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

Abstract

We consider the Cauchy problem for the gradient flow \begin{equation} \label{eq:81} \tag{$\star$} u'(t)=-\nablaϕ(u(t)),\quad t\ge 0;\quad u(0)=u_0, \end{equation} generated by a continuously differentiable function $ϕ:\mathbb H \to \mathbb R$ in a Hilbert space $\mathbb H$ and study the reverse approximation of solutions to ($\star$) by the De Giorgi Minimizing Movement approach. We prove that if $\mathbb H$ has finite dimension and $ϕ$ is quadratically bounded from below (in particular if $ϕ$ is Lipschitz) then for every solution $u$ to ($\star$) (which may have an infinite number of solutions) there exist perturbations $ϕ_τ:\mathbb H \to \mathbb R \ (τ>0)$ converging to $ϕ$ in the Lipschitz norm such that $u$ can be approximated by the Minimizing Movement scheme generated by the recursive minimization of $Φ(τ,U,V):=\frac 1{2τ}|V-U|^2+ ϕ_τ(V)$: \begin{equation} \label{eq:abstract} \tag{$\star\star$} U_τ^n\in \operatorname{argmin}_{V\in \mathbb H} Φ(τ,U_τ^{n-1},V)\quad n\in\mathbb N, \quad U_τ^0:=u_0. \end{equation} We show that the piecewise constant interpolations with time step $τ> 0$ of all possible selections of solutions $(U_τ^n)_{n\in\mathbb N}$ to ($\star\star$) will converge to $u$ as $τ\downarrow 0$. This result solves a question raised by Ennio De Giorgi. We also show that even if $\mathbb H$ has infinite dimension the above approximation holds for the distinguished class of minimal solutions to ($\star$), that generate all the other solutions to ($\star$) by time reparametrization.