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October  2021, 41(10): 4987-5008. doi: 10.3934/dcds.2021065

Convergence of nonlocal geometric flows to anisotropic mean curvature motion

Annalisa Cesaroni 1,, and  Valerio Pagliari 2,

1. 

Department of Statistical Sciences, Università di Padova, Via Battisti 241/243, 35121 Padova, Italy
2. 

Institute of Analysis and Scientific Computing, TU Wien, Wiedner Hauptstraße 8 - 10, 1040 Vienna, Austria

* Corresponding author: Annalisa Cesaroni

Received  December 2020 Revised  February 2021 Published  October 2021 Early access  March 2021

Fund Project: The authors are members and were supported by the INDAM/GNAMPA

We consider nonlocal curvature functionals associated with positive interaction kernels, and we show that local anisotropic mean curvature functionals can be retrieved in a blow-up limit from them. As a consequence, we prove that the viscosity solutions to the rescaled nonlocal geometric flows locally uniformly converge to the viscosity solution to the anisotropic mean curvature motion. The result is achieved by combining a compactness argument and a set-theoretic approach related to the theory of De Giorgi's barriers for evolution equations.

Keywords: Nonlocal curvature flow, anisotropic mean curvature flow, geometric equations, De Giorgi's barriers for geometric evolutions, level-set method, viscosity solutions.
Mathematics Subject Classification: Primary: 53E10; Secondary: 35D40, 35K93, 35R11.
Citation: Annalisa Cesaroni, Valerio Pagliari. Convergence of nonlocal geometric flows to anisotropic mean curvature motion. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4987-5008. doi: 10.3934/dcds.2021065
References:
[1]

N. Abatangelo and E. Valdinoci, A notion of nonlocal curvature, Numerical Functional Analysis and Optimization, 35 (2014), 793-815.  doi: 10.1080/01630563.2014.901837.  Google Scholar

[2]

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L. Ambrosio, Geometric evolution problems, distance function and viscosity solutions, Springer Berlin Heidelberg, Berlin, Heidelberg, 2000, 5–93. Google Scholar

[5]

L. Ambrosio, G. De Philippis and L. Martinazzi, $\Gamma$-convergence of nonlocal perimeter functionals, Manuscripta Math., 134 (2011), 377-403.  doi: 10.1007/s00229-010-0399-4.  Google Scholar

[6]

G. Barles and C. Georgelin, A simple proof of convergence for an approximation scheme for computing motions by mean curvature, SIAM J. Numer. Anal., 32 (1995), 484-500.  doi: 10.1137/0732020.  Google Scholar

[7]

G. Barles and O. Ley, Nonlocal first-order Hamilton-Jacobi equations modelling dislocations dynamics, Commun. Partial Differ. Equations, 31 (2006), 1191-1208.  doi: 10.1080/03605300500361446.  Google Scholar

[8]

G. Bellettini, Alcuni risultati sulle minime barriere per movimenti geometrici di insiemi, Bollettino UMI, 7 (1997), 485-512.   Google Scholar

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G. Bellettini and M. Novaga, Comparison results between minimal barriers and viscosity solutions for geometric evolutions, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 97-131.   Google Scholar

[10]

G. Bellettini and M. Novaga, Some aspects of {D}e {G}iorgi's barriers for geometric evolutions, Springer Berlin Heidelberg, Berlin, Heidelberg, 2000,115–151. Google Scholar

[11]

G. Bellettini and M. Paolini, Some results on minimal barriers in the sense of {D}e {G}iorgi applied to driven motion by mean curvature, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5), 19 (1995), 43-67.   Google Scholar

[12]

J. K. Bence, B. Merriman and S. Osher, Diffusion generated motion by mean curvature, Amer. Math. Soc., Providence, RI, 1992. Google Scholar

[13]

J. Berendsen and V. Pagliari, On the asymptotic behaviour of nonlocal perimeters, ESAIM Control Optim. Calc. Var., 25 (2019), Paper No. 48, 27pp. doi: 10.1051/cocv/2018038.  Google Scholar

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J. Bourgain, H. Brezis and P. Mironescu, Another look at {S}obolev spaces, In Optimal control and partial differential equations, IOS, Amsterdam, 2001,439–455.  Google Scholar

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L. A. Caffarelli and P. E. Souganidis, Convergence of nonlocal threshold dynamics approximations to front propagation, Arch. Ration. Mech. Anal., 195 (2010), 1-23.  doi: 10.1007/s00205-008-0181-x.  Google Scholar

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L. A. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations, 41 (2011), 203-240.  doi: 10.1007/s00526-010-0359-6.  Google Scholar

[17]

A. Cesaroni, S. Dipierro, M. Novaga and E. Valdinoci, Fattening and nonfattening phenomena for planar nonlocal curvature flows, Math. Ann., 375 (2019), 687-736.  doi: 10.1007/s00208-018-1793-6.  Google Scholar

[18]

A. Cesaroni, L. De Luca, M. Novaga and M. Ponsiglione, Stability results for nonlocal geometric evolutions and limit cases for fractional mean curvature flows, Comm. Partial Differential Equations, 2020, arXiv: 2003.02248. Google Scholar

[19]

A. Chambolle, M. Morini and M. Ponsiglione, Nonlocal curvature flows, Arch. Ration. Mech. Anal., 218 (2015), 1263-1329.  doi: 10.1007/s00205-015-0880-z.  Google Scholar

[20]

A. Chambolle and M. Novaga, Convergence of an algorithm for the anisotropic and crystalline mean curvature flow, SIAM J. Math. Anal., 37 (2006), 1978-1987.  doi: 10.1137/050629641.  Google Scholar

[21]

A. Chambolle, M. Novaga and B. Ruffini, Some results on anisotropic fractional mean curvature flows, Interfaces Free Bound, 19 (2017), 393-415.  doi: 10.4171/IFB/387.  Google Scholar

[22]

Y.-G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom., 33 (1991), 749-786.  doi: 10.4310/jdg/1214446564.  Google Scholar

[23]

E. Cinti, C. Sinestrari and E. Valdinoci, Neckpinch singularities in fractional mean curvature flows, Proc. Amer. Math. Soc., 146 (2018), 2637-2646.  doi: 10.1090/proc/14002.  Google Scholar

[24]

F. Da Lio, N. Forcadel and R. Monneau, Convergence of a non-local eikonal equation to anisotropic mean curvature motion. application to dislocations dynamics, J. Eur. Math. Soc. (JEMS), 10 (2008), 1061-1104.  doi: 10.4171/JEMS/140.  Google Scholar

[25]

E. De Giorgi, Barriers, Boundaries, Motion of Manifolds, Conference held at Dipartimento di Matematica, Univ. of Pavia, March 18, 1994. Google Scholar

[26]

L. C. Evans, Convergence of an algorithm for mean curvature motion,, Indiana Univ. Math. J., 42 (1993), 533–557. doi: 10.1512/iumj.1993.42.42024.  Google Scholar

[27]

N. Forcadel, C. Imbert and R. Monneau, Homogenization of some particle systems with two-body interactions and of the dislocation dynamics, DCDS-A, 23 (2009), 785-826.  doi: 10.3934/dcds.2009.23.785.  Google Scholar

[28]

P. Hajłasz, Sobolev Spaces on Metric-Measure Spaces, volume 338 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2003. Google Scholar

[29]

C. Imbert, Level set approach for fractional mean curvature flows, Interfaces Free Bound., 11 (2009), 153-176.  doi: 10.4171/IFB/207.  Google Scholar

[30]

C. Imbert, R. Monneau and E. Rouy-Mironescu, Homogenization of first order equations with $u/ \varepsilon$-periodic Hamiltonians. part ii: application to dislocations dynamics, Comm. in PDEs, 33 (2008), 479-516.  doi: 10.1080/03605300701318922.  Google Scholar

[31]

H. Ishii, A generalization of the Bence, Merriman and Osher algorithm for motion by mean curvature, Proceedings of the International Conference on Curvature Flows and Related Topics Held in Levico, Italy, June 27-July 2nd, 1994, 5 (1995), 111–127.  Google Scholar

[32]

H. Ishii, G. E. Pires and P. E. Souganidis, Threshold dynamics type approximation schemes for propagating fronts, J. Math. Soc. Japan, 51 (1999), 267-308.  doi: 10.2969/jmsj/05120267.  Google Scholar

[33]

J. M. Mazon, J. D. Rossi and J. Toledo, Nonlocal perimeter, curvature and minimal surfaces for measurable sets, J. Anal. Math., 138 (2019), 235-279.  doi: 10.1007/s11854-019-0027-5.  Google Scholar

[34]

V. Pagliari, Halfspaces minimise nonlocal perimeter: A proof via calibrations, Ann. Mat. Pura Appl., 199 (2020), 1685-1696.  doi: 10.1007/s10231-019-00937-7.  Google Scholar

[35]

O. Savin and E. Valdinoci, $\Gamma$-convergence for nonlocal phase transitions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 479-500.  doi: 10.1016/j.anihpc.2012.01.006.  Google Scholar

[36]

D. Slepčev, Approximation schemes for propagation of fronts with nonlocal velocities and Neumann boundary conditions, Nonlinear Anal., 52 (2003), 79-115.  doi: 10.1016/S0362-546X(02)00098-6.  Google Scholar

show all references

References:
[1]

N. Abatangelo and E. Valdinoci, A notion of nonlocal curvature, Numerical Functional Analysis and Optimization, 35 (2014), 793-815.  doi: 10.1080/01630563.2014.901837.  Google Scholar

[2]

O. Alvarez, P. Cardaliaguet and R. Monneau, Existence and uniqueness for dislocation dynamics with nonnegative velocity, Interfaces Free Bound., 7 (2005), 415-434.  doi: 10.4171/IFB/131.  Google Scholar

[3]

O. Alvarez, P. Hoch, Y. Le Bouar and R. Monneau, Dislocation dynamics: Short-time existence and uniqueness of the solution, Arch. Ration. Mech. Anal., 181 (2006), 449-504.  doi: 10.1007/s00205-006-0418-5.  Google Scholar

[4]

L. Ambrosio, Geometric evolution problems, distance function and viscosity solutions, Springer Berlin Heidelberg, Berlin, Heidelberg, 2000, 5–93. Google Scholar

[5]

L. Ambrosio, G. De Philippis and L. Martinazzi, $\Gamma$-convergence of nonlocal perimeter functionals, Manuscripta Math., 134 (2011), 377-403.  doi: 10.1007/s00229-010-0399-4.  Google Scholar

[6]

G. Barles and C. Georgelin, A simple proof of convergence for an approximation scheme for computing motions by mean curvature, SIAM J. Numer. Anal., 32 (1995), 484-500.  doi: 10.1137/0732020.  Google Scholar

[7]

G. Barles and O. Ley, Nonlocal first-order Hamilton-Jacobi equations modelling dislocations dynamics, Commun. Partial Differ. Equations, 31 (2006), 1191-1208.  doi: 10.1080/03605300500361446.  Google Scholar

[8]

G. Bellettini, Alcuni risultati sulle minime barriere per movimenti geometrici di insiemi, Bollettino UMI, 7 (1997), 485-512.   Google Scholar

[9]

G. Bellettini and M. Novaga, Comparison results between minimal barriers and viscosity solutions for geometric evolutions, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 97-131.   Google Scholar

[10]

G. Bellettini and M. Novaga, Some aspects of {D}e {G}iorgi's barriers for geometric evolutions, Springer Berlin Heidelberg, Berlin, Heidelberg, 2000,115–151. Google Scholar

[11]

G. Bellettini and M. Paolini, Some results on minimal barriers in the sense of {D}e {G}iorgi applied to driven motion by mean curvature, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5), 19 (1995), 43-67.   Google Scholar

[12]

J. K. Bence, B. Merriman and S. Osher, Diffusion generated motion by mean curvature, Amer. Math. Soc., Providence, RI, 1992. Google Scholar

[13]

J. Berendsen and V. Pagliari, On the asymptotic behaviour of nonlocal perimeters, ESAIM Control Optim. Calc. Var., 25 (2019), Paper No. 48, 27pp. doi: 10.1051/cocv/2018038.  Google Scholar

[14]

J. Bourgain, H. Brezis and P. Mironescu, Another look at {S}obolev spaces, In Optimal control and partial differential equations, IOS, Amsterdam, 2001,439–455.  Google Scholar

[15]

L. A. Caffarelli and P. E. Souganidis, Convergence of nonlocal threshold dynamics approximations to front propagation, Arch. Ration. Mech. Anal., 195 (2010), 1-23.  doi: 10.1007/s00205-008-0181-x.  Google Scholar

[16]

L. A. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations, 41 (2011), 203-240.  doi: 10.1007/s00526-010-0359-6.  Google Scholar

[17]

A. Cesaroni, S. Dipierro, M. Novaga and E. Valdinoci, Fattening and nonfattening phenomena for planar nonlocal curvature flows, Math. Ann., 375 (2019), 687-736.  doi: 10.1007/s00208-018-1793-6.  Google Scholar

[18]

A. Cesaroni, L. De Luca, M. Novaga and M. Ponsiglione, Stability results for nonlocal geometric evolutions and limit cases for fractional mean curvature flows, Comm. Partial Differential Equations, 2020, arXiv: 2003.02248. Google Scholar

[19]

A. Chambolle, M. Morini and M. Ponsiglione, Nonlocal curvature flows, Arch. Ration. Mech. Anal., 218 (2015), 1263-1329.  doi: 10.1007/s00205-015-0880-z.  Google Scholar

[20]

A. Chambolle and M. Novaga, Convergence of an algorithm for the anisotropic and crystalline mean curvature flow, SIAM J. Math. Anal., 37 (2006), 1978-1987.  doi: 10.1137/050629641.  Google Scholar

[21]

A. Chambolle, M. Novaga and B. Ruffini, Some results on anisotropic fractional mean curvature flows, Interfaces Free Bound, 19 (2017), 393-415.  doi: 10.4171/IFB/387.  Google Scholar

[22]

Y.-G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom., 33 (1991), 749-786.  doi: 10.4310/jdg/1214446564.  Google Scholar

[23]

E. Cinti, C. Sinestrari and E. Valdinoci, Neckpinch singularities in fractional mean curvature flows, Proc. Amer. Math. Soc., 146 (2018), 2637-2646.  doi: 10.1090/proc/14002.  Google Scholar

[24]

F. Da Lio, N. Forcadel and R. Monneau, Convergence of a non-local eikonal equation to anisotropic mean curvature motion. application to dislocations dynamics, J. Eur. Math. Soc. (JEMS), 10 (2008), 1061-1104.  doi: 10.4171/JEMS/140.  Google Scholar

[25]

E. De Giorgi, Barriers, Boundaries, Motion of Manifolds, Conference held at Dipartimento di Matematica, Univ. of Pavia, March 18, 1994. Google Scholar

[26]

L. C. Evans, Convergence of an algorithm for mean curvature motion,, Indiana Univ. Math. J., 42 (1993), 533–557. doi: 10.1512/iumj.1993.42.42024.  Google Scholar

[27]

N. Forcadel, C. Imbert and R. Monneau, Homogenization of some particle systems with two-body interactions and of the dislocation dynamics, DCDS-A, 23 (2009), 785-826.  doi: 10.3934/dcds.2009.23.785.  Google Scholar

[28]

P. Hajłasz, Sobolev Spaces on Metric-Measure Spaces, volume 338 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2003. Google Scholar

[29]

C. Imbert, Level set approach for fractional mean curvature flows, Interfaces Free Bound., 11 (2009), 153-176.  doi: 10.4171/IFB/207.  Google Scholar

[30]

C. Imbert, R. Monneau and E. Rouy-Mironescu, Homogenization of first order equations with $u/ \varepsilon$-periodic Hamiltonians. part ii: application to dislocations dynamics, Comm. in PDEs, 33 (2008), 479-516.  doi: 10.1080/03605300701318922.  Google Scholar

[31]

H. Ishii, A generalization of the Bence, Merriman and Osher algorithm for motion by mean curvature, Proceedings of the International Conference on Curvature Flows and Related Topics Held in Levico, Italy, June 27-July 2nd, 1994, 5 (1995), 111–127.  Google Scholar

[32]

H. Ishii, G. E. Pires and P. E. Souganidis, Threshold dynamics type approximation schemes for propagating fronts, J. Math. Soc. Japan, 51 (1999), 267-308.  doi: 10.2969/jmsj/05120267.  Google Scholar

[33]

J. M. Mazon, J. D. Rossi and J. Toledo, Nonlocal perimeter, curvature and minimal surfaces for measurable sets, J. Anal. Math., 138 (2019), 235-279.  doi: 10.1007/s11854-019-0027-5.  Google Scholar

[34]

V. Pagliari, Halfspaces minimise nonlocal perimeter: A proof via calibrations, Ann. Mat. Pura Appl., 199 (2020), 1685-1696.  doi: 10.1007/s10231-019-00937-7.  Google Scholar

[35]

O. Savin and E. Valdinoci, $\Gamma$-convergence for nonlocal phase transitions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 479-500.  doi: 10.1016/j.anihpc.2012.01.006.  Google Scholar

[36]

D. Slepčev, Approximation schemes for propagation of fronts with nonlocal velocities and Neumann boundary conditions, Nonlinear Anal., 52 (2003), 79-115.  doi: 10.1016/S0362-546X(02)00098-6.  Google Scholar

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