Journal of Mathematical Sciences

, Volume 143, Issue 2, pp 2875–2882 | Cite as

Weak first-order interior estimates and Hessian equations

  • N. M. Ivochkina


Development of the modern theory of fully nonlinear, second-order partial differential equations has amazingly enriched the classical collection of ideas and methods. In this paper, we construct first-order interior a priori estimates of new type for solutions of Hessian equations. These estmates are applied to present the most transparent version of Krylov’s method and its tight connection with fully nonlinear equations. Bibliography: 11 titles.


Dirichlet Problem Viscosity Solution Bellman Equation Nonlinear Elliptic Equation Admissible Solution 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    N. V. Krylov, “Unconditional solvability of the Bellman equations with constant coefficients in convex domains,” Mat. Sb., 34, 65–95 (1989).Google Scholar
  2. 2.
    N. V. Krylov, “Weak interior second-order derivatives estimates for degenerate nonlinear elliptic equations,” Diff. Int. Eqns., 7, 133–156 (1994).MATHMathSciNetGoogle Scholar
  3. 3.
    N. V. Krylov, Lectures on Fully Nonlinear Second Order Elliptic Equations, Rudolph-Lipschitz-Vorlesung, No. 29, Vorlesungsreihe, Rheinische Friedrich-Wilhelms-Universitat, Bonn (1994).Google Scholar
  4. 4.
    N. M. Ivochkina, N. Trudinger, and X.-J. Wand, “The Dirichlet problem for degenerate Hessian equations,” Comm. Part. Diff. Eqns., 29, 219–235 (2004).MATHCrossRefGoogle Scholar
  5. 5.
    L. Caffarelli, L. Nirenberg, and J. Spruck, “Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian,” Acta Math., 155, 261–301 (1985).MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    N. S. Trudinger, “On the Dirichlet problem for Hessian equations,” Acta Math., 175, 151–164 (1995).MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    N. M. Ivochkina, “A description of stability cones generated by differential operators of Monge-Ampere type,” Mat. Sb., 122, 265–275 (1983).MathSciNetGoogle Scholar
  8. 8.
    L. Gärding, “An inequality for hyperbolic polynomials,” J. Math. Mech., 8, 957–965 (1959).MathSciNetGoogle Scholar
  9. 9.
    M. G. Crandall, “Semidifferential quadratic forms and fully nonlinear elliptic equations of second order,” Ann. Inst. H. Poincaré, Anal. Non-Lineaire, 6, 419–435 (1989).MATHMathSciNetGoogle Scholar
  10. 10.
    N. M. Ivochkina, “Mini review of fundamental notions in the theory of fully nonlinear, elliptic, second-order differential equations,” Zap. Nauch. Semin. POMI, 249, 199–211 (1997).Google Scholar
  11. 11.
    N. M. Ivochkina, “The Dirichlet principle in the theory of equations of Monge-Ampere type,” Algebra Analiz, 4, 131–156 (1992).MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • N. M. Ivochkina
    • 1
  1. 1.St.Petersburg State University of Architecture and Civil EngineeringRussia

Personalised recommendations