A numerical approach to variational problems subject to convexity constraint

Summary.

We describe an algorithm to approximate the minimizer of an elliptic functional in the form $\int_\Omega j(x, u, \nabla u)$ on the set ${\cal C}$ 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 $u_0$. Let $(T_n)$ be any quasiuniform sequence of meshes whose diameter goes to zero, and $I_n$ the corresponding affine interpolation operators. We prove that the minimizer over ${\cal C}$ is the limit of the sequence $(u_n)$, where $u_n$ minimizes the functional over $I_n({\cal C})$. We give an implementable characterization of $I_n({\cal C})$. Then the finite dimensional problem turns out to be a minimization problem with linear constraints.