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 and 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 $A$
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