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Minimizing movements for oscillating energies: the critical regime

Part of: Partial differential equations Optimality conditions Existence theories

Published online by Cambridge University Press:  27 December 2018

Nadia Ansini ,
Andrea Braides  and
Johannes Zimmer
Nadia Ansini
Affiliation:
Dept. of Mathematics, Sapienza University of Rome, P.le Aldo Moro 2, 00185 Rome, Italy
Andrea Braides
Affiliation:
Dept. of Mathematics, University of Rome ‘Tor Vergata’, Via della Ricerca Scientifica, 00133 Rome, Italy
Johannes Zimmer
Affiliation:
Dept. of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK (zimmer@maths.bath.ac.uk)
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Abstract

Minimizing movements are investigated for an energy which is the superposition of a convex functional and fast small oscillations. Thus a minimizing movement scheme involves a temporal parameter τ and a spatial parameter ε, with τ describing the time step and the frequency of the oscillations being proportional to 1/ε. The extreme cases of fast time scales τ ≪ ε and slow time scales ε ≪ τ have been investigated in [4]. In this paper, the intermediate (critical) case of finite ratio ε/τ > 0 is studied. It is shown that a pinning threshold exists, with initial data below the threshold being a fixed point of the dynamics. A characterization of the pinning threshold is given. For initial data above the pinning threshold, the equation and velocity describing the homogenized motion are determined.

Keywords

Gradient flowwiggly energyΓ-convergenceminimizing movements

MSC classification

Primary: 35B27: Homogenization; equations in media with periodic structure
Secondary: 49K40: Sensitivity, stability, well-posedness 49J10: Free problems in two or more independent variables 49J45: Methods involving semicontinuity and convergence; relaxation
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 3 , June 2019 , pp. 719 - 737
Copyright
Copyright © Royal Society of Edinburgh 2018 

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References

1Abeyaratne, R., Chu, C. and James, R. D.. Kinetics of materials with wiggly energies: theory and application to the evolution of twinning microstructures in a Cu-Al-Ni shape memory alloy. Philos. Mag. A 73 (1996), 457–497.Google Scholar
2Abeyaratne, R. and Vedantam, S.. Propagation of a front by kink motion from a discrete model to a continuum model. In Variations of domain and free-boundary problems in solid mechanics (Paris, 1997), volume 66 of Solid Mech. Appl., pp. 77–84 (Dordrecht: Kluwer Acad. Publ., 1999).Google Scholar
3Ambrosio, L., Gigli, N. and Savaré, G.. Gradient flows in metric spaces and in the space of probability measures. Lectures in Mathematics ETH Zürich, 2nd edn (Basel: Birkhäuser Verlag, 2008).Google Scholar
4Braides, A.. Local minimization, variational evolution and Γ-convergence, volume 2094 of Lecture Notes in Mathematics (Cham: Springer, 2014).Google Scholar
5Braides, A. and Truskinovsky, L.. Asymptotic expansions by Γ-convergence. Continuum Mech. Thermodyn. 20 (2008), 21–62.Google Scholar
6De Giorgi, E.. New problems on minimizing movements. In Boundary value problems for partial differential equations and applications, volume 29 of RMA Res. Notes Appl. Math., pp. 81–98 (Paris: Masson, 1993.Google Scholar
7Dupuis, P. and Spiliopoulos, K.. Large deviations for multiscale diffusion via weak convergence methods. Stochastic Process. Appl. 122 (2012), 1947–1987.Google Scholar
8Ibrahim, H. and Monneau, R.. On the rate of convergence in periodic homogenization of scalar first-order ordinary differential equations. SIAM J. Math. Anal. 42 (2010), 2155–2176.Google Scholar
9Katok, A. and Hasselblatt, B.. Introduction to the modern theory of dynamical systems, volume 54 of Encyclopedia of Mathematics and its Applications (Cambridge: Cambridge University Press, 1995), With a supplementary chapter by Katok and Leonardo Mendoza.Google Scholar
10Menon, G.. Gradient systems with wiggly energies and related averaging problems. Arch. Ration. Mech. Anal. 162 (2002), 193–246.Google Scholar