Calculus of Variations and Geometric Measure Theory

A. Tribuzio

Perturbations of minimizing movements and curves of maximal slope

created by tribuzio on 13 Jun 2017
modified on 07 Jan 2020


Published Paper

Inserted: 13 jun 2017
Last Updated: 7 jan 2020

Journal: Networks and Heterogeneous Media
Volume: 13
Number: 3
Pages: 423-448
Year: 2017
Doi: 10.3934/nhm.2018019

The topics discussed in this paper are a reworking of those contained in part of my master thesis. Thus, I thank Professor Andrea Braides for his careful help and his ideas that made this work possible.


We modify the De Giorgi's minimizing movements scheme for a functional, by perturbing the dissipation term, and find a condition on the perturbations which ensures the convergence of the scheme to an absolutely continuous perturbed minimizing movement. The perturbations produce a variation of the metric derivative of the minimizing movement. This process is formalized by the introduction of the notion of curve of maximal slope with a given rate. We show that if we relax the condition on the perturbations we may have many different meaningful effects; in particular, some perturbed minimizing movements may explore different potential wells.

Keywords: Gradient flows, minimizing movements, perturbations