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Motivated by a problem arising from Density Functional Theory, we provide thesharp next-order asymptotics for a class of multimarginal optimal transportproblems with cost given by singular, long-range pairwise interactionpotentials. More precisely, we consider an $N$-marginal optimal transportproblem with $N$ equal marginals supported on $\mathbb R^d$ and with cost ofthe form $\sum_{i\neq j}|x_i-x_j|^{-s}$. In this setting we determine thesecond-order term in the $N\to\infty$ asymptotic expansion of the minimumenergy, for the long-range interactions corresponding to all exponents $0<s<d$.We also prove a small oscillations property for this second-order energy term.Our results can be extended to a larger class of models than power-law-typeradial costs, such as non-rotationally-invariant costs. The key ingredient andmain novelty in our proofs is a robust extension and simplification of theFefferman-Gregg decomposition (Fefferman 1985, Gregg 1989), extended here toour class of kernels, and which provides a unified method valid across our fullrange of exponents. Our first result generalizes a recent work of Lewin, Lieband Seiringer (2017), who dealt with the second-order term for the Coulomb case$s=1,d=3$, by different methods.

Next-order asymptotic expansion for N-marginal optimal transport with Coulomb and Riesz costs