Title: Smoothing of transport plans with fixed marginals and rigorous semiclassical limit of the Hohenberg-Kohn functional,
Abstract: We prove rigorously that the exact N-electron Hohenberg-Kohn density
functional converges in the strongly interacting limit to the strictly
correlated electrons (SCE) functional, and that the absolute value squared of
the associated constrained-search wavefunction tends weakly in the sense of
probability measures to a minimizer of the multi-marginal optimal transport
problem with Coulomb cost associated to the SCE functional. This extends our
previous work for N=2 [CFK11]. The correct limit problem has been derived in
the physics literature by Seidl [Se99] and Seidl, Gori-Giorgi and Savin
[SGS07]; in these papers the lack of a rigorous proof was pointed out.
We also give a mathematical counterexample to this type of result, by
replacing the constraint of given one-body density -- an infinite-dimensional
quadratic expression in the wavefunction -- by an infinite-dimensional
quadratic expression in the wavefunction and its gradient. Connections with the
Lawrentiev phenomenon in the calculus of variations are indicated.