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

Abstract

We consider the Cauchy problem for the gradient flow u ′ (t) = −∇φ(u(t)), t ≥ 0; u(0) = u 0 , (⋆) generated by a continuously differentiable function φ : H → R 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 φ is quadratically bounded from below (in particular if φ is Lipschitz) then for every solution u to (⋆) (which may have an infinite number of solutions) there exist perturbations φτ : H → R (τ > 0) converging to φ in the Lipschitz norm such that u can be approximated by the Minimizing Movement scheme generated by the recursive minimization of Φ(τ, U, V ) :We show that the piecewise constant interpolations with time step τ > 0 of all possible selections of solutions (U n τ ) n∈N to (⋆⋆) will converge to u as τ ↓ 0. This result solves a question raised by Ennio De Giorgi in [9].We also show that even if H has infinite dimension the above approximation holds for the distinguished class of minimal solutions to (⋆), that generate all the other solutions to (⋆) by time reparametrization.

Perturbations of minimizing movements and curves of maximal slope

“…These perturbations may also be seen as a variation of the functional, considering φ(u)/a τ n in the minimization problem. For the interested readers we suggest the work by Fleissner and Savaré [11].…”

A minimizing Movement approach to a class of scalar reaction–diffusion equations

“…The necessary condition of first order leads to a discrete version of the classical gradient flow equation u (t) = −∇E(u(t)), t ≥ 0, (1.3) and indeed, it is not difficult to see that every u ∈ GMM(Φ; u 0 ) (which is a nonempty set) is a solution to (1.3) with initial datum u 0 ∈ H. The statement also holds good if we replace E by E τ in Φ, with E τ : H → R converging to E in the Lipschitz semi-norm as τ ↓ 0. Conversely, for every solution u ∈ C 1 ([0, +∞); H) to (1.3) there exist functions E τ : H → R (τ > 0) such that Lip[E τ − E] → 0 as τ ↓ 0 and MM(Φ; u(0)) = {u} = GMM(Φ; u(0)) for Φ(τ, v, x) := E τ (x) + 1 2τ |x − v| 2 , see [12]. This gives a full characterization of solutions to (1.3) as (Generalized) Minimizing Movements.…”

Perturbed minimizing movements of families of functionals

“…In the present paper we offer a contribution to the general problem of understanding the interaction between energy and dissipation terms in variational approaches to gradient-flow type evolutions from the standpoint of minimizing movements (see also e.g. [11,12,13] for related work).…”