Regularity of solutions for some variational problems subject to a convexity constraint

We first study the minimizers, in the class of convex functions, of an elliptic functional with nonhomogeneous Dirichlet boundary conditions. We prove C¹ regularity of the minimizers under the assumption that the upper envelope of admissible functions is C¹. This condition is optimal at least when the functional depends only on the gradient [3]. We then give various extensions of this result. In Particular, we consider a problem without boundary conditions arising in an economic model introduced by Rochet and Choné in [4]. © 2001 John Wiley & Sons, Inc.