Professor of Mathematics, University of British Columbia

Details

We expose and exploit a surprising relationship between elliptic gradient systems of PDEs and a multi-marginal Monge-Kantorovich optimal transport problem. We show that the notion of an "orientable" elliptic system (Fazly-Ghoussoub) conjectured to imply that stable solutions are essentially 1-dimensional, is equivalent to the definition of a "compatible" cost function (Carlier-Pass), known to imply uniqueness and structural results for optimal measures to certain Monge-Kantorovich problems. We use this equivalence to show that solutions to these elliptic PDEs, with appropriate monotonicity properties, are related to optimal measures in the Monge-Kantorovich problem. We also prove a decoupling result for solutions to elliptic PDEs and show that under the orientability condition, the decoupling has additional properties, due to the connection to optimal transport.