Integral functionals that are continuous with respect to the weak topology on

✍ Scribed by Robert Černý; Stanislav Hencl; Jan Kolář

• Publisher: Elsevier Science
• Year: 2009
• Tongue: English
• Weight: 685 KB
• Volume: 71
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2009.01.117

✦ Synopsis

Let Ω ⊂ R N be a bounded open set and let g: Ω × R → R be a Carathéodory function that satisfies standard growth conditions. Then the functional Φ(u) = Ω g (x, u(x)) dx is weakly continuous on W 1,p 0 (Ω), 1 ≤ p ≤ ∞, if and only if g is linear in the second variable.