  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.
