Tonelli's Theorem (functional Analysis) - Statement of The Theorem

Statement of The Theorem

Let Ω be an open domain in n-dimensional Euclidean space Rn and let f : Rm → R ∪ {±∞} be a continuous extended real-valued function. Define a nonlinear functional F on functions u : Ω → Rm by

Then F is sequentially weakly lower semicontinuous on the Lp space Lp(Ω; Rm) for 1 < p < +∞ and weakly-∗ lower semicontinuous on L∞(Ω; Rm) if and only if the function f

is convex.

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