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Journal of Convex Analysis 11 (2004), No. 2, 401--411
Copyright Heldermann Verlag 2004

A Necessary and Sufficient Optimality Condition for a Class of Nonconvex Scalar Variational Problems

Guillaume Carlier
Universite Bordeaux IV, GRAPE, UMR CNRS 5113, Avenue Leon Duguit, 33608 Pessac, France,


This article studies the minimization of the functional $$ u\mapsto\int_{0}^{1}f(\dot{u}) $$ among all convex functions $u$ that satisfy the additional obstacle constraint $u\geq \ovu$, $u(0)=\ovu(0)$, $u(1)=\ovu(1)$ where $\ovu$ is a given convex function. We first show that this nonconvex problem is in fact equivalent to a linear programming problem. This enables us to establish a necessary and sufficient optimality condition.

Keywords: convexity constraint, monotone rearrangements, duality.

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