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Reverse approximation of gradient flows as Minimizing Movements

Banff International Research Station for Mathematical Innovation and Discovery

The Minimizing Movement (MM) scheme is a variational method introduced by E. De Giorgi to solve gradient flows in a quite general setting. In finite dimensional Euclidean spaces, when the driving function f is continuously differentiable, it is not difficult to see that all the limit curves are solutions to the ODE system generated by the gradient of f. However, since this vector field is only continuous, solutions may be not unique and there are solutions which cannot be obtained as a direct limit of the MM scheme. In his inspiring 1993 paper â New problems on Minimizing Movementsâ , De Giorgi raised the conjecture that all the solutions can be obtained as limit of a modifed scheme, obtained by a Lipschitz perturbation of f, converging to f in the Lipschitz norm as the time step goes to 0. We present some ideas of the proof of this conjecture related to nonsmooth calculus and to a new distinguished class of solutions to gradient flows. We will also discuss a partial extension of this result to infinite dimensional Hilbert spaces. In collaboration with Florentine Fleissner.

Mathematics; Calculus of variations and optimal control; optimization; Partial differential equations

video/mp4

Author affiliation: University of Pavia

2019-03-21

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/69006

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/