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Title: On New Structures in the Theory of Fully Nonlinear Equations

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.

Authors:
 [1];  [2]
  1. St. Petersburg State University (Russian Federation)
  2. St. Petersburg State University of Architecture and Civil Engineering (Russian Federation)
Publication Date:
OSTI Identifier:
22773858
Resource Type:
Journal Article
Journal Name:
Journal of Mathematical Sciences
Additional Journal Information:
Journal Volume: 233; Journal Issue: 4; Conference: 7. international conference on differential and functional differential equations, Moscow (Russian Federation), 22-29 Aug 2014, International workshop on spatio-temporal dynamical systems, Moscow (Russian Federation), 22-29 Aug 2014; Other Information: Copyright (c) 2018 Springer Science+Business Media, LLC, part of Springer Nature; http://www.springer-ny.com; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1072-3374
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICAL METHODS AND COMPUTING; ASYMPTOTIC SOLUTIONS; BOUNDARY-VALUE PROBLEMS; EVOLUTION EQUATIONS; GEOMETRY; MATRICES; NONLINEAR PROBLEMS

Citation Formats

Ivochkina, N. M., E-mail: ninaiv@NI1570.spb.edu, and Filimonenkova, N. V., E-mail: nf33@yandex.ru. On New Structures in the Theory of Fully Nonlinear Equations. United States: N. p., 2018. Web. doi:10.1007/S10958-018-3939-1.
Ivochkina, N. M., E-mail: ninaiv@NI1570.spb.edu, & Filimonenkova, N. V., E-mail: nf33@yandex.ru. On New Structures in the Theory of Fully Nonlinear Equations. United States. doi:10.1007/S10958-018-3939-1.
Ivochkina, N. M., E-mail: ninaiv@NI1570.spb.edu, and Filimonenkova, N. V., E-mail: nf33@yandex.ru. Sat . "On New Structures in the Theory of Fully Nonlinear Equations". United States. doi:10.1007/S10958-018-3939-1.
@article{osti_22773858,
title = {On New Structures in the Theory of Fully Nonlinear Equations},
author = {Ivochkina, N. M., E-mail: ninaiv@NI1570.spb.edu and Filimonenkova, N. V., E-mail: nf33@yandex.ru},
abstractNote = {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.},
doi = {10.1007/S10958-018-3939-1},
journal = {Journal of Mathematical Sciences},
issn = {1072-3374},
number = 4,
volume = 233,
place = {United States},
year = {2018},
month = {9}
}