Regularity of solutions for some variational problems subject to convexityconstraint

We study the minimizers, in the class of convex functions, of an elliptic f unctional with nonhomogeneous Dirichlet boundary conditions. We prove C-1 r egularity of the minimizers under the assumption that the upper envelope of admissible functions is C-1. This condition is optimal at least when the f unctional depends only on the gradient [4]. We then extend this result to a problem without boundary conditions arising in an economic model introduced by Rochet and Chone in [5]. (C) 2001 Acade mie des sciences/Editions scientifiques et medicales Elsevier SAS.