Reverse approximation of gradient flows as Minimizing Movements: a conjecture by De Giorgi

We consider the Cauchy problem for a gradient flow generated by a continuously differentiable function phi in a Hilbert space H and study the reverse approximation of solutions to by the De Giorgi Minimizing Movement approach. We prove that if H has finite dimension and phi is quadratically bounded from below (in particular if phi is Lipschitz) then for every solution u there exist perturbations converging to phi in the Lipschitz norm such that u can be approximated by the Minimizing Movement scheme generated by the recursive minimization associated to the perturbations. We show that the piecewise constant interpolations will converge to u as the time step vanishes. This result solves a question raised by Ennio De Giorgi We also show that even if H has infinite dimension the above approximation holds for the distinguished class of minimal solutions, that generate all the other solutions to by time reparametrization.