Extremal points of a functional on the set of convex functions

Abstract:

We investigate the extremal points of a functional $\int f(\nabla u)$, for a convex or concave function $f$. The admissible functions $u:\Omega \subset \mathbf {R}^N\to \mathbf {R}$ are convex themselves and satisfy a condition $u_2\leq u \leq u_1$. We show that the extremal points are exactly $u_1$ and $u_2$ if these functions are convex and coincide on the boundary $\partial \Omega$. No explicit regularity condition is imposed on $f$, $u_1$, or $u_2$. Subsequently we discuss a number of extensions, such as the case when $u_1$ or $u_2$ are non-convex or do not coincide on the boundary, when the function $f$ also depends on $u$, etc.