G. Carlier et T. Lachand-robert, Regularity of solutions for some variational problems subject to a convexity constraint, COM PA MATH, 54(5), 2001, pp. 583-594
We first study the minimizers, in the class of convex functions, of an elli
ptic functional with nonhomogeneous Dirichlet boundary conditions. We prove
C-1 regularity of the minimizers under the assumption that the upper envel
ope of admissible functions is C-1. 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 introduc
ed by Rochet and Chone in [4]. (C) 2001 John Wiley & Sons, Inc.