J. Math. Pures Appl. 89 (2008) 107–113 www.elsevier.com/locate/matpur Singular viscosity solutions to fully nonlinear elliptic equations Nikolai Nadirashvili a , Serge Vl˘ adu¸ t b,∗ a LATP, CMI, 39, rue F. Joliot-Curie, 13453 Marseille, France b IML, Luminy, case 907, 13288 Marseille Cedex, France Received 19 March 2007 Available online 1 November 2007 Abstract We prove the existence of a viscosity solution of a fully nonlinear elliptic equation in 24 dimensions with blowing up second derivative.  2007 Elsevier Masson SAS. All rights reserved. Résumé Nous démontrons l’existence d’une solution d’une équation elliptique complètement non linéaire en dimension 24 dont la seconde dérivée explose.  2007 Elsevier Masson SAS. All rights reserved. MSC: 35J60 Keywords: Fully nonlinear elliptic equations; Viscosity solutions; Dirichlet problem 1. Introduction In this paper we study the regularity of solutions of fully nonlinear elliptic equations of the form F (D 2 u) = 0, (1) where function u defined in a domain of R n and D 2 u denotes the Hessian of the function u. We assume that F is uniformly elliptic, i.e. there exists a constant Λ  1 (called an ellipticity constant) such that: Λ −1 |ξ | 2  F u ij ξ i ξ j  Λ|ξ | 2 , ∀ξ ∈ R n . (2) Here, u ij denotes the partial derivative ∂ 2 u/∂x i ∂x j . A function u is called a classical solution of (1) if u ∈ C 2 (Ω) and u satisfies (1). Actually, any classical solution of (1) is a smooth (C α+3 ) solution, provided that F is a smooth * Corresponding author. E-mail addresses: nicolas@cmi.univ-mrs.fr (N. Nadirashvili), vladut@iml.univ-mrs.fr (S. Vl˘ adu¸ t). 0021-7824/$ – see front matter  2007 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.matpur.2007.10.004