Extremal Points of a Functional on the Set of Convex Functions
نویسندگان
چکیده
We investigate the extremal points of a functional ∫ f(∇u), for a convex or concave function f . The admissible functions u : Ω ⊂ RN → R are convex themselves and satisfy a condition u2 ≤ u ≤ u1. We show that the extremal points are exactly u1 and u2 if these functions are convex and coincide on the boundary ∂Ω. No explicit regularity condition is imposed on f , u1, or u2. Subsequently we discuss a number of extensions, such as the case when u1 or u2 are non-convex or do not coincide on the boundary, when the function f also depends on u, etc.
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