a1 Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, U.K.
a2 Department of Mathematics, University of Reading, Whiteknights, P.O. Box 220, Reading RG6 6AX, U.K.
This paper proves some extensions of Brenier's theorem that an integrable vector-valued function u, satisfying a nondegeneracy condition, admits a unique polar factorisation u = u# ° s. Here u# is the monotone rearrangement of u, equal to the gradient of a convex function almost everywhere on a bounded connected open set Y with smooth boundary, and s is a measure-preserving mapping. We show that two weaker alternative hypotheses are sufficient for the existence of the factorisation; that u# be almost injective (in which case s is unique), or that u be countably degenerate (which allows u to have level sets of positive measure). We allow Y to be any set of finite positive Lebesgue measure. Our construction of the measure-preserving map s is especially simple.
(Received November 18 1996)
(Revised July 08 1997)