The geometry of the space of Cauchy data of nonlinear PDEs
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.
KeywordsGeometry of PDEs Cauchy data Lagrangian formalism Fiber bundles Jet spaces Flags
MSC14M15 35A99 53B15 53C80 57R99 58A20
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- 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, 1999Google Scholar
- van Brunt B., The Calculus of Variations, Universitext, Springer, New York, 2004Google Scholar
- Giaquinta M., Hildebrandt S., Calculus of Variations. I, Grundlehren Math. Wiss., 310, Springer, Berlin, 1996Google Scholar
- Moreno G., A C-spectral sequence associated with free boundary variational problems, In: Geometry, Integrability and Quantization, Avangard Prima, Sofia, 2010, 146–156Google Scholar
- 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, 359–379Google Scholar
- Vinogradov A.M., Cohomological Analysis of Partial Differential Equations and Secondary Calculus, Transl. Math. Monogr., 204, American Mathematical Society, Providence, 2001Google Scholar
- Vitagliano L., private communication, 2010Google Scholar