Minimisation de fonctionnelles dans un ensemble de fonctions convexes [Minimizing functionals on a set of convex functions]

We investigate the minima of functionals of the form ¿gWƒ(u), where O 2 is a bounded domain and ƒ a smooth function. The admissible functions are convex and satisfy on O, where and are fixed functions on O. An important example is the problem of the body of least resistance formulated by Newton (see [2]). If ƒ is convex or concave, we show that the minimum is attained by either or if these functions are equal on ¿O. In the case where ƒ is nonconvex, we prove that any minimizer u has a special structure in the region where it is different from and : in any open set where u is differentiable, u is not strictly convex. Convex functions with this property are ‘rare’ in the sense of Baire (see [8]). A consequence of this result is that the radial minimizer calculated by Newton does not attain the global minimum for this problem.