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On Riemann integration with respect to a continuous increment

✍ Scribed by Rosalind Cecily Young

Publisher
Springer-Verlag
Year
1929
Tongue
French
Weight
809 KB
Volume
29
Category
Article
ISSN
0025-5874

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Let Ω βŠ‚ R N be a bounded open set and let g: Ω Γ— R β†’ R be a CarathΓ©odory function that satisfies standard growth conditions. Then the functional Ξ¦(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.