Extremal points of a functional on the set of convex functions

Abstract

We investigate the extremal points of a functional ∫ f ( ∇ u ) \int f(\nabla u) , for a convex or concave function f f . The admissible functions u : Ω ⊂ R N → R u:\Omega \subset \mathbf {R}^N\to \mathbf {R} are convex themselves and satisfy a condition u 2 ≤ u ≤ u 1 u_2\leq u \leq u_1 . We show that the extremal points are exactly u 1 u_1 and u 2 u_2 if these functions are convex and coincide on the boundary ∂ Ω \partial \Omega . No explicit regularity condition is imposed on f f , u 1 u_1 , or u 2 u_2 . Subsequently we discuss a number of extensions, such as the case when u 1 u_1 or u 2 u_2 are non-convex or do not coincide on the boundary, when the function f f also depends on u u , etc.