Matrix Hölder's inequality and divergence formulation of optimal transport of vector measures

Abstract

We characterise equality cases in matrix Hölder's inequality and develop a divergence formulation of optimal transport of vector measures. As an application, we reprove the representation formula for measures in the polar cone to monotone maps. We generalise the last result to a wide class of polar cones, including polar cones to tangent cones to the unit ball in the space of differentiable functions and in the Sobolev spaces.

accepted in SIAM Journal on Mathematical Analysis; 27 pages, comments are welcome