Given two probability measures $\mu$ and $\nu$, the Kantorovich Dual problem for quadratic cost is to $$ \text{minimize} \quad \int \phi(x)d\mu + \int \psi(y)d\nu $$ over pairs $(\phi,\psi)\in L^1(d\mu)\times L^1(d\nu)$ such that $xy \leq \phi(x) + \psi(y)$. In Villani's book Topics in Optimal Transportation, P71 section 2.1.6, a variational argument is used to derived the Euler-Lagrange Equation for this problem, which turns out to be $$ \nabla \phi_{\#} \mu = \nu. $$ The author also refers to a paper by Gangbo for the derivation.
In both the book and the paper, it is assume that measures $\mu$ and $\nu$ are supported by compact sets. In Gangbo's paper, $\mu$ and $\nu$ are simply the Lebesgue measure.
My question: is there an argument for the general case without any assumption on $\mu$ and $\nu$?
It is possible to weaken the assumptions on $\mu$ and $\nu$ significantly, but it does not seem to be true without some sort of technical assumption on the measures. See Theorem 1.22 in Santambrogio's book, which he describes as "the sharpest result in the unbounded case" - there it is assumed that both measures have finite second moment, and that $\mu$ satisfies a rectifiability assumption which is weaker than absolute continuity.
On the previous page, he cites a paper by Gigli which purports to show that this is, in fact the sharpest result possible.