Abstract
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).
Similar content being viewed by others
References
Caffarelli, L., Interior a priory estimates for solutions of fully nonlinear equations, Ann. Math., 130, 1989, 189–213.
Caffarelli, L. and Cabre, X., Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications, 43, Amer. Math. Soc., Providence, RI,1995.
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.
Nadirashvili, N. and Vlăduţ, S., Singular solutions of Hessian elliptic equations in five dimensions, J. Math. Pures Appl., 100(9), 2013, 769–784.
Nadirashvili, N. and Vlăduţ, S., Singular solutions of Hessian fully nonlinear elliptic equations, Adv. Math., 228, 2011, 1718–1741.
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.
Tkachev, V. G., A Jordan algebra approach to the eiconal, J. of Algebra, 419, 2014, 34–51.
Trudinger, N., Hölder gradient estimates for fully nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A, 108, 1988, 57–65.
Trudinger, N., Fully nonlinear elliptic equations in geometry. CBMS Lectures, October 2004 draft. http://maths-people.anu.edu.au/neilt/RecentPapers/notes1.pdf
Acknowledgments
The authors are deeply grateful to the anonimous referee whose advise permitted to ameliorate significantly our exposition.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Professor Haim Brezis on the occasion of his 70th birthday
Rights and permissions
About this article
Cite this article
Nadirashvili, N., Vlăduţ, S. Singular solutions to conformal Hessian equations. Chin. Ann. Math. Ser. B 38, 591–600 (2017). https://doi.org/10.1007/s11401-017-1085-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11401-017-1085-6