lin, Chun-Chi},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {3},
pages = {609-623},
publisher = {Scuola Normale Superiore, Pisa},
title = {Rigidity for the hyperbolic Monge-Ampère equation},
url = {http://eudml.org/doc/84542},
volume = {3},
year = {2004},
}

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
PB - Scuola Normale Superiore, Pisa
VL - 3
IS - 3
SP - 609
EP - 623
AB - 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 ${\mathcal {H}}_{-1}$, where ${\mathcal {H}}_{-1}\subset {\mathbb {M}}_{\rm sym}^{2\times 2}$, the set of symmetric $2\times 2$ matrices. The hyperboloid ${\mathcal {H}}_{-1}$ is generated by two families of rank-one lines and related to the hyperbolic Monge-Ampère equation $\det \nabla ^{2}u=-1$. For some compact subsets $K\subset {\mathcal {H}}_{-1}$ 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.
LA - eng
UR - http://eudml.org/doc/84542
ER -

References

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  1. [1] R. Aumann – S. Hart, Bi-convexity and bi-martingales, Israel J. Math. 54 (1986), 159-180. Zbl0607.52001MR852476
  2. [2] J. M. Ball – R. D. James, Fine phase mixtures as minimizers of energy, Arch. Ration. Mech. Anal. 100 (1987), 13-52. Zbl0629.49020MR906132
  3. [3] E. Casadio-Tarabusi, An algebraic characterization of quasi-convex functions, Ricerche Mat. 42 (1993), 11-24. Zbl0883.26011MR1283802
  4. [4] N. Chaudhuri – S. Müller, Rank-one convexity implies quasiconvexity on certain hypersurfaces, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003), 1263-1272. Zbl1054.49018MR2027644
  5. [5] M. Chlebík – B. Kirchheim, Rigidity for the four gradient problem, J. Reine Angew. Math. 551 (2002), 1-9. Zbl1019.49022MR1932170
  6. [6] M. Chipot – D. Kinderlehrer, Equilibrium configurations of crystals, Arch. Ration. Mech. Anal. 103 (1988), 237-277. Zbl0673.73012MR955934
  7. [7] G. Dolzmann, “Variational Methods for Crystalline Microstructure–Analysis and Computation”, Springer Lecture Notes in Mathematics 1803, 2003. Zbl1016.74002MR1954274
  8. [8] L. C. Evans, “Partial differential equations”, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998. Zbl0902.35002MR1625845
  9. [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. Zbl0954.49024MR1731471
  10. [10] H. Federer, “Geometric Measure Theory”, Springer-verlag, New York, 1969. Zbl0176.00801MR257325
  11. [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. Zbl1021.74024MR1916989
  12. [12] E. Heinz, Über die Lösungen der Minimalflä chengleichung, Nach. Akad. Wissensch. in Göttingen Math.-Phys. Kl. II (1952), 51-56. Zbl0048.15401MR54182
  13. [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. Zbl0292.53003MR344392
  14. [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. Zbl0850.73037MR970527
  15. [15] B. Kirchheim, “Habilitation thesis”, University of Leipzig, 2001. 
  16. [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. Zbl0814.35075MR1296381
  17. [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. Zbl0968.74050MR1731640
  18. [18] S. Müller, Rank-one convexity implies quasiconvexity on diagonal matrices, Internat. Math. Res. Notices (20) (1999), 1087-1095. Zbl1055.49506MR1728018
  19. [19] S. Müller – V. Šverák, Convex integration for Lipschitz mappings and counterexamples to regularity, Ann. of Math. (2-3) 157 (2003), 715-742. Zbl1083.35032MR1983780
  20. [20] V. Nesi – G.W. Milton, Polycrystalline configurations that maximize electrical resistivity, J. Mech. Phys. Solids (4) 39 (1991), 525-542. Zbl0734.73068MR1106125
  21. [21] V. Scheffer, “Regularity and irregularity of solutions to nonlinear second order elliptic systems of partial differential equations and inequalities”, Dissertation, Princeton University, 1974. MR2624766
  22. [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. Zbl1031.53112MR1662755
  23. [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. Zbl0709.35038MR1079936
  24. [24] V. Šverák, “On regularity for the Monge-Ampère equations”, preprint, Heriot-Watt University, 1991. 
  25. [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. Zbl0823.26008MR1320538

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