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2013 | 11 | 11 | 1960-1981
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The geometry of the space of Cauchy data of nonlinear PDEs

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First-order jet bundles can be put at the foundations of the modern geometric approach to nonlinear PDEs, since higher-order jet bundles can be seen as constrained iterated jet bundles. The definition of first-order jet bundles can be given in many equivalent ways - for instance, by means of Grassmann bundles. In this paper we generalize it by means of flag bundles, and develop the corresponding theory for higher-oder and infinite-order jet bundles. We show that this is a natural geometric framework for the space of Cauchy data for nonlinear PDEs. As an example, we derive a general notion of transversality conditions in the Calculus of Variations.
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Bibliografia
  • [1] Bocharov A.V., Chetverikov V.N., Duzhin S.V., Khor’kova N.G., Krasil’shchik I.S., Samokhin A.V., Torkhov Yu.N., Verbovetsky A.M., Vinogradov A.M., Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, Transl. Math. Monogr., 182, American Mathematical Society, Providence, 1999
  • [2] Bott R., Tu L.W., Differential Forms in Algebraic Topology, Grad. Texts in Math., 82, Springer, New York-Berlin, 1982 http://dx.doi.org/10.1007/978-1-4757-3951-0
  • [3] van Brunt B., The Calculus of Variations, Universitext, Springer, New York, 2004
  • [4] Bryant R.L., Chern S.S., Gardner R.B., Goldschmidt H.L., Griffiths P.A., Exterior Differential Systems, Math. Sci. Res. Inst. Publ., 18, Springer, New York, 1991 http://dx.doi.org/10.1007/978-1-4613-9714-4
  • [5] Giaquinta M., Hildebrandt S., Calculus of Variations. I, Grundlehren Math. Wiss., 310, Springer, Berlin, 1996
  • [6] Kijowski J., A simple derivation of canonical structure and quasi-local Hamiltonians in general relativity, Gen. Relativity Gravitation, 1997, 29(3), 307–343 http://dx.doi.org/10.1023/A:1010268818255
  • [7] Krasil’shchik J., Verbovetsky A., Geometry of jet spaces and integrable systems, J. Geom. Phys., 2011, 61(9), 1633–1674 http://dx.doi.org/10.1016/j.geomphys.2010.10.012
  • [8] Krupka D., Of the structure of the Euler mapping, Arch. Math. (Brno), 1974, 10(1), 55–61
  • [9] Michor P.W., Manifolds of Differentiable Mappings, Shiva Mathematics Series, 3, Shiva Publishing, Nantwich, 1980
  • [10] Moreno G., A C-spectral sequence associated with free boundary variational problems, In: Geometry, Integrability and Quantization, Avangard Prima, Sofia, 2010, 146–156
  • [11] Vinogradov A.M., Many-valued solutions, and a principle for the classification of nonlinear differential equations, Dokl. Akad. Nauk SSSR, 1973, 210, 11–14 (in Russian)
  • [12] Vinogradov A.M., The C-spectral sequence, Lagrangian formalism, and conservation laws. I. The linear theory, J. Math. Anal. Appl., 1984, 100(1), 1–40 http://dx.doi.org/10.1016/0022-247X(84)90071-4
  • [13] Vinogradov A.M., The C-spectral sequence, Lagrangian formalism, and conservation laws. II. The nonlinear theory, J. Math. Anal. Appl., 1984, 100(1), 41–129 http://dx.doi.org/10.1016/0022-247X(84)90072-6
  • [14] Vinogradov A.M., Geometric singularities of solutions of nonlinear partial differential equations, In: Differential Geometry and its Applications, Brno, 1986, Math. Appl. (East European Ser.), 27, Reidel, Dordrecht, 1987, 359–379
  • [15] Vinogradov A.M., Cohomological Analysis of Partial Differential Equations and Secondary Calculus, Transl. Math. Monogr., 204, American Mathematical Society, Providence, 2001
  • [16] Vinogradov A.M., Moreno G., Domains in infinite jet spaces: the C-spectral sequence, Dokl. Math., 2007, 75(2), 204–207 http://dx.doi.org/10.1134/S1064562407020081
  • [17] Vitagliano L., Secondary calculus and the covariant phase space, J. Geom. Phys., 2009, 59(4), 426–447 http://dx.doi.org/10.1016/j.geomphys.2008.12.001
  • [18] Vitagliano L., private communication, 2010
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bwmeta1.element.doi-10_2478_s11533-013-0292-y
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