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September  2018, 13(3): 423-448. doi: 10.3934/nhm.2018019

## Perturbations of minimizing movements and curves of maximal slope

 Via della Ricerca Scientifica 1, Rome, 00133, Italy

Received  October 2017 Revised  April 2018 Published  July 2018

We modify the De Giorgi's minimizing movements scheme for a functional $φ$, 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 for $φ$ 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.

Citation: Antonio Tribuzio. Perturbations of minimizing movements and curves of maximal slope. Networks & Heterogeneous Media, 2018, 13 (3) : 423-448. doi: 10.3934/nhm.2018019
##### References:
 [1] N. Ansini, Gradient flows with wiggly potential: a variational approach to the dynamics, in Mathematical Analysis of Continuum Mechanics and Industrial Applications Ⅱ, CoMFoS16 (Springer), 30 (2017), 139-151. doi: 10.1007/978-981-10-6283-4_12. [2] N. Ansini, A. Braides and J. Zimmer, Minimising movements for oscillating energies: The critical regime, Proceedings of the Royal Society of Edinburgh Section A (in press), 2016, arXiv: 1605.01885. [3] F. Almgren, J. E. Taylor and L. Wang, Curvature-driven flows: a variational approach, SIAM Journal of Control and Optimization, 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, Birkhauser Verlag, second edition, 2008. [5] A. Braides, Γ-convergence for Beginners, Oxford University Press, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001. [6] A. Braides, Local Minimization, Variational Evolution and Γ-convergence, Springer Cham, 2014. doi: 10.1007/978-3-319-01982-6. [7] A. Braides, M. Colombo, M. Gobbino and M. Solci, Minimizing movements along a sequence of functionals and curves of maximal slope, Comptes Rendus Matematique, 354 (2005), 685-689. doi: 10.1016/j.crma.2016.04.011. [8] M. Colombo and M. Gobbino, Passing to the limit in maximal slope curves: from a regularized Perona-Malik to the total variation flow, Mathematical Models and Methods in Applied Sciences, 22 (2012), 1250017, 19pp. doi: 10.1142/S0218202512500170. [9] L. C. Evans, Partial Differential Equations, American Mathematical Society, second edition, 2010. doi: 10.1090/gsm/019. [10] F. Fleissner, Γ-convergence and relaxations for gradient flows in metric spaces: A minimizing movement approach, ESAIM Control, Optimisation and Calculus of Variations (to appear), preprint, arXiv: 1603.02822, (2016). [11] F. Fleissner and G. Savaré, Reverse approximation of gradient flows as minimizing movements: A conjecture by De Giorgi, Forthcoming Articles, (2018), 30pp, arXiv: 1711.07256. doi: 10.2422/2036-2145.201711_008. [12] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Plank equation, SIAM Journal of Mathematical Analysis, 29 (1998), 1-17. doi: 10.1137/S0036141096303359. [13] E. Sandier and S. Serfaty, Gamma-convergence of gradient flows and application to Ginzburg-Landau, Communications on Pure and Applied Mathematics, 57 (2004), 1627-1672. doi: 10.1002/cpa.20046.

show all references

##### References:
 [1] N. Ansini, Gradient flows with wiggly potential: a variational approach to the dynamics, in Mathematical Analysis of Continuum Mechanics and Industrial Applications Ⅱ, CoMFoS16 (Springer), 30 (2017), 139-151. doi: 10.1007/978-981-10-6283-4_12. [2] N. Ansini, A. Braides and J. Zimmer, Minimising movements for oscillating energies: The critical regime, Proceedings of the Royal Society of Edinburgh Section A (in press), 2016, arXiv: 1605.01885. [3] F. Almgren, J. E. Taylor and L. Wang, Curvature-driven flows: a variational approach, SIAM Journal of Control and Optimization, 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, Birkhauser Verlag, second edition, 2008. [5] A. Braides, Γ-convergence for Beginners, Oxford University Press, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001. [6] A. Braides, Local Minimization, Variational Evolution and Γ-convergence, Springer Cham, 2014. doi: 10.1007/978-3-319-01982-6. [7] A. Braides, M. Colombo, M. Gobbino and M. Solci, Minimizing movements along a sequence of functionals and curves of maximal slope, Comptes Rendus Matematique, 354 (2005), 685-689. doi: 10.1016/j.crma.2016.04.011. [8] M. Colombo and M. Gobbino, Passing to the limit in maximal slope curves: from a regularized Perona-Malik to the total variation flow, Mathematical Models and Methods in Applied Sciences, 22 (2012), 1250017, 19pp. doi: 10.1142/S0218202512500170. [9] L. C. Evans, Partial Differential Equations, American Mathematical Society, second edition, 2010. doi: 10.1090/gsm/019. [10] F. Fleissner, Γ-convergence and relaxations for gradient flows in metric spaces: A minimizing movement approach, ESAIM Control, Optimisation and Calculus of Variations (to appear), preprint, arXiv: 1603.02822, (2016). [11] F. Fleissner and G. Savaré, Reverse approximation of gradient flows as minimizing movements: A conjecture by De Giorgi, Forthcoming Articles, (2018), 30pp, arXiv: 1711.07256. doi: 10.2422/2036-2145.201711_008. [12] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Plank equation, SIAM Journal of Mathematical Analysis, 29 (1998), 1-17. doi: 10.1137/S0036141096303359. [13] E. Sandier and S. Serfaty, Gamma-convergence of gradient flows and application to Ginzburg-Landau, Communications on Pure and Applied Mathematics, 57 (2004), 1627-1672. doi: 10.1002/cpa.20046.
Graphs of the discrete solutions with different values of $\tau$. The smaller jumps of $u^\tau$ correspond to the larger parameter $\beta$
Pinned motion produced by perturbations diverging in the interval $(1, 2)$
The graphs represent two discrete solutions for the same value of $\tau$, corresponding respectively to perturbations as in (25) and (26). Note the discontinuous behavior on the left, while on the right jumps are going to disappear
On the left the time chart of $u^\tau$, on the right the plot of $\phi(u^\tau)$, corresponding to perturbations with $\delta = 1$. Note how the motion exits from the lower energy state when $t = 1$
Graphs as in Figure 4 with $\delta = 8$. It is grater than the critical value $e^2$, indeed when $t = 1$ the motion does not exit the first well
Graphs for $\delta = 8$. As in Figure 5, the motion does not exit the well when $t = 1$, but when $t = 2$ it does
Graph for $\delta = 1$. The motion always passes to the very next potential well
Graphs of a discrete solution passing through two potential wells at every jump discontinuity
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