Chinese Annals of Mathematics, Series B

, Volume 38, Issue 2, pp 591–600 | Cite as

Singular solutions to conformal Hessian equations

  • Nikolai Nadirashvili
  • Serge Vlăduţ


The authors show that for any ε ∈]0, 1[, there exists an analytic outside zero solution to a uniformly elliptic conformal Hessian equation in a ball B ⊂ ℝ5 which belongs to C 1,ε (B) \C 1,ε +(B).


Viscosity solutions Conformal Hessian equation Cartan’s cubic 

2000 MR Subject Classification

35J60 53C38 


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The authors are deeply grateful to the anonimous referee whose advise permitted to ameliorate significantly our exposition.


  1. [1]
    Caffarelli, L., Interior a priory estimates for solutions of fully nonlinear equations, Ann. Math., 130, 1989, 189–213.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    Caffarelli, L. and Cabre, X., Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications, 43, Amer. Math. Soc., Providence, RI,1995.Google Scholar
  3. [3]
    Crandall, M. G., Ishii, H. and Lions, P.-L., User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27, 1992, 1–67.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    Nadirashvili, N. and Vlăduţ, S., Singular solutions of Hessian elliptic equations in five dimensions, J. Math. Pures Appl., 100(9), 2013, 769–784.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    Nadirashvili, N. and Vlăduţ, S., Singular solutions of Hessian fully nonlinear elliptic equations, Adv. Math., 228, 2011, 1718–1741.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    Nadirashvili, N., Tkachev, V. G. and Vlăduţ, S., Nonlinear elliptic equations and nonassociative algebras, Math. Surv. and Monogr., 200, Amer. Math. Soc., Providence, RI, 2014.Google Scholar
  7. [7]
    Tkachev, V. G., A Jordan algebra approach to the eiconal, J. of Algebra, 419, 2014, 34–51.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    Trudinger, N., Hölder gradient estimates for fully nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A, 108, 1988, 57–65.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    Trudinger, N., Fully nonlinear elliptic equations in geometry. CBMS Lectures, October 2004 draft. Scholar

Copyright information

© Fudan University and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Aix Marseille Université, CNRS, Centrale Marseille, I2M UMR 7373MarseilleFrance
  2. 2.IITP RASMoscowRussia

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