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On the Dirichlet Problem of Mixed Type for Lower Hybrid Waves in Axisymmetric Cold Plasmas

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Abstract

For a class of linear second order partial differential equations of mixed elliptic-hyperbolic type, which includes a well known model for analyzing possible heating in axisymmetric cold plasmas, we give results on the weak well-posedness of the Dirichlet problem and show that such solutions are characterized by a variational principle. The weak solutions are shown to be saddle points of natural functionals suggested by the divergence form of the PDEs. Moreover, the natural domains of the functionals are the weighted Sobolev spaces to which the solutions belong. In addition, all critical levels will be characterized in terms of global extrema of the functionals restricted to suitable infinite dimensional linear subspaces. These subspaces are defined in terms of a robust spectral theory with weights which is associated to the linear operator and is developed herein. Similar characterizations for the weighted eigenvalue problem and nonlinear variants will also be given. Finally, topological methods are employed to obtain existence results for nonlinear problems including perturbations in the gradient which are then applied to the well-posedness of the linear problem with lower order terms.

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Authors and Affiliations

  1. Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133, Milan, Italy

    Daniela Lupo

  2. Dipartimento di Matematica “F. Enriques”, Università di Milano, Via C. Saldini 50, 20133, Milan, Italy

    Dario D. Monticelli & Kevin R. Payne

Authors
  1. Daniela Lupo
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  2. Dario D. Monticelli
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  3. Kevin R. Payne
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Corresponding author

Correspondence to Kevin R. Payne.

Additional information

Communicated by P. Rabinowitz

Daniela Lupo: The authors have been partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

Dario D. Monticelli: Monticelli has also been supported by the GNAMPA project “Equazioni differenziali con invarianze in analisi globale” and by the MIUR project “Metodi variazionali e topologici nello studio di fenomeni nonlineari”.

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Lupo, D., Monticelli, D.D. & Payne, K.R. On the Dirichlet Problem of Mixed Type for Lower Hybrid Waves in Axisymmetric Cold Plasmas. Arch Rational Mech Anal 217, 37–69 (2015). https://doi.org/10.1007/s00205-014-0828-8

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  • DOI: https://doi.org/10.1007/s00205-014-0828-8

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