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We study the JKO scheme for the total variation, characterize the optimizers,prove some of their qualitative properties (in particular a form of maximumprinciple and in some cases, a minimum principle as well). Finally, weestablish a convergence result as the time step goes to zero to a solution of afourth-order nonlinear evolution equation, under the additional assumption thatthe density remains bounded away from zero. This lower bound is shown indimension one and in the radially symmetric case.

On the total variation Wasserstein gradient flow and the TV-JKO scheme