Perturbed minimizing movements of families of functionals

Andrea Braides; Antonio Tribuzio

doi:10.3934/dcdss.2020324

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2021, Volume 14, Issue 1: 373-393. Doi: 10.3934/dcdss.2020324

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Perturbed minimizing movements of families of functionals

Andrea Braides^, and
Antonio Tribuzio

Department of Mathematics, University of Rome Tor Vergata, via della Ricerca Scientifica, 00133 Rome, Italy

^* Corresponding author: Andrea Braides
^* Corresponding author: Andrea Braides

Dedicated to Alexander Mielke on the occasion of his 60th birthday

Received: February 28, 2019

Revised: September 30, 2019

Early access: April 2020

Published: January 2021

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Abstract

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.

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Mathematics Subject Classification: Primary: 47J30, 35K90, 49J45; Secondary: 47J35, 35B27.

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Figure 1. The dark line represents the graph of $\gamma\mapsto1/a^\gamma$, the light line is the constant $1/a^*$. On the left $\alpha>\beta/2$ so the sup is reached in $\gamma_1^\beta$, on the right $\alpha < \beta/2$ and the sup is reached in $\gamma_1^\alpha$

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