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On New Structures in the Theory of Fully Nonlinear Equations

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Abstract

We describe the current state of the theory of equations with m-Hessian stationary and evolution operators. It is quite important that new algebraic and geometric notions appear in this theory. In the present work, a list of those notions is provided. Among them, the notion of m-positivity of matrices is quite important; we provide a proof of an analog of Sylvester’s criterion for such matrices. From this criterion, we easily obtain necessary and sufficient conditions for existence of classical solutions of the first initial boundary-value problem for m-Hessian evolution equations. The asymptotic behavior of m-Hessian evolutions in a semibounded cylinder is considered as well.

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Authors and Affiliations

  1. St. Petersburg State University, Saint Petersburg, Russia

    N. M. Ivochkina

  2. St. Petersburg State University of Architecture and Civil Engineering, Saint Petersburg, Russia

    N. V. Filimonenkova

  3. Peter the Great St. Petersburg Polytechnic University, Saint Petersburg, Russia

    N. V. Filimonenkova

Authors
  1. N. M. Ivochkina
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  2. N. V. Filimonenkova
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Correspondence to N. M. Ivochkina.

Additional information

Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 58, Proceedings of the Seventh International Conference on Differential and Functional Differential Equations and InternationalWorkshop “Spatio-Temporal Dynamical Systems” (Moscow, Russia, 22–29 August, 2014). Part 1, 2015.

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Ivochkina, N.M., Filimonenkova, N.V. On New Structures in the Theory of Fully Nonlinear Equations. J Math Sci 233, 480–494 (2018). https://doi.org/10.1007/s10958-018-3939-1

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  • DOI: https://doi.org/10.1007/s10958-018-3939-1

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