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