An Aronson-B\'enilan / Li-Yau estimate in the JKO scheme in small dimension

We derive an Aronson-B\'enilan / Li-Yau estimate in the JKO scheme associated to the porous-medium, heat, and fast-diffusion equations, in dimensions $1$ and $2$, and on simple domains (cubes, quarter-space, half-spaces, whole space, and the torus). Our method is based on a maximum principle for the determinant of the Hessian of Brenier potentials, iterated as a one-step improvement along the scheme. As a consequence, we obtain local $L^\infty$ bounds on the density, uniform in the time step, consistent with the continuous-time result. As a byproduct, we rigorously derive the optimality conditions in the fast-diffusion case, filling a gap in the literature.