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Perturbed minimizing movements of families of functionals

  • * Corresponding author: Andrea Braides

    * Corresponding author: Andrea Braides 

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

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  • 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.

    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 $\gamma_1^\alpha$

  • [1] R. AlicandroA. Braides and M. Cicalese, Phase and anti-phase boundaries in binary discrete systems: A variational viewpoint, Netw. Heterog. Media, 1 (2006), 85-107.  doi: 10.3934/nhm.2006.1.85.
    [2] F. Almgren and J. E. Taylor, Flat flow is motion by mean curvature for curves with crystalline energy, J. Differential Geom., 42 (1995), 1-22.  doi: 10.4310/jdg/1214457030.
    [3] F. AlmgrenJ. L. Taylor and L. Wang, Curvature-driven flows: A variational approach, SIAM J. Control Optim., 31 (1993), 387-438.  doi: 10.1137/0331020.
    [4] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, $2^{nd}$ edition, Birkhäuser, Basel, 2008.
    [5] N. AnsiniA. Braides and J. Zimmer, Minimizing movements for oscillating energies: The critical regime, Proc. Royal Soc. Edin. A, 149 (2019), 719-737.  doi: 10.1017/prm.2018.46.
    [6] A. Braides$\Gamma$-convergence for Beginners, Oxford University Press, Oxford, 2002.  doi: 10.1093/acprof:oso/9780198507840.001.0001.
    [7] A. Braides, Local Minimization, Variational Evolution and $\Gamma$-convergence, Springer, Cham, 2014. doi: 10.1007/978-3-319-01982-6.
    [8] A. BraidesM. ColomboM. Gobbino and M. Solci, Minimizing movements along a sequence of functionals and curves of maximal slope, C. R. Math. Acad. Sci. Paris, 354 (2005), 685-689.  doi: 10.1016/j.crma.2016.04.011.
    [9] A. BraidesM. S. Gelli and M. Novaga, Motion and pinning of discrete interfaces, Arch. Ration. Mech. Anal., 195 (2010), 469-498.  doi: 10.1007/s00205-009-0215-z.
    [10] E. De Giorgi, New problems on minimizing movements, in Boundary Value Problems for Partial Differential Equations and Applications (C. Baiocchi and J. L. Lions, eds.) Masson, Paris, 29 (1993), 81–98.
    [11] P. Dondl, T. Frenzel and A. Mielke, A gradient system with a wiggly energy and relaxed EDP-convergence,, ESAIM Control Optim. Calc. Var., 25 (2019), Art. 68, 45 pp. doi: 10.1051/cocv/2018058.
    [12] F. Fleissner, $\Gamma$-convergence and relaxation of gradient flows in metric spaces: A minimizing movement approach,, ESAIM Control Optim. Calc. Var., 25 (2019), Art. 28, 29pp. doi: 10.1051/cocv/2017035.
    [13] F. Fleissner and G. Savaré, Reverse approximation of gradient flows as minimizing movements: A conjecture by de Giorgi, Ann. Sc. Norm. Super. Pisa Cl. Sci., in press, (2017), 1–30.
    [14] E. Sandier and S. Serfaty, $\Gamma$-convergence of gradient flows with applications to Ginzburg-Landau, Commun. Pure Appl. Math., 57 (2004), 1627-1672.  doi: 10.1002/cpa.20046.
    [15] A. Tribuzio, Perturbations of minimizing movements and curves of maximal slope, Netw. Heterog. Media, 13 (2018), 423-448.  doi: 10.3934/nhm.2018019.
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