# 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:

- St. Petersburg State University (Russian Federation)
- 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}

}

Other availability

Save to My Library

You must Sign In or Create an Account in order to save documents to your library.