## 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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## References

L. Caffarelli, L. Nirenberg, and J. Spruck, “The Dirichlet problem for nonlinear second order elliptic equations, III. Functions of the eigenvalues of the Hessian,”

*Acta Math.*,**155**, 261–301 (1985).K.-S. Chou and X.-J. Wang, “A variational theory of the Hessian equations,”

*Commun. Pure Appl. Math.*,**54**, 1029–1064 (2001).L.C. Evans, “Classical solutions of fully nonlinear convex second order elliptic equations,”

*Commun. Pure Appl. Math.*,**25**, 333–363 (1982).N. V. Filimonenkova, “On the classical solvability of the Dirichlet problem for nondegenerate

*m*-Hessian equations,”*J. Math. Sci.*(*N. Y.*),**178**, No. 6, 666–694 (2011).N. V. Filimonenkova,

*Sylvester’s criterion for m-positive matrices*, St. Petersburg Math. Soc., Preprint No. 7, St. Petersburg (2014).L. Gårding, “An inequality for hyperbolic polynomials,”

*J. Math. Mech.*,**8**, 957–965 (1959).G. H. Hardy, J. E. Littlewood, and G. Pólya,

*Inequalities*, Cambridge Univ. Press, Cambridge (1934).N. M. Ivochkina, “The integral method of barrier functions and the Dirichlet problem for equations with operators of the Monge–Ampère type,”

*Mat. Sb.*(*N.S.*),**112**(**154**), No. 2 (6), 193–206 (1980).N. M. Ivochkina, “Description of cones of stability generated by differential operators of Monge–Ampère type,”

*Mat. Sb.*(*N.S.*),**122**(**164**), No. 2, 265–275 (1983).N. M. Ivochkina, “The Dirichlet problem for the curvature equation of order

*m,*”*Leningrad Math. J.*,**2**, No. 3, 631–654 (1991).N. M. Ivochkina, “On the Dirichlet problem for fully nonlinear parabolic equations,”

*J. Math. Sci.*(*N. Y.*),**93**, No. 5, 689–696 (1999).N. M. Ivochkina, “Mini survey of the principal notions in the theory of fully nonlinear elliptic second-order differential equations,”

*J. Math. Sci.*(*N. Y.*),**101**, No. 5, 3503–3511 (2000).N. M. Ivochkina, “Weakly first-order interior estimates and Hessian equations,”

*J. Math. Sci.*(*N. Y.*),**143**, No. 2, 2875–2882 (2007).N.M. Ivochkina, “On approximate solutions to the first initial-boundary value problem for the

*m*-Hessian evolution equations,”*J. Fixed Point Theory Appl.*,**4**, No. 1, 47–56 (2008).N. M. Ivochkina, “On classic solvability of the

*m*-Hessian evolution equation,”*Amer. Math. Soc. Transl. Ser.*2,**229**, 119–129 (2010).N. M. Ivochkina, “On some properties of the positive

*m*-Hessian operators in*C*^{2}(Ω)*,*”*J. Fixed Point Theory Appl.*,**14**, No. 1, 79–90 (2014).N. M. Ivochkina, “From Gårding cones to

*p*-convex hypersurfaces,”*J. Math. Sci.*(*N. Y.*),**201**, No. 5, 634–644 (2014).N. M. Ivochkina and N. V. Filimonenkova, “On the backgrounds of the theory of

*m*-Hessian equations,”*Commun. Pure Appl. Anal.*,**12**, No. 4, 1687–1703 (2013).N. M. Ivochkina and N. V. Filimonenkova, “On algebraic and geometric conditions in the theory of Hessian equations,”

*J. Fixed Point Theory Appl.*,**16**, No. 1-2, 11–25 (2014).N. M. Ivochkina and N. V. Filimonenkova, “On attractors of

*m*-Hessian evolutions,”*J. Math. Sci.*(*N. Y.*),**207**, No. 2, 226–235 (2015).N. M. Ivochkina and O. A. Ladyzhenskaya, “On parabolic problems generated by some symmetric functions of the Hessian,”

*Topol. Methods Nonlinear Anal.*,**4**, 19–29 (1994).N.M. Ivochkina, N. Trudinger, and X.-J. Wang, “The Dirichlet problem for degenerate Hessian equations,”

*Commun. Part. Differ. Equ.*,**29**, 219–235 (2004).N.M. Ivochkina, G.V. Yakunina, and S. I. Prokof’eva, “The Gårding cones in the modern theory of fully nonlinear second order differential equations,”

*J. Math. Sci.*(*N. Y.*),**184**, No. 3, 295–315 (2012).N. V. Krylov, “Boundedly inhomogeneous elliptic and parabolic equations in a domain,”

*Izv. Akad. Nauk SSSR Ser. Mat.*,**47**, No. 1, 75–108 (1983).N. V. Krylov,

*Nonlinear Second-Order Elliptic and Parabolic Equations*[in Russian], Nauka, Moscow (1985).G. M. Lieberman,

*Second Order Parabolic Differential Equations*, World Scientific Publishing Co., Inc., River Edge (1996).M. Lin and N. Trudinger, “On some inequalities for elementary symmetric functions,”

*Bull. Aust. Math. Soc.*,**50**, 317–326 (1994).S. I. Prokof’eva and G.V. Yakunina, “On the concept of ellipticity for second-order fully nonlinear partial differential equations,”

*Differ. Uravn. Prots. Upr.*, No. 1, 142–145 (2012).M. V. Safonov, “Smoothness near the boundary of solutions of elliptic Bellman equations,”

*Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)*,**147**, No. 17, 150–154 (1985).N. Trudinger, “On the Dirichlet problem for Hessian equations,”

*Acta Math.*,**175**, 151–164 (1995).N. Trudinger and X.-J. Wang, “A Poincaré type inequality for Hessian integrals,”

*Calc. Var. Part. Differ. Equ.*,**6**, 315–328 (1998).K. Tso, “On an Aleksandrov–Bakel’man type maximum principle for second-order parabolic equations,”

*Commun. Part. Differ. Equ.*,**10**, 543–553 (1985).X.-J. Wang, “A class of fully nonlinear elliptic equations and related functionals,”

*Indiana Univ. Math. J.*,**43**, 25–54 (1994).X.-J. Wang, “The

*k*-Hessian equation,”*Lecture Notes in Math.*,**1977**, 177–252 (2009).

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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