Decoupling of DeGiorgi-type systems via multi-marginal optimal transport

TL;DR

This paper reveals a surprising link between elliptic PDE systems, multi-marginal optimal transport, and Hardy-Littlewood inequalities, leading to a decoupling result for solutions under certain conditions.

Contribution

It establishes an equivalence between orientable elliptic systems and compatible cost functions in optimal transport, enabling new decoupling results for PDE solutions.

Findings

Equivalence between orientable elliptic systems and compatible cost functions.

Decoupling of solutions to elliptic PDEs under orientability.

Connection of submodular functions to rearrangement inequalities.

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 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.