Calculus of variations and descriptive set theory

Abstract

AbstractIf X is a locally compact Polish space, then LSC(X, ℝ) denotes the compact Polish space of lower semi‐continuous real‐valued functions on X equipped with the topology of epi‐convergence.Our purpose in this article is to prove the following: if –∞ < α < β < ∞ and –∞ < a < b < ∞, while r ∈ ℕ \ {0}, then the set CV of all f ∈ LSC([α, β ] × [a, b ] × ℝ, ℝ) for which there is u ∈ Cr ([α, β ], [a, b ]) such that for any v ∈ Cr ([α, β ], [a, b ]) we have that ∫αβ f (x, u (x), v ′(x))dx ≥ ∫αβ f (x, v (x), v (x))dx is not Borel (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)