The modern theory of fully nonlinear second order partial differential equations is based on some algebraic facts and, in particular, on the theory of a-hyperbolic polynomials created by L. Gårding in 1959. The goal of this paper is to describe the Gårding cones in this context. Bibliography: 30 titles.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L.C. Evans, “Classical solutions of fully nonlinear convex second order elliptic equations,” Commun. Pure Appl. Math. 25, 333–363 (1982).
N. V. Krylov, “Boundedly nonhomogeneous elliptic and parabolic equations in a domain” [in Russian], Izv. Akad. Nauk SSSR, Ser. Mat. 47, No. 1, 75–108 (1983); English transl.: Math. USSR, Izv. 22, 67–97 (1984).
M. V. Safonov, “Harnack’s inequality for elliptic equations and the Hölder property of their solutions” [in Russian], Zap. Nauchn. Semin. LOMI 96, 272–287 (1980); English transl.: J. Sov. Math. 21, 851-863 (1983).
M. V. Safonov, “Smoothness near the boundary of the solutions of the elliptic Bellman equations,” [in Russian], Zap. Nauchn. Semin. POMI 147, 150–154 (1985); English transl.: J. Sov. Math. 37, 885-888 (1987).
N. M. Ivochkina, “On second order differential equations with d-elliptic operators,” [in Russian], Tr. Mat. Inst. Steklova 147, 40–56 (1980); English transl.: Proc. Steklov Inst. Math. 147, 37-54 (1981).
N. M. Ivochkina, “Integral method of barrier functions and the DIrichlet problem for equations with operators of Monge–Ampère type” [in Russian], Mat. Sb. 112, No. 2, 193–206 (1980); English transl.: Math. USSR Sb. 40, No. 2, 179–192 (1981).
N. M. Ivochkina, “A description of the stability cones generated by differential operators of Monge–Ampère type” [in Russian], Mat. Sb. 122, No. 2, 265–275 (1983); English transl.: Math. USSR, Sb. 50, 259-268 (1985).
N. M. Ivochkina, “Solution of the Dirichlet problem for some equations of Monge–Ampère type” [in Russian], Mat. Sb. 128, No. 3, 403–415 (1985); English transl.: Math. USSR-Sb. 56, No. 2, 403-415 (1987).
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. Gårding, “An inequality for hyperbolic polynomials,” J. Math. Mech. 8, 957–965 (1959).
L. Caffarelli, L. Nirenberg, and J. Spruck, “Nonlinear second-order elliptic equations V. The Dirichlet problem for Weingarten hypersurfaces,” Commun. Pure Appl. Math. 41, 47–70 (1988).
N. M. Ivochkina, “Solution of the Dirichlet problem for curvature equations of order m” [in Russian], Mat. Sb. 180, No. 7, 867–887 (1989); English transl.: Math. USSR, Sb. 67, No. 2, 317–339 (1990).
N. M. Ivochkina, “The Dirichlet problem for the equations of curvature of order m” [in Russian], Algebra Anal. 2, No. 3, 192–217 (1990); English transl.: Leningr. Math. J. 2, No. 3, 631–654 (1991).
N. S. Trudinger, “The Dirichlet problem for the prescribed curvature equations,” Arch. Ration. Mech. Anal. 111, 153–179 (1990).
N. M. Ivochkina, Th. Nehring, and F. Tomi, “Evolution of starshaped hypersurfaces by nonhomogeneous curvature functions” [in Russian], Algebra Anal. 12, 185-203 (2000); English transl.: St. Petersburg Math. J. 12, 145-160 (2001).
N. M. Ivochkina, “Geometric evolution equations preserving convexity,” Am. Math. Soc. Transl. (2) 220 191–121 (2007).
M. C. Crandall, H. Ishii, and P.L. Lions, “User’s guide to viscosity solutions of second order partial differential equations,” Bull. Am. Math. Soc. 27 1-67 (1992).
N. M. Ivochkina, “The Dirichlet principle in the theory of equations of Monge–Ampere type” [in Russian], 4, No. 6. 131–156 (1992); English transl.: St. Petersbg. Math. J. 4, No. 6, 1169-1189 (1993).
N. S. Trudinger and X.-J. Wang, “Hessian measures II,” Ann. Math. 150 579–604 (1999).
N. S. Trudinger, “Week solutions of Hessian equations,” Commun. Partial Different. Equ. 22, 1251–1261 (1997).
N. V. Filimonenkova, “On the classical solvability of the Dirichlet problem for nondegenerate m-Hessian equations” [in Russian], Probl. Mat. Anal. 60, 89-110 (2011); English transl.: [ J. Math. Sci. 178, No. 6, 666-694 (2011).
L. Caffarelli and L. Silvestre, “Smooth approximations to solutions of nonconvex fully nonlinear elliptic equations,” Am. Math. Soc. Transl., Series 2 229 67-85 (2010).
H. Dong, N. V. Krylov, and X. Li, “On fully nonlinear elliptic and parabolic equations in domains with VMO coefficients” [in Russian], Algebra Anal. 24, No. 1, 53–94 (2012).
M. Lin and N. S. Trudinger, “On some inequalities for elementary symmetric functions,” Bull. Austr. Math. Soc. 50, 317–326 (1994).
N. V. Krylov, Nonlinear Elliptic and Parabolic Equations of the Second Order [in Russian], Nauka, Moscow (1985); English transl.: Reidel, Dordrecht (1987).
N. M. Ivochkina, N. S. Trudinger, and X.-J. Wang, “The Dirichlet problem for degenerate Hessian equations,” Commun. Partial Different. Equ. 29, 219–235 (2004).
C. Gerhardt, “Flow of nonconvex hypersurfaces into spheres,” J. Diff. Geom. 32, 299-314 (1990).
J. Urbas, “On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures,” Math. Z. 205, 355-372 (1990).
J. Urbas, “An expansion of convex hypersurfaces,” J. Diff. Geom. 33 91-125 (1991).
B. Andrews, “Contraction of convex hypersurfaces in Euclidian space,” Calc. Var. 2, 151-171 (1994).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Problems in Mathematical Analysis 64, 2012, p. 63–80.
Rights and permissions
About this article
Cite this article
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). https://doi.org/10.1007/s10958-012-0869-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-012-0869-1