Geodesic
On new structures in the theory of fully nonlinear equations
Contemporary Mathematics. Fundamental Directions, Proceedings of the Seventh International Conference on Differential and Functional-Differential Equations (Moscow, August 22–29, 2014). Part 1, Tome 58 (2015), pp. 82-95

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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 mpositivity 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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     author = {N. M. Ivochkina and N. V. Filimonenkova},
     title = {On new structures in the theory of fully nonlinear equations},
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     volume = {58},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2015_58_a4/}
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N. M. Ivochkina; N. V. Filimonenkova. On new structures in the theory of fully nonlinear equations. Contemporary Mathematics. Fundamental Directions, Proceedings of the Seventh International Conference on Differential and Functional-Differential Equations (Moscow, August 22–29, 2014). Part 1, Tome 58 (2015), pp. 82-95. http://geodesic.mathdoc.fr/item/CMFD_2015_58_a4/