Calculus of Variations and Geometric Measure Theory

A. Braides - A. Tribuzio

Perturbed minimizing movements of families of functionals

created by braidesa on 31 Mar 2019
modified on 08 Dec 2020


Published Paper

Inserted: 31 mar 2019
Last Updated: 8 dec 2020

Journal: Discrete and Continuous Dynamical Systems Series S
Volume: 14
Number: 1
Pages: 373-393
Year: 2021

ArXiv: 1910.03260 PDF
Links: link to the paper on the journal site


We consider the well-known minimizing-movement approach to the definition of a solution of gradient-flow type equations by means of an implicit Euler scheme depending on an energy and a dissipation term. We perturb the energy by considering a ($\Gamma$-converging) sequence and the dissipation by varying multiplicative terms. The scheme depends on two small parameters $\varepsilon$ and $\tau$, governing energy and time scales, respectively. We characterize the extreme cases when $\varepsilon/\tau$ and $\tau/\varepsilon$ converges to $0$ sufficiently fast, and exhibit a sufficient condition that guarantees that the limit is indeed independent of $\varepsilon$ and $\tau$. We give examples showing that this in general is not the case, and apply this approach to study some discrete approximations, the homogenization of wiggly energies and geometric crystalline flows obtained as limits of ferromagnetic energies.

Keywords: Homogenization, Gamma-convergence, Gradient Flow, minimizing movements, Variational evolution