A numerical approach to variational problems subject to convexity constraint

We describe an algorithm to approximate the minimizer of an elliptic functional in the form R Ω j(x, u,∇u) on the set C of convex functions u in an appropriate functional space X. Such problems arise for instance in mathematical economics [4]. A special case gives the convex envelope u∗∗ 0 of a given function u0. Let (Tn) be any quasiuniform sequence of meshes whose diameter goes to zero, and In the corresponding affine interpolation operators. We prove that the minimizer over C is the limit of the sequence (un), where un minimizes the functional over In(C). We give an implementable characterization of In(C). Then the finite dimensional problem turns out to be a minimization problem with linear constraints.ou