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UBC Theses and Dissertations

On a polar factorization theorem Millien, Pierre


We study the link between two different factorization theorems and their proofs : Brenier's Theorem which states that for any u ∈L^p(Ω), where Ω is a bounded domain in R ^d and 1 ≤ p ≤ ∞, u can be written as u=∇ ɸ ο s where ɸ is a convex function, and s a measure preserving transformation, and on the other hand Ghoussoub and Moameni's theorem which states that for any u ∈ L∞ (Ω), u(x) = ∇₁H(S(x),x), where H is a convex concave anti-symmetric function, and S is a measure preserving involution. In a second time we prove that Ghoussoub and Moameni's theorem is true in L², and find the decomposition for particular example : u(x) = |x-1/2|.

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