Applied Mathematics Seminar

Abstract: We exhibit a surprising relationship between elliptic gradient systems of PDEs, multi-marginal Monge-Kantorovich optimal transport problem, and multivariable Hardy-Littlewood inequalities. We show that the notion of an "orientable" elliptic system, conjectured to imply that --at least in low dimensions-- solutions with certain monotonicity properties are essentially $1$-dimensional, is equivalent to the definition of a "compatible" cost function, known to imply uniqueness and structural results for optimal measures to certain Monge-Kantorovich problems. Orientable nonlinearities and compatible cost functions turn out to be also related to "sub-modular" functions, which appear in rearrangement inequalities of Hardy-Littlewood type.
We use this equivalence to establish a decoupling result for certain solutions to elliptic PDEs and show that under the orientability condition, the decoupling has additional properties, due to the connection to optimal transport.

Bio: Nassif Ghoussoub obtained his Doctorat d'état in 1979 from the Université Pierre et Marie Curie in Paris, France. His present research interests are in non-linear analysis and partial differential equations. He is currently a Professor of Mathematics, a "Distinguished University Scholar", and an elected member of the Board of Governors of the University of British Columbia. He was the founding Director of PIMS (Pacific Institute for the Mathematical Sciences), a co-founder of the MITACS Network of Centres of Excellence (Mathematics of Information Technology and Complex Systems) and a member of its Board of Directors for the periods. He is also the founder of BIRS (Banff International Research Station) and its Scientific Director. In 2011, he became the Scientific Director of the MPrime network of Centres of Excellence.