Abstract
In this work, we continue the investigation of algebraic properties of Gårding cones in the space of symmetric matrices. Based on this theory, we propose a new approach to study fully nonlinear differential operators and second-order partial differential equations. We prove comparison theorems of a new type for evolution Hessian operators and establish a relation between Hessian and Bellman equations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
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).
L. C. Evans, “Classical solutions of fully nonlinear convex second order elliptic equations,” Commun. Pure Appl. Math., 25, 333–363 (1982).
N. V. Filimonenkova and P. A. Bakusov, “Hyperbolic polynomials and Gårding’s cones,” Mat. Prosveshch. Tret’ya Ser., 20, 143–166 (2016).
L. Gårding, “An inequality for hyperbolic polynomials,” J. Math. Mech., 8, No. 2, 957–965 (1959).
N. M. Ivochkina, “Description of stability cones generated by differential operators of the Monge–Ampère type,” Mat. Sb., 22, 265–275 (1983).
N. M. Ivochkina, “Solution of the Dirichlet problem for some equations of the Monge–Ampère type,” Mat. Sb., 128, 403–415 (1985).
N. M. Ivochkina, “On classic solvability of the m-Hessian evolution equation,” Am. Math. Soc. Transl., 229, 119–129 (2010).
N. M. Ivochkina, “From Gårding’s cones to p-convex hypersurfaces,” Sovrem. Mat. Fundam. Napravl., 45, 94–104 (2012).
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, “Attractors of m-Hessian evolutions,” J. Math. Sci. (N. Y.), 207, No. 2, 226–235 (2015).
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, 11–25 (2015).
N. M. Ivochkina and N. V. Filimonenkova, “On new structures in the theory of fully nonlinear equations,” Sovrem. Mat. Fundam. Napravl., 58, 82–95 (2015).
N. M. Ivochkina, S. I. Prokof’eva, and G. V. Yakunina, “Gårding’s cones in contemporary theory of fully nonlinear second-order differential equations,” Problemy Mat. Anal., 64, 63–80 (2012).
N. V. Krylov, “Boundedly nonhomogeneous elliptic and parabolic equations in a domain,” Izv. AN SSSR. Ser. Mat., 47, No. 1, 75–108 (1983).
N. V. Krylov, Nonlinear Second-Order Elliptic and Parabolic Equations [in Russian], Nauka, Moscow (1985).
N. V. Krylov, “On the first boundary-value problem for nonlinear degenerating elliptic equations,” Izv. AN SSSR, 51, No. 2, 242–269 (1987).
N. V. Krylov, “On general notion of fully nonlinear second order elliptic equation,” Trans. Am. Math. Soc., 347, No. 3, 857–895 (1995).
M. Lin and N. S. Trudinger, “On some inequalities for elementary symmetric functions,” Bull. Aust. Math. Soc., 50, 317–326 (1994).
A. I. Nazarov and N. N. Uraltseva, “Convex-monotone hulls and an estimate of the maximum of the solution of a parabolic equation,” J. Soviet Math., 37, 851–859 (1987).
A. V. Pogorelov, Multidimensional Minkowski Problem [in Russian], Nauka, Moscow (1975).
M. V. Safonov, “Harnack’s inequality for elliptic equations and the Hölder property of their solutions,” Zap. Nauch. Sem. LOMI, 12, 272–287 (1983).
M. V. Safonov, “On smoothness of solutions of elliptic Bellman equations near the boundary,” Zap. Nauch. Sem. LOMI, 17, 150–154 (1985).
N. S. Trudinger, “The Dirichlet problem for the prescribed curvature equations,” Arch. Ration. Mech. Anal., 111, 153–179 (1990).
N. S. Trudinger, “On the Dirichlet problem for Hessian equations,” Acta Math., 175, 151–164 (1995).
J. I. Urbas, “Nonlinear oblique boundary value problems for Hessian equations in two dimensions,” Ann. Inst. H. Poincaré Anal. Non Linéaire, 12, 507–575 (1995).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 63, No. 4, Differential and Functional Differential Equations, 2017.
Rights and permissions
About this article
Cite this article
Ivochkina, N.M., Filimonenkova, N.V. Gårding Cones and Bellman Equations in the Theory of Hessian Operators and Equations. J Math Sci 259, 833–844 (2021). https://doi.org/10.1007/s10958-021-05665-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-021-05665-z