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On the Solvability of Certain Degenerate Partial Differential Operators

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Shocks, Singularities and Oscillations in Nonlinear Optics and Fluid Mechanics

Part of the book series: Springer INdAM Series ((SINDAMS,volume 17))

Abstract

In this paper I will give a survey of the problem of solvability of degenerate operators that are not of principal type. In the first place, I will describe some recent results obtained with C. Parenti about semi-global solvability of degenerate operators with symplectic multiple characteristics. I will then describe some other recent results obtained with S. Federico about local solvability in the Sobolev spaces of a class of degenerate operators which are an elaboration of the class considered by Colombini-Cordaro-Pernazza (in turn, an elaboration of the Kannai operator).

Dedicated to Guy Métivier

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Acknowledgements

I wish to thank the organizers of the conference, Ferruccio Colombini, Daniele Del Santo and David Lennes. I wish also to thank M. Petrucci for his efficiency and kindness, and the Istituto Nazionale di Alta Matematica “F. Severi” (Roma) for the kind hospitality.

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Authors and Affiliations

  1. Department of Mathematics, University of Bologna, Piazza di Porta San Donato 5, 40126, Bologna, Italy

    Alberto Parmeggiani

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  1. Alberto Parmeggiani
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Correspondence to Alberto Parmeggiani .

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Editors and Affiliations

  1. Dipartimento di Matematica, Università di Pisa, Pisa, Italy

    Ferruccio Colombini

  2. Dipartimento di Matematica e Geoscienze, Università di Trieste, Trieste, Italy

    Daniele Del Santo

  3. Institut de Mathématiques de Bordeaux, Université de Bordeaux - UMR5251, Talence, France

    David Lannes

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Parmeggiani, A. (2017). On the Solvability of Certain Degenerate Partial Differential Operators. In: Colombini, F., Del Santo, D., Lannes, D. (eds) Shocks, Singularities and Oscillations in Nonlinear Optics and Fluid Mechanics. Springer INdAM Series, vol 17. Springer, Cham. https://doi.org/10.1007/978-3-319-52042-1_7

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