%0 Journal Article
%T Decoupling of DeGiorgi-type systems via multi-marginal optimal transport
%A Nassif Ghoussoub
%A Brendan Pass
%J Mathematics
%D 2013
%I arXiv
%X 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 in [6] to imply that (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 [11]. Orientable nonlinearities and compatible cost functions turned out to be also related to submodular 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.
%U http://arxiv.org/abs/1302.3280v2