Question

Assume that $u$ is a function in some domain $\Omega$ in $\mathbb{R}^d$ satisfying restrictions like $u(0)=0$ and $\nabla u\in P$ in any point, where $P$ is a given polytope (for example, constraints are $\partial u/\partial x_i\in [0,1]$ and $\sum \partial u/\partial x_i\leq 2$ or like so). We have to minimize the integral functional like $\int_{\Omega} \langle\nabla u(x),F(x)\rangle dx$ with some function $F$, which may change sign and so on. A priori minimum should be attained in an extreme point of the convex set of our admissible functions $u$. But is there any nice description of the set of extreme points? Clearly, in any inner point of $\Omega$ the gradient must lie on the boundary of $P$, because else we may vary $u$ by a small function supported in the small neighborhood of such a point. But this looks like not sufficient at all, because one equality like $\partial u/\partial x_1=0$ still leaves many degrees of freedom for possible variations, the problem is that they are no longer local.