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The General Optimal L^pEuclidean logarithmic Sobolev inequality by HamiltonJacobi equations
, 2002
"... We prove a general optimal L^pEuclidean logarithmic Sobolev inequality by using PrekopaLeindler inequality and a special HamiltonJacobi equation. In particular we generalize the inequality proved by Del Pino and Dolbeault in (DPD02a). ..."
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Cited by 13 (4 self)
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We prove a general optimal L^pEuclidean logarithmic Sobolev inequality by using PrekopaLeindler inequality and a special HamiltonJacobi equation. In particular we generalize the inequality proved by Del Pino and Dolbeault in (DPD02a).
Rearrangements of vector valued functions, with application to atmospheric and oceanic ows
"... This paper establishes the equivalence of four de nitions of two vector valued functions being rearrangements, and gives a characterisation of the set of rearrangements of a prescribed function. The theory of monotone rearrangement ofavector valued function is used to show the existence and uniquene ..."
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Cited by 5 (2 self)
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This paper establishes the equivalence of four de nitions of two vector valued functions being rearrangements, and gives a characterisation of the set of rearrangements of a prescribed function. The theory of monotone rearrangement ofavector valued function is used to show the existence and uniqueness of the minimiser of an energy functional arising from a model for atmospheric and oceanic ow. At each xed time solutions are shown to be equal to the gradient of a convex function, verifying the conjecture of Cullen, Norbury and Purser. Key words Rearrangement of functions, semigeostrophic, variational problems, generalised solution.
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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Cited by 3 (0 self)
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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.