Extremal problems related to convexity.

We consider the extremal problem of maximizing functions u in the class of real-valued biconvex functions satisfying a boundary condition ψ on a product of the unit ball with itself, with the $\ell^{p}$ -norm. In 1986, Burkholder explicitly found the maximal function for $p=2$ . In this paper, we find some characterizations of such extremal functions. We establish that sufficiently smooth solutions to the convex extremal problems with given boundary values are affine on line segments and the domain D is foliated by such segments. [ABSTRACT FROM AUTHOR]