On the existence of heteroclinic connections

Abstract

Assume that \(W:\mathbb {R}^m\rightarrow \mathbb {R}\) is a nonnegative potential that vanishes only on a finite set A with at least two elements. By direct minimization of the action functional on a suitable set of maps we give a new elementary proof of the existence of a heteroclinic orbit that connects any given \(a_-\in A\) to some \(a_{+}\in A{\setminus }\{a_{-}\}\).

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Acknowledgements

G.F.G. has been partially supported by the University of Pisa via Grant PRA-2017 “Sistemi dinamici in analisi, geometria, logica, e meccanica celeste”. M.N. was partially supported by GNAMPA and by the University of Pisa via Grant PRA-2017 “Problemi di ottimizzazione ed evoluzione in ambito variazionale”.

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Correspondence to Giorgio Fusco.

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Fusco, G., Gronchi, G.F. & Novaga, M. On the existence of heteroclinic connections. São Paulo J. Math. Sci. 12, 68–81 (2018). https://doi.org/10.1007/s40863-017-0080-x

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Keywords

  • Heteroclinic orbit
  • Action functional
  • Jacobi functional
  • Minimization

Mathematics Subject Classification

  • 37J50
  • 37J45