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Geometrical proofs for the global solvability of systems

Author

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  • Adalberto Panobianco Bergamasco
  • Alberto Parmeggiani
  • Sérgio Luís Zani
  • Giuliano Angelo Zugliani

Abstract

We study a linear operator associated with a closed non‐exact 1‐form b defined on a smooth closed orientable surface M of genus g>1. Here we present two proofs that reveal the interplay between the global solvability of the operator and the global topology of the surface. The first result brings an answer for the global solvability when the system is defined by a generic Morse 1‐form. Necessary conditions for the global solvability bearing on the sublevel and superlevel sets of primitives of a smooth 1‐form b have already been established; we also present a more intuitive proof of this result.

Suggested Citation

  • Adalberto Panobianco Bergamasco & Alberto Parmeggiani & Sérgio Luís Zani & Giuliano Angelo Zugliani, 2018. "Geometrical proofs for the global solvability of systems," Mathematische Nachrichten, Wiley Blackwell, vol. 291(16), pages 2367-2380, November.
    • Handle: RePEc:bla:mathna:v:291:y:2018:i:16:p:2367-2380
    DOI: 10.1002/mana.201700300
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    Cited by:

    1. Fernando de Ávila Silva, 2023. "Globally hypoelliptic triangularizable systems of periodic pseudo‐differential operators," Mathematische Nachrichten, Wiley Blackwell, vol. 296(6), pages 2293-2320, June.

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