Inserted: 22 nov 2017
Last Updated: 12 feb 2020
Journal: Annali della Scuola Normale Superiore di Pisa, Classe di Scienze
We consider the Cauchy problem for a gradient ﬂow (GF) generated by a continuously diﬀerentiable function in a Hilbert space H and study the reverse approximation of its solutions by the De Giorgi Minimizing Movement approach.
We prove that if H has ﬁnite dimension and the driving potential is quadratically bounded from below (in particular if it is Lipschitz) then for every solution u of (GF) (which may have an infinite number of solutions) there exist perturbations depending on the time step and converging to the potential in the Lipschitz norm such that u can be approximated by the perturbed Minimizing Movement scheme.
This result solves a question raised by E. De Giorgi, New problems on minimizing movements, in Boundary Value Problems for PDE and Applications, C. Baiocchi and J. L. Lions, eds., Masson, 1993, pp. 81–98.
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 the gradient flow by time reparametrization.
Keywords: Gradient flows, minimizing movements, Nonuniqueness, Reverse approximation