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Asymptotic speed of spread for a nonlocal evolutionary-epidemic system
Convergence of nonlocal geometric flows to anisotropic mean curvature motion
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 |
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.
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. |
[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. |
[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. |
[4] |
L. Ambrosio, Geometric evolution problems, distance function and viscosity solutions, Springer Berlin Heidelberg, Berlin, Heidelberg, 2000, 5–93. |
[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. |
[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. |
[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. |
[8] |
G. Bellettini,
Alcuni risultati sulle minime barriere per movimenti geometrici di insiemi, Bollettino UMI, 7 (1997), 485-512.
|
[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.
|
[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. |
[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.
|
[12] |
J. K. Bence, B. Merriman and S. Osher, Diffusion generated motion by mean curvature, Amer. Math. Soc., Providence, RI, 1992. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[25] |
E. De Giorgi, Barriers, Boundaries, Motion of Manifolds, Conference held at Dipartimento di Matematica, Univ. of Pavia, March 18, 1994. |
[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. |
[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. |
[28] |
P. Hajłasz, Sobolev Spaces on Metric-Measure Spaces, volume 338 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2003. |
[29] |
C. Imbert,
Level set approach for fractional mean curvature flows, Interfaces Free Bound., 11 (2009), 153-176.
doi: 10.4171/IFB/207. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[4] |
L. Ambrosio, Geometric evolution problems, distance function and viscosity solutions, Springer Berlin Heidelberg, Berlin, Heidelberg, 2000, 5–93. |
[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. |
[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. |
[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. |
[8] |
G. Bellettini,
Alcuni risultati sulle minime barriere per movimenti geometrici di insiemi, Bollettino UMI, 7 (1997), 485-512.
|
[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.
|
[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. |
[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.
|
[12] |
J. K. Bence, B. Merriman and S. Osher, Diffusion generated motion by mean curvature, Amer. Math. Soc., Providence, RI, 1992. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[25] |
E. De Giorgi, Barriers, Boundaries, Motion of Manifolds, Conference held at Dipartimento di Matematica, Univ. of Pavia, March 18, 1994. |
[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. |
[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. |
[28] |
P. Hajłasz, Sobolev Spaces on Metric-Measure Spaces, volume 338 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2003. |
[29] |
C. Imbert,
Level set approach for fractional mean curvature flows, Interfaces Free Bound., 11 (2009), 153-176.
doi: 10.4171/IFB/207. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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