The paper is a kind of survey which contains a description of new algebraic and geometric structures together with fragments of a technique developed in the theory of Hessian equations. Our principal concern here is the m-Hessian evolutionary equations, and we formulate some existence and nonexistence theorems of solutions to the first initial boundary value problems in C 2(Q̄ T ) for such equations.
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Caffarelli L., Nirenberg L., Spruck J.: The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian. Acta Math. 155, 261–301 (1985)
Caffarelli L., Nirenberg L., Spruck J.: Nonlinear second-order elliptic equations. V. The Dirichlet problem for Weingarten hypersurfaces. Comm. Pure Appl. Math. 41, 47–70 (1988)
Evans L.C.: Classical solutions of fully nonlinear, convex, second-order elliptic equations. Comm. Pure Appl. Math. 35, 333–363 (1982)
N. V. Filimonenkova, Sylvester criterion for m-positive matrices. J. Math. Sci., to appear.
Gårding L.: An inequality for hyperbolic polynomials. J. Math. Mech. 8, 957–965 (1959)
N. M. Ivochkina, Second order equations with d-elliptic operators. Tr. Mat. Inst. Steklova 147 (1980), 40–56 (in Russian); English transl.: Proc. Steclov Inst. Math. 147 (1981), 37–54.
N. M. Ivochkina, A description of the stability cones generated by differential operators of Monge-Ampère type. Mat. Sb. (N.S.) 122 (1983), 265–275 (in Russian); English transl.: Math. USSR Sb. 50 (1985), 259–268.
N. M. Ivochkina, Solution of the Dirichlet problem for curvature equations of order m. Math. USSR Sb. 67 (1990), 317–339 (in Russian); English transl.: Leningrad Math. J. 2 (1991), 192–217.
Ivochkina N.M.: On approximate solutions to the first initial boundary value problem for the m-Hessian evolution equations. J. Fixed Point Theory Appl. 4, 47–56 (2008)
N. M. Ivochkina, On classic solvability of the m-Hessian evolution equation. In: Nonlinear Partial Differential Equations and Related Topics, Amer. Math. Soc. Transl. Ser. 2, vol. 229, Amer. Math. Soc., Providence, RI, 2010, 119–129.
Ivochkina N.M.: From Gårding’s cones to p-convex hypersurfaces. J. Math. Sci. 201, 634–644 (2014)
Ivochkina N.M.: On some properties of the positive m-Hessian operators in C 2(Ω). J. Fixed Point Theory Appl. 14, 79–90 (2013)
Ivochkina N.M., Filimonenkova N.V.: On the backgrounds of the theory of m-Hessian equations. Commun. Pure Appl. Anal. 12, 1687–1703 (2013)
Ivochkina N.M., Prokof’eva S.I., Yakunina G.V.: The Gårding cones in the modern theory of fully nonlinear second order differential equations. J. Math. Sci. 184, 295–315 (2012)
N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain. Izv. Akad. Nauk. SSSR Ser. Mat. 47 (1983), 75–108 (in Russian); English transl.: Math. USSR Izv. 22 (1984), 67–97.
A. V. Pogorelov, The Minkowski Multidimensional Problem. Nauka, Moscow, 1975 (in Russian); English transl.: John Wiley & Sons, New York, 1978.
Trudinger N.S.: The Dirichlet problem for the prescribed curvature equations. Arch. Ration. Mech. Anal. 111, 153–179 (1990)
To Professor Andrzej Granas
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Ivochkina, N.M., Filimonenkova, N.V. On algebraic and geometric conditions in the theory of Hessian equations. J. Fixed Point Theory Appl. 16, 11–25 (2014). https://doi.org/10.1007/s11784-015-0217-4
MathemMathematics Subject Classification
- Primary 35K61
- Secondary 15B48
- m-Hessian equation
- m-positive matrix
- m-admissible function
- boundary barrier
- m-convex hypersurface