September 2018, 13(3): 493-513. doi: 10.3934/nhm.2018022

Crystalline evolutions in chessboard-like microstructures

1. 

Dipartimento di Matematica, Sapienza Università di Roma, Piazzale Aldo Moro 2, 00185 Roma, Italy

2. 

Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56217 Pisa, Italy

Received  November 2017 Revised  March 2018 Published  July 2018

We describe the macroscopic behavior of evolutions by crystalline curvature of planar sets in a chessboard-like medium, modeled by a periodic forcing term. We show that the underlying microstructure may produce both pinning and confinement effects on the geometric motion.

Citation: Annalisa Malusa, Matteo Novaga. Crystalline evolutions in chessboard-like microstructures. Networks & Heterogeneous Media, 2018, 13 (3) : 493-513. doi: 10.3934/nhm.2018022
References:
[1]

F. Almgren and J. E. Taylor, Flat flow is motion by crystalline curvature for curves with crystalline energies, J. Differential Geometry, 42 (1995), 1-22. doi: 10.4310/jdg/1214457030.

[2]

G. BarlesA. Cesaroni and M. Novaga, Homogenization of fronts in highly heterogeneous media, SIAM J. Math. Anal., 43 (2011), 212-227. doi: 10.1137/100800014.

[3]

G. BellettiniR. Goglione and M. Novaga, Approximation to driven motion by crystalline curvature in two dimensions, Adv. Math. Sci. and Appl., 10 (2000), 467-493.

[4]

G. BellettiniM. Novaga and M. Paolini, Characterization of facet breaking for nonsmooth mean curvature flow in the convex case, Interfaces Free Bound., 3 (2001), 415-446. doi: 10.4171/IFB/47.

[5]

G. BellettiniM. Novaga and M. Paolini, On a crystalline variational problem, part Ⅰ: First variation and global $L^∞$ regularity, Arch. Rational Mech. Anal, 57 (2001), 165-191. doi: 10.1007/s002050010127.

[6]

G. BellettiniM. Novaga and M. Paolini, On a crystalline variational problem, part Ⅱ: $BV$ regularity and structure of minimizers on facets, Arch. Rational Mech. Anal., 157 (2001), 193-217. doi: 10.1007/s002050100126.

[7]

A. Braides, $Γ$-convergence for Beginners, Oxford University Press, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001.

[8]

A. Braides, Local Minimization, Variational Evolution and Γ–convergence, Lecture Notes in Mathematics, Springer, Berlin, 2014. doi: 10.1007/978-3-319-01982-6.

[9]

A. BraidesM. Cicalese and N. K. Yip, Crystalline Motion of Interfaces Between Patterns, J. Stat. Phys., 165 (2016), 274-319. doi: 10.1007/s10955-016-1609-6.

[10]

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.

[11]

A. Braides, A. Malusa and M. Novaga, Crystalline evolutions with rapidly oscillating forcing terms, to appear on Ann. Scuola Norm. Sci. doi: 10.2422/2036-2145.201707_011.

[12]

A. Braides and G. Scilla, Motion of discrete interfaces in periodic media, Interfaces Free Bound., 15 (2013), 451-476. doi: 10.4171/IFB/310.

[13]

A. Braides and M. Solci, Motion of discrete interfaces through mushy layers, J. Nonlinear Sci., 26 (2016), 1031-1053. doi: 10.1007/s00332-016-9297-6.

[14]

A. CesaroniN. Dirr and M. Novaga, Homogenization of a semilinear heat equation, J. Éc. polytech. Math., 4 (2017), 633-660. doi: 10.5802/jep.54.

[15]

A. CesaroniM. Novaga and E. Valdinoci, Curve shortening flow in heterogeneous media, Interfaces and Free Bound., 13 (2011), 485-505. doi: 10.4171/IFB/269.

[16]

A. ChambolleM. Morini and M. Ponsiglione, Existence and uniqueness for a crystalline mean curvature flow, Comm. Pure Appl. Math., 70 (2017), 1084-1114. doi: 10.1002/cpa.21668.

[17]

A. Chambolle, M. Morini, M. Novaga and M. Ponsiglione, Existence and uniqueness for anisotropic and crystalline mean curvature flows, preprint, arXiv: 1702.03094.

[18]

A. Chambolle and M. Novaga, Approximation of the anisotropic mean curvature flow, Math. Models Methods Appl. Sci., 17 (2007), 833-844. doi: 10.1142/S0218202507002121.

[19]

J. Cortes, Discontinuous Dynamical Systems: A tutorial on solutions, nonsmooth analysis, and stability, IEEE Control Systems Magazine, 28 (2008), 36-73. doi: 10.1109/MCS.2008.919306.

[20]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, vol. 18 of Mathematics and Its Applications. Dordrecht, The Netherlands, Kluwer Academic Publishers, 1988. doi: 10.1007/978-94-015-7793-9.

[21]

Y. Giga, Surface Evolution Equations. A Level Set Approach, vol. 99 of Monographs in Mathematics. Birkhäuser Verlag, Basel, 2006.

[22]

Y. Giga and M. E. Gurtin, A comparison theorem for crystalline evolution in the plane, Quarterly of Applied Mathematics, 54 (1996), 727-737. doi: 10.1090/qam/1417236.

[23]

Y. Giga and P. Rybka, Facet bending in the driven crystalline curvature flow in the plane, J. Geom. Anal., 18 (2008), 109-147. doi: 10.1007/s12220-007-9004-9.

[24]

Y. Giga and P. Rybka, Facet bending driven by the planar crystalline curvature with a generic nonuniform forcing term, J. Differential Equations, 246 (2009), 2264-2303. doi: 10.1016/j.jde.2009.01.009.

[25]

M. E. Gurtin, Thermomechanics of Evolving Phase Boundaries in the Plane, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1993.

[26]

M. Novaga and E. Valdinoci, Closed curves of prescribed curvature and a pinning effect, Netw. Heterog. Media, 6 (2011), 77-88. doi: 10.3934/nhm.2011.6.77.

[27]

J. E. Taylor, Crystalline variational problems, Bull. Amer. Math. Soc., 84 (1978), 568-588. doi: 10.1090/S0002-9904-1978-14499-1.

[28]

J. E. TaylorJ. Cahn and C. Handwerker, Geometric Models of Crystal Growth, Acta Metall. Mater., 40 (1992), 1443-1474.

show all references

References:
[1]

F. Almgren and J. E. Taylor, Flat flow is motion by crystalline curvature for curves with crystalline energies, J. Differential Geometry, 42 (1995), 1-22. doi: 10.4310/jdg/1214457030.

[2]

G. BarlesA. Cesaroni and M. Novaga, Homogenization of fronts in highly heterogeneous media, SIAM J. Math. Anal., 43 (2011), 212-227. doi: 10.1137/100800014.

[3]

G. BellettiniR. Goglione and M. Novaga, Approximation to driven motion by crystalline curvature in two dimensions, Adv. Math. Sci. and Appl., 10 (2000), 467-493.

[4]

G. BellettiniM. Novaga and M. Paolini, Characterization of facet breaking for nonsmooth mean curvature flow in the convex case, Interfaces Free Bound., 3 (2001), 415-446. doi: 10.4171/IFB/47.

[5]

G. BellettiniM. Novaga and M. Paolini, On a crystalline variational problem, part Ⅰ: First variation and global $L^∞$ regularity, Arch. Rational Mech. Anal, 57 (2001), 165-191. doi: 10.1007/s002050010127.

[6]

G. BellettiniM. Novaga and M. Paolini, On a crystalline variational problem, part Ⅱ: $BV$ regularity and structure of minimizers on facets, Arch. Rational Mech. Anal., 157 (2001), 193-217. doi: 10.1007/s002050100126.

[7]

A. Braides, $Γ$-convergence for Beginners, Oxford University Press, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001.

[8]

A. Braides, Local Minimization, Variational Evolution and Γ–convergence, Lecture Notes in Mathematics, Springer, Berlin, 2014. doi: 10.1007/978-3-319-01982-6.

[9]

A. BraidesM. Cicalese and N. K. Yip, Crystalline Motion of Interfaces Between Patterns, J. Stat. Phys., 165 (2016), 274-319. doi: 10.1007/s10955-016-1609-6.

[10]

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.

[11]

A. Braides, A. Malusa and M. Novaga, Crystalline evolutions with rapidly oscillating forcing terms, to appear on Ann. Scuola Norm. Sci. doi: 10.2422/2036-2145.201707_011.

[12]

A. Braides and G. Scilla, Motion of discrete interfaces in periodic media, Interfaces Free Bound., 15 (2013), 451-476. doi: 10.4171/IFB/310.

[13]

A. Braides and M. Solci, Motion of discrete interfaces through mushy layers, J. Nonlinear Sci., 26 (2016), 1031-1053. doi: 10.1007/s00332-016-9297-6.

[14]

A. CesaroniN. Dirr and M. Novaga, Homogenization of a semilinear heat equation, J. Éc. polytech. Math., 4 (2017), 633-660. doi: 10.5802/jep.54.

[15]

A. CesaroniM. Novaga and E. Valdinoci, Curve shortening flow in heterogeneous media, Interfaces and Free Bound., 13 (2011), 485-505. doi: 10.4171/IFB/269.

[16]

A. ChambolleM. Morini and M. Ponsiglione, Existence and uniqueness for a crystalline mean curvature flow, Comm. Pure Appl. Math., 70 (2017), 1084-1114. doi: 10.1002/cpa.21668.

[17]

A. Chambolle, M. Morini, M. Novaga and M. Ponsiglione, Existence and uniqueness for anisotropic and crystalline mean curvature flows, preprint, arXiv: 1702.03094.

[18]

A. Chambolle and M. Novaga, Approximation of the anisotropic mean curvature flow, Math. Models Methods Appl. Sci., 17 (2007), 833-844. doi: 10.1142/S0218202507002121.

[19]

J. Cortes, Discontinuous Dynamical Systems: A tutorial on solutions, nonsmooth analysis, and stability, IEEE Control Systems Magazine, 28 (2008), 36-73. doi: 10.1109/MCS.2008.919306.

[20]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, vol. 18 of Mathematics and Its Applications. Dordrecht, The Netherlands, Kluwer Academic Publishers, 1988. doi: 10.1007/978-94-015-7793-9.

[21]

Y. Giga, Surface Evolution Equations. A Level Set Approach, vol. 99 of Monographs in Mathematics. Birkhäuser Verlag, Basel, 2006.

[22]

Y. Giga and M. E. Gurtin, A comparison theorem for crystalline evolution in the plane, Quarterly of Applied Mathematics, 54 (1996), 727-737. doi: 10.1090/qam/1417236.

[23]

Y. Giga and P. Rybka, Facet bending in the driven crystalline curvature flow in the plane, J. Geom. Anal., 18 (2008), 109-147. doi: 10.1007/s12220-007-9004-9.

[24]

Y. Giga and P. Rybka, Facet bending driven by the planar crystalline curvature with a generic nonuniform forcing term, J. Differential Equations, 246 (2009), 2264-2303. doi: 10.1016/j.jde.2009.01.009.

[25]

M. E. Gurtin, Thermomechanics of Evolving Phase Boundaries in the Plane, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1993.

[26]

M. Novaga and E. Valdinoci, Closed curves of prescribed curvature and a pinning effect, Netw. Heterog. Media, 6 (2011), 77-88. doi: 10.3934/nhm.2011.6.77.

[27]

J. E. Taylor, Crystalline variational problems, Bull. Amer. Math. Soc., 84 (1978), 568-588. doi: 10.1090/S0002-9904-1978-14499-1.

[28]

J. E. TaylorJ. Cahn and C. Handwerker, Geometric Models of Crystal Growth, Acta Metall. Mater., 40 (1992), 1443-1474.

Figure 1.  Microscopic and macroscopic nontrivial equilibrium ($\alpha+\beta <0$)
Figure 2.  The breaking and recomposing phenomenon
Figure 3.  The cutting phenomenon
Figure 4.  The effective evolution in Case (ⅱ) of confinement
Figure 5.  How the mixed case starts
Figure 6.  How the mixed case carries on
Figure 7.  Effective evolutions, case (ⅲ) and $U_0\leq 0$
Figure 8.  Left: short-time effective evolution, case (ⅲ), $U_0> 0$. Right: phase portrait of (18), with the region $A$
[1]

Laura Sigalotti. Homogenization of pinning conditions on periodic networks. Networks & Heterogeneous Media, 2012, 7 (3) : 543-582. doi: 10.3934/nhm.2012.7.543

[2]

Shin Kiriki, Ming-Chia Li, Teruhiko Soma. Geometric Lorenz flows with historic behavior. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7021-7028. doi: 10.3934/dcds.2016105

[3]

Shouwen Fang, Peng Zhu. Differential Harnack estimates for backward heat equations with potentials under geometric flows. Communications on Pure & Applied Analysis, 2015, 14 (3) : 793-809. doi: 10.3934/cpaa.2015.14.793

[4]

Gianni Dal Maso, Alexander Mielke, Ulisse Stefanelli. Preface: Rate-independent evolutions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : i-ii. doi: 10.3934/dcdss.2013.6.1i

[5]

François Gay-Balmaz, Cesare Tronci, Cornelia Vizman. Geometric dynamics on the automorphism group of principal bundles: Geodesic flows, dual pairs and chromomorphism groups. Journal of Geometric Mechanics, 2013, 5 (1) : 39-84. doi: 10.3934/jgm.2013.5.39

[6]

Carolina Mendoza, Jean Bragard, Pier Luigi Ramazza, Javier Martínez-Mardones, Stefano Boccaletti. Pinning control of spatiotemporal chaos in the LCLV device. Mathematical Biosciences & Engineering, 2007, 4 (3) : 523-530. doi: 10.3934/mbe.2007.4.523

[7]

Matteo Novaga, Enrico Valdinoci. Closed curves of prescribed curvature and a pinning effect. Networks & Heterogeneous Media, 2011, 6 (1) : 77-88. doi: 10.3934/nhm.2011.6.77

[8]

Mickaël Dos Santos, Oleksandr Misiats. Ginzburg-Landau model with small pinning domains. Networks & Heterogeneous Media, 2011, 6 (4) : 715-753. doi: 10.3934/nhm.2011.6.715

[9]

Michael Shearer, Nicholas Giffen. Shock formation and breaking in granular avalanches. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 693-714. doi: 10.3934/dcds.2010.27.693

[10]

Freddy Dumortier, Robert Roussarie. Canard cycles with two breaking parameters. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 787-806. doi: 10.3934/dcds.2007.17.787

[11]

Maksim Maydanskiy, Benjamin P. Mirabelli. Semisimplicity of the quantum cohomology for smooth Fano toric varieties associated with facet symmetric polytopes. Electronic Research Announcements, 2011, 18: 131-143. doi: 10.3934/era.2011.18.131

[12]

Stefano Bosia, Michela Eleuteri, Elisabetta Rocca, Enrico Valdinoci. Preface: Special issue on rate-independent evolutions and hysteresis modelling. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : i-i. doi: 10.3934/dcdss.2015.8.4i

[13]

Francis Michael Russell, J. C. Eilbeck. Persistent mobile lattice excitations in a crystalline insulator. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1267-1285. doi: 10.3934/dcdss.2011.4.1267

[14]

Jean-Marie Souriau. On Geometric Mechanics. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 595-607. doi: 10.3934/dcds.2007.19.595

[15]

Yilun Shang. Group pinning consensus under fixed and randomly switching topologies with acyclic partition. Networks & Heterogeneous Media, 2014, 9 (3) : 553-573. doi: 10.3934/nhm.2014.9.553

[16]

Francesco Della Pietra, Ireneo Peral. Breaking of resonance for elliptic problems with strong degeneration at infinity. Communications on Pure & Applied Analysis, 2011, 10 (2) : 593-612. doi: 10.3934/cpaa.2011.10.593

[17]

Fanni M. Sélley. Symmetry breaking in a globally coupled map of four sites. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3707-3734. doi: 10.3934/dcds.2018161

[18]

Lucio Cadeddu, Giovanni Porru. Symmetry breaking in problems involving semilinear equations. Conference Publications, 2011, 2011 (Special) : 219-228. doi: 10.3934/proc.2011.2011.219

[19]

Hwai-Chiuan Wang. Stability and symmetry breaking of solutions of semilinear elliptic equations. Conference Publications, 2005, 2005 (Special) : 886-894. doi: 10.3934/proc.2005.2005.886

[20]

Claudia Anedda, Giovanni Porru. Symmetry breaking and other features for Eigenvalue problems. Conference Publications, 2011, 2011 (Special) : 61-70. doi: 10.3934/proc.2011.2011.61

2017 Impact Factor: 1.187

Article outline

Figures and Tables

[Back to Top]