MINIMIZING FUNCTIONALS ON A SET OF CONVEX -FUNCTIONS

We investigate the minima of functionals of the form integral(Omega) f (del u), where Omega subset of R-2 is a bounded domain anti f a smooth function. The admissible functions u : <(Omega)over bar> --> R are co nvex and satisfy (u) under bar less than or equal to u less than or eq ual to (u) over bar on Omega, where (u) under bar and (u) over bar are fixed functions on Omega. An important example is the problem of the body of least resistance formulated by Newton (see [2]). If f is conve x or concave, we show that the minimum is attained by either (u) under bar or (u) over bar if these functions are equal on partial derivativ e>Omega. Irt the case where f is nonconvex, we pr-eve that any minimiz er u, has a special structure in the region where it is different from (u) under bar and (u) over bar: in any open set where u is differenti able, u is not strictly convex. Convex functions with this property ar e '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 far this problem.