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Remarks on the semi-classical Hohenberg-Kohn functional

In this note, we study an optimal transportation problem arising in density functional theory. We derive an upper bound on the semi-classical Hohenberg-Kohn functional derived by Cotar, Friesecke and Klüppelberg (2012) which can be computed in a straightforward way for a given single particle density. This complements a lower bound derived by the aforementioned authors. We also show that for radially symmetric densities the optimal transportation problem arising in the semi-classical Hohenberg-Kohn functional can be reduced to a 1-dimensional problem. This yields a simple new proof of the explicit solution to the optimal transport problem for two particles found by Cotar, Friesecke and Klüppelberg (2012). For more particles, we use our result to demonstrate two new qualitative facts: first, that the solution can concentrate on higher dimensional submanifolds and second that the solution can be non-unique, even with an additional symmetry constraint imposed.