Calculus of variations and descriptive set theory

✍ Scribed by Nikolaos E. Sofronidis

• Publisher: John Wiley and Sons
• Year: 2009
• Tongue: English
• Weight: 89 KB
• Volume: 55
• Category: Article
• ISSN: 0044-3050
• DOI: 10.1002/malq.200810003

✦ Synopsis

Abstract

If 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 ∈ C^r^ ([α, β ], [a, b ]) such that for any v ∈ C^r^ ([α, β ], [a, b ]) we have that ∫~α~^β^ f (x, u (x), v ′(x))d__x__ ≥ ∫~α~^β^ f (x, v (x), v (x))d__x__ is not Borel (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)