@leomonsaingeon Thank you! And obviously, they are also uniformly bounded with a bound which depends only on $c$, right?

@leomonsaingeon You said "Indeed a Kantorovich potential $\varphi$ is always Lipschitz (at least in bounded domains)". Why? Can you tell me a reference where to find such a result? Thank you

Yes, i mean that. Of course it should be true, but I cannot prove them. In particular, i can compute all the derivatives but I cannot turn out to an equality between $\partial_tm$ and $-\operatorname{div}(\beta m)$. Btw, thank you anyway.

Thank you for replying. Yes, indeed i proved the claim $(\star)$ just using the weak formulation. But i was asking myself if we can obtain $(\star)$ via $(\triangle)$, but maybe i'm wrong..

Thank you. Could not be sufficient that $\pi\in C^1$ and $\|\nabla\pi\|>0$?

@PierrePC I posted the question both here and in math.SE. You're right, the answers in math.SE are rather detailed but they were posted after my question on mathoverflow. Anyway, your comments were useful for me! Thank you!

@erz I don't know, maybe yes. But on the set $K$ too. I was asking if there are special cases in which the condition $(\star)$ holds.