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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[17]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[21]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[25]

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

[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. Google Scholar

[27]

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

[28]

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

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[17]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[21]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[25]

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

[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. Google Scholar

[27]

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

[28]

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

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]

Ruiqi Jiang, Youde Wang, Jun Yang. Vortex structures for some geometric flows from pseudo-Euclidean spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1745-1777. doi: 10.3934/dcds.2019076

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[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]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

2018 Impact Factor: 0.871

Metrics

  • PDF downloads (29)
  • HTML views (182)
  • Cited by (0)

Other articles
by authors

[Back to Top]