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Some Rigidity Results Related to Monge–Ampèere Functions

  • Robert L. Jerrard (a1)
Abstract

The space of Monge–Ampère functions, introduced by J. H. G. Fu, is a space of rather rough functions in which the map u ⟼ DetD 2 u is well defined and weakly continuous with respect to a natural notion of weak convergence. We prove a rigidity theorem for Lagrangian integral currents that allows us to extend the original definition of Monge–Ampère functions. We also prove that if a Monge–Ampère function u on a bounded set Ω ⊂ ℝ2 satisfies the equation DetD 2 u = 0 in a particular weak sense, then the graph of u is a developable surface, and moreover u enjoys somewhat better regularity properties than an arbitrary Monge–Ampère function of 2 variables.

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References
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[1] Ambrosio, L., Fusco, N., and Pallara, D., Functions of Bounded Variation and Free Discontinuity Problems. The Clarendon Press, Oxford University Press, New York, 2000.
[2] Bernig, A., Support functions, projections and Minkowski addition of Legendrian cycles. Indiana Univ. Math. J. 55(2006), no. 2, 443–464. doi:10.1512/iumj.2006.55.2684
[3] Bernig, A. and L. Bröcker, Lipschitz-Killing invariants. Math. Nachr. 245(2002), 5–25. doi:10.1002/1522-2616(200211)245:1h5::AID-MANA5i3.0.CO;2-E
[4] Cohen-Steiner, D. and Morvan, J. M., Second fundamental measure of geometric sets and local approximation of curvatures. J. Differential Geom. 74(2006), no. 3, 363–394.
[5] Federer, H., Geometric Measure Theory. Die Grundlehren des mathematischen Wissenschaften 153. Springer-Verlag, New York, 1969.
[6] Fonseca, I. and Maly, J., From Jacobian to Hessian: distributional form and relaxation. Riv. Mat. Univ. Parma 4*(2005), 45–74.
[7] Fu, J. H. G., Monge–Ampère functions. I. Indiana Univ. Math. J. 38(1989), no. 3, 745–771. doi:10.1512/iumj.1989.38.38035
[8] Fu, J. H. G., Monge–Ampère functions. II. Indiana Univ. Math. J. 38(1989), no. 3, 773–789. doi:10.1512/iumj.1989.38.38035
[9] Fu, J. H. G., Curvature measures of subanalytic sets. Amer. J. Math. 116(1994), no. 4, 819–880. doi:10.2307/2375003
[10] Giaquinta, M., Modica, G., and Soucek, J., Cartesian Currents in the Calculus of Variations I, II. Ergebnisse der Mathematik und ihrer Grenzgebiete 37, 38. Springer-Verlag, Berlin, 1998.
[11] Hartman, P. and Nirenberg, L., On spherical image maps whose Jacobians do not change sign. Amer. J. Math. 81(1959), 901–920. doi:10.2307/2372995
[12] Iwaniec, T.. On the concept of the weak Jacobian and Hessian. In: Papers on Analysis. Rep. Univ. Jyväkylä Dep. Math. Stat. 83. University Jyväkylä, 2001, pp. 181–205.
[13] Jerrard, R. L., Some remarks on Monge–Ampère functions. In: Singularities in PDE and the Calculus of Variations. CR M Proc. Lecture Notes 44. American Mathematical Society, Providence, RI, 2008, pp. 89–112.
[14] Jerrard, R. L. and N. Jung. Strict convergence and minimal liftings in BV. Proc. Roy. Soc. Edinburgh Sect. A 134(2004), no. 6, 1163–1176. doi:10.1017/S0308210500003681
[15] Kirchheim, B., Geometry and Rigidity of Microstructures. Habilitation Thesis, Leipzig, 2001.
[16] Pakzad, M. R., On the Sobolev space of isometric immersions. J. Differential Geom. 66(2004), no. 1, 47–69.
[17] Pogorelov, A. V., Vneshnyaya geometriya vypuklykh poverkhnosteı. [The extrinsic geometry of convex surfaces] Izdat. “Nauka”, Moscow 1969.
[18] Rataj, C. and M. Zähle, Normal cycles of Lipschitz manifolds by approximation with parallel sets. Differential Geom. Appl. 19(2003), no. 1, 113–126. doi:10.1016/S0926-2245(03)00020-2
[19] Solomon, B., A new proof of the closure theorem for integral currents. Indiana Univ. Math. J. 33(1984), no. 3, 393–418. doi:10.1512/iumj.1984.33.33022
[20] B.White, Rectifiablity of flat chains. Ann. of Math. 150(1999), no. 1, 165–184. doi:10.2307/121100
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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