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Journal of Mathematical Sciences

, Volume 201, Issue 5, pp 634–644 | Cite as

From Gårding’s Cones to p-Convex Hypersurfaces

  • N. M. IvochkinaEmail author
Article

Abstract

We consider cones discovered by Gårding in 1959. They play the fundamental role in the modern theory of fully nonlinear second-order partial differential equations. A new classification of symmetric matrices is presented based on the m-positiveness property. Such a classification establishes a new trend in geometry, generating a notion of m-convex hypersurfaces.

Keywords

Dirichlet Problem Principal Curvature Symmetric Matrice Convex Hypersurface Hyperbolic Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    B. Andrews, “Contraction of convex hypersurfaces in Euclidian space,” Calc. Var. Part. Differ. Equ., 2, 151–171 (1994).CrossRefzbMATHGoogle Scholar
  2. 2.
    L. Caffarelli, L. Nirenberg, and J. Spruck, “The Dirichlet problem for nonlinear second-order elliptic equations. III,” Acta Math., 155, 261–301 (1985).CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    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).CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    L. Caffarelli and L. Silvestre, “Smooth approximations to solutions of nonconvex fully nonlinear elliptic equations,” In: AMS Transl. — Ser. 2. Adv. Math. Sci., 229, 67–85 (2010).Google Scholar
  5. 5.
    H. Dong, N. V. Krylov, and X. Li, “On fully nonlinear elliptic and parabolic equations with VMO coefficients in domains,” Algebra i Analiz, 24, No. 1, 53–94 (2012).MathSciNetGoogle Scholar
  6. 6.
    L. C. Evans, “Classical solutions of fully nonlinear convex second-order elliptic equations,” Commun. Pure Appl. Math., 25, 333–363 (1982).CrossRefGoogle Scholar
  7. 7.
    N. V. Filimonenkova, “On the classical solvability of the Dirichlet problem for nondegenerate m-Hessian equations,” J. Math. Sci., 178, 666–694 (2011).CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    L. Gårding, “An inequality for hyperbolic polynomials,” J. Math. Mech., 8, 957–965 (1959).zbMATHMathSciNetGoogle Scholar
  9. 9.
    N. M. Ivochkina, “A description of the stability cones generated by differential operators of Monge–Ampere type,” Sb. Math., 50, No. 1, 259–268 (1985).CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    N. M. Ivochkina, “Solution of the Dirichlet problem for some equations of Monge–Ampere type,” Sb. Math., 56, No. 2, 403–415 (1987).CrossRefzbMATHGoogle Scholar
  11. 11.
    N. M. Ivochkina, “The integral method of barrier functions and the Dirichlet problem for equations with operator of Monge–Ampere type,” Sb. Math., 40, No. 2, 179–192 (1981).CrossRefzbMATHGoogle Scholar
  12. 12.
    N. M. Ivochkina, “Solution of the Dirichlet problem for curvature equations of order m,” Sb. Math., 67, No. 2, 317–339 (1990).CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    N. M. Ivochkina, “The Dirichlet problem for the equations of curvature of order m,” Leningr. Math. J., 2, No. 3, 631–654 (1991).zbMATHMathSciNetGoogle Scholar
  14. 14.
    N. M. Ivochkina, “Geometric evolution equations preserving convexity,” AMS Transl. — Ser. 2. Adv. Math. Sci., 220, 191–121 (2007).MathSciNetGoogle Scholar
  15. 15.
    N. M. Ivochkina, Th. Nehring, and F. Tomi. “Evolution of star-shaped hypersurfaces by nonhomogeneous curvature functions,” St. Petersburg Math. J., 12, No. 1, 145–160 (2001).MathSciNetGoogle Scholar
  16. 16.
    N. M. Ivochkina, N. Trudinger, X.-J. Wang, “The Dirichlet problem for degenerate Hessian equations,” Commun. Part. Differ. Equ., 29, 219–235 (2004).CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    N. V. Krylov, “Boundedly inhomogeneous elliptic and parabolic equations in a domain,” Izv. Akad. Nauk. SSSR. Ser. Mat., 22, 67–97 (1984).zbMATHGoogle Scholar
  18. 18.
    N. V. Krylov, Nonlinear Elliptic and Parabolic Equations of Second Order, Reidel, Dordrecht (1987).CrossRefzbMATHGoogle Scholar
  19. 19.
    M. Lin and N. Trudinger, “On some inequalities for elementary symmetric functions,” Bull. Aust. Math. Soc., 50, 317–326 (1994).CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    N. S. Trudinger, “The Dirichlet problem for the prescribed curvature equations,” Arch. Ration. Mech. Anal., 111, 153–179 (1990).CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    J. Urbas, “On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures,” Math. Z., 205, 355–372 (1990).CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    J. Urbas, “An expansion of convex hypersurfaces,” J. Differ. Geom., 33, 91–125 (1991).zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.St.-Petersburg State University of Architecture and Civil Engineering 4St.-PetersburgRussia

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