Uniqueness of Kantorovich potentials?

Question

$\newcommand{\R}{\mathbb R}$Take $\Omega\subset \R^d$ bounded, convex, and smooth. Consider the optimal transport problem with cost $c(x,y)=\lvert x-y\rvert^2$, leading to the quadratic Wassersein distance. Are there known optimal (?) conditions on the probability measures $\mu,\nu\in \mathcal P(\Omega)$ guaranteeing uniqueness (up to additive constants) of the Kantorovich potentials $\varphi$ from $\mu$ to $\nu$?

One standard answer is to require that $\mu$ be absolutely continuous w.r.t the $d$-dimensional Lebesgue measure $\mathcal L^d_{\Omega}$. Indeed a Kantorovich potential $\varphi$ is always Lipschitz (at least in bounded domains) hence differentiable Lebesgue-almost everywhere (Rademacher's theorem), and therefore also $\mu$-almost everywhere. Then one can prove that $y-x=\nabla\varphi(x)$ for any $(x,y)$ in the support of any optimal plan, which gives uniqueness of both the plans and potentials (and as a byproduct that $T(x)=x+\nabla\varphi(x)$ is the optimal Monge's map s.t. $T_\#\mu=\nu$). This is well explained e.g. in Filippo Santambrogio's book [1, theorem 1.17 pp. 15].

However, for my specific purpose I'm interested in a case where $\mu=f_\mu \mathcal L^d\rvert_\Omega+g_\mu\mathcal L^{d-1}\rvert_{\partial\Omega}$ and $\nu=f_\nu \mathcal L^d\rvert_\Omega+g_\nu\mathcal L^{d-1}\rvert_{\partial\Omega}$ typically have a nice smooth absolutely continuous part in the interior $\Omega$ but also a $(d-1)$-dimensional component on the boundary ($\mathcal L^k$ denotes here the $k$-dimensional Lebesgue measure). The above sufficient condition obviously fails, and I am left wondering what can be said? I still expect somehow that the Kantorovich potential (which always exists) is unique, but I have no real clue how to proceed and I was hoping someone out there would have a "black-box reference" that I could use before reinventing the wheel.

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[1] Santambrogio, Filippo, Optimal transport for applied mathematicians. Calculus of variations, PDEs, and modeling, Progress in Nonlinear Differential Equations and Their Applications 87. Cham: Birkhäuser/Springer (ISBN 978-3-319-20827-5/hbk; 978-3-319-20828-2/ebook). xxvii, 353 p. (2015). ZBL1401.49002.

Answer 1

Uniqueness of Kantorovich potentials (up to a constant shift) has been analyzed in a very general framework in this work of ours: "On the Uniqueness of Kantorovich Potentials" - https://arxiv.org/pdf/2201.08316.pdf .

Your setting is encompassed in Corollary 2. The key idea is that by optimality, the gradient of the Kantorovich potential is uniquely determined on a subset of $\text{int}(\Omega)$ with full Lebesgue measure. Since for your setting, the Kantorovich potential is locally Lipschitz on $\Omega$ it follows that it is uniquely characterized on the (connected) domain $\Omega$. Let me emphasize that there is no need for $\mu$ or $\nu$ to be absolutely continuous, uniqueness rather depends on the topology of the support of the underlying measures.

Our work also provides sufficient conditions for Kantorovich potentials to be unique if the measures have disconnected support. This could be helpful to extend the scope of your work.