TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 361, Number 9, September 2009, Pages 4581–4591 S 0002-9947(09)04640-6 Article electronically published on April 14, 2009 REGULARITY OF SUBELLIPTIC MONGE-AMP ` ERE EQUATIONS IN THE PLANE PENGFEI GUAN AND ERIC SAWYER Abstract. We establish a C ∞ regularity result for C 1,1 solutions of degen- erate Monge-Amp` ere equation in R 2 , under the assumption that the trace of the Hessian is bounded from below. 1. Introduction There is a vast body of elliptic regularity results for the Monge-Amp` ere equation det D 2 u (x) = k (x) , x ∈ Ω, (1.1) u (ξ ) = ϕ (ξ ) , ξ ∈ ∂Ω. If the data k, ϕ, ∂Ω are smooth and ∂Ω is strictly convex, then there is a unique solution u of (1.1), smooth up to the boundary of Ω (e.g., [2]). In the event that ellipticity degenerates, i.e. k ≥ 0 vanishes in Ω, existence and uniqueness persist with some restrictions, but the optimal regularity of solutions is now C 1,1 (e.g., see [5, 7]). For the case n = 2, the Monge-Amp` ere equation is closely related to Weyl’s isometric embedding of surfaces ([10]). If the Gauss curvature is only assumed to be nonnegative, a C 1,1 isometric embedding was obtained in [6, 8]. It is desirable to know under what conditions the solution is smooth. From an analytic point of view, detecting smoothness from the data is problematic even for the simplest degeneracies: the nonsmooth function u (x)= c n |x| 2+ 2 n solves (1.1) with polynomial data k (x)= |x| 2 , ϕ (ξ )= c n and ∂Ω= S n−1 . A similar global example for the Weyl isometric embedding problem can be found in [1, 9], where the metric is analytic with nonnegative Gauss curvature but the isometric embedding is not C 3 . The problem is that the mean curvature of the isometric embedding vanishes along with the Gauss curvature at a point. That is, the surface is umbilical at a point where the Gauss curvature vanishes. An alternative approach to detecting smoothness in the degenerate case has been introduced by one of the authors that involves geometric quantities associated with the solution u. For example ([4]), if in n = 2 dimensions a C 1,1 convex solution u to (1.2) det D 2 u (x)= k (x) , x ∈ Ω, Received by the editors April 26, 2007. 2000 Mathematics Subject Classification. Primary 35J60, 35B65. Research of the authors was supported in part by NSERC Discovery Grants. c 2009 American Mathematical Society Reverts to public domain 28 years from publication 4581 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use