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Uniqueness of the Polar Factorisation and Projection of a VectorValued Mapping
"... This paper proves some results concerning the polar factorisation of an integrable vectorvalued function u into the composition u = u s, where u is equal almost everywhere to the gradient of a convex function, and s is a measurepreserving mapping. It is shown that the factorisation is un ..."
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This paper proves some results concerning the polar factorisation of an integrable vectorvalued function u into the composition u = u s, where u is equal almost everywhere to the gradient of a convex function, and s is a measurepreserving mapping. It is shown that the factorisation is unique (i.e. the measurepreserving mapping s is unique) precisely when u is almost injective. Not every integrable function has a polar factorisation; we introduce a class of counterexamples. It is further
Uniqueness of the polar factorisation and projection of a vectorvalued mapping
"... abstract. This paper proves some results concerning the polar factorisation of an integrable vectorvalued function u into the composition u = u # ◦ s, where u # is equal almost everywhere to the gradient of a convex function, and s is a measurepreserving mapping. It is shown that the factorisati ..."
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abstract. This paper proves some results concerning the polar factorisation of an integrable vectorvalued function u into the composition u = u # ◦ s, where u # is equal almost everywhere to the gradient of a convex function, and s is a measurepreserving mapping. It is shown that the factorisation is unique (i.e. the measurepreserving mapping s is unique) precisely when u # is almost injective. Not every integrable function has a polar factorisation; we introduce a class of counterexamples. It is further shown that if u is square integrable, then measurepreserving mappings s which satisfy u = u # ◦s are exactly those, if any, which are closest to u in the L2norm.
vectorvalued
"... Nonexistence of polar factorisations and polar inclusion of a ..."
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