= Solution
A standard sufficient form of <Brenier theorem> assumes $\mu,\nu\in\mathcal P(\mathbb R^d)$ have finite second <moments> and $\mu\ll\mathcal L^d$, that is, <absolute continuity of measures> with respect to <Lebesgue measure>. For the cost $|x-y|^2$, there exists a unique optimal <transport plan>, and it is induced by a <transport map>:
$$
\boxed{\pi^\dagger=(\operatorname{Id},\nabla\Phi)_\#\mu,\qquad T^\dagger=\nabla\Phi,\qquad(\nabla\Phi)_\#\mu=\nu.}
$$
Here $\Phi$ is a <proper convex function>, which may be chosen <sequentially lower semicontinuous>, and its <gradient> exists $\mu$-almost everywhere. The map $\nabla\Phi$ is unique $\mu$-almost everywhere and is the unique minimizer of the <Monge optimal transport problem>; its cost equals the <Kantorovich optimal transport problem> minimum. Equivalently, it is the unique <gradient> of a <convex function> transporting $\mu$ to $\nu$.
The uniqueness claim concerns the map and the <transport plan>, not a globally unique potential. The potential may be shifted by a constant, and additional nonuniqueness away from the source can occur. No density assumption is required on $\nu$. A primary reference is https://www.ceremade.dauphine.fr/~carlier/Brenier91.pdf[Brenier's Polar factorization and monotone rearrangement of vector-valued functions].
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