2017/06/19 by Codina Cotar, Cotar, Codina, Mircea Petrache +1 · 1 citation
Mathematics · Engineering · #Nonlinear Partial Differential Equations #Nuclear reactor physics and engineering #Markov Chains and Monte Carlo Methods
paper · pdf · doi:10.48550/arxiv.1706.06008
Motivated by a problem arising from Density Functional Theory, we provide the\nsharp next-order asymptotics for a class of multimarginal optimal transport\nproblems with cost given by singular, long-range pairwise interaction\npotentials. More precisely, we consider an N-marginal optimal transport\nproblem with N equal marginals supported on mathbb Rd and with cost of\nthe form \∑i\≠ j|xi-xj|-s. In this setting we determine the\nsecond-order term in the N\→\∞ asymptotic expansion of the minimum\nenergy, for the long-range interactions corresponding to all exponents 0<s<d.\nWe also prove a small oscillations property for this second-order energy term.\nOur results can be extended to a larger class of models than power-law-type\nradial costs, such as non-rotationally-invariant costs. The key ingredient and\nmain novelty in our proofs is a robust extension and simplification of the\nFefferman-Gregg decomposition (Fefferman 1985, Gregg 1989), extended here to\nour class of kernels, and which provides a unified method valid across our full\nrange of exponents. Our first result generalizes a recent work of Lewin, Lieb\nand Seiringer (2017), who dealt with the second-order term for the Coulomb case\ns=1,d=3, by different methods.\n