Mean curvature flow with obstacles
Almeida, L. ; Chambolle, A. ; Novaga, M.
Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012), p. 667-681 / Harvested from Numdam
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We consider the evolution of fronts by mean curvature in the presence of obstacles. We construct a weak solution to the flow by means of a variational method, corresponding to an implicit time-discretization scheme. Assuming the regularity of the obstacles, in the two-dimensional case we show existence and uniqueness of a regular solution before the onset of singularities. Finally, we discuss an application of this result to the positive mean curvature flow.

Publié le : 2012-01-01
DOI : https://doi.org/10.1016/j.anihpc.2012.03.002
Classification:  35R37,  35R45,  49J40,  49Q20,  53A10 
@article{AIHPC_2012__29_5_667_0,
     author = {Almeida, L. and Chambolle, A. and Novaga, M.},
     title = {Mean curvature flow with obstacles},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {29},
     year = {2012},
     pages = {667-681},
     doi = {10.1016/j.anihpc.2012.03.002},
     mrnumber = {2971026},
     zbl = {1252.49072},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2012__29_5_667_0}
}

Almeida, L.; Chambolle, A.; Novaga, M. Mean curvature flow with obstacles. Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) pp. 667-681. doi : 10.1016/j.anihpc.2012.03.002. http://gdmltest.u-ga.fr/item/AIHPC_2012__29_5_667_0/

[1] L. Almeida, P. Bagnerini, A. Habbal, S. Noselli, F. Serman, Tissue repair modeling, Singularities in Nonlinear Evolution Phenomena and Applications, CRM Series vol. 9 (2009) | MR 2528697 | Zbl 1190.35218

[2] L. Almeida, P. Bagnerini, A. Habbal, S. Noselli, F. Serman, A mathematical model for dorsal closure, J. Theoret. Biol. 268 no. 1 (2011), 105-119

[3] L. Almeida, P. Bagnerini, A. Habbal, Modeling actin cable contraction, preprint, 2011; Comput. Math. Appl. (March 2012), http://dx.doi.org/10.1016/j.camwa.2012.02.041, in press. | MR 2944813

[4] L. Almeida, J. Demongeot, Predictive power of “a minima” models in biology, preprint, 2011; Acta Biotheor. (9 February 2012), pp. 1–17, http://dx.doi.org/10.1007/s10441-012-9146-4, in press.

[5] F. Almgren, J.E. Taylor, L.-H. Wang, Curvature-driven flows: a variational approach, SIAM J. Control Optim. 31 no. 2 (1993), 387-438 | MR 1205983 | Zbl 0783.35002

[6] L. Ambrosio, Movimenti minimizzanti, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5) 19 (1995), 191-246 | MR 1387558

[7] L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Math. Monogr., The Clarendon Press/Oxford University Press, New York (2000) | MR 1857292 | Zbl 0957.49001

[8] G. Barles, A. Cesaroni, M. Novaga, Homogenization of fronts in highly heterogeneous media, SIAM J. Math. Anal. 43 no. 1 (2011), 212-227 | MR 2765689 | Zbl 1228.35026

[9] G. Barles, F. Da Lio, Remarks on the Dirichlet and state-constraint problems for quasilinear parabolic equations, Adv. Differential Equations 8 no. 8 (2003), 897-922 | MR 1989355 | Zbl 1073.35120

[10] G. Bellettini, M. Novaga, Comparison results between minimal barriers and viscosity solutions for geometric evolutions, Ann. Sc. Norm. Super. Pisa Cl. Sci. 26 no. 1 (1998), 97-131 | Numdam | MR 1632984 | Zbl 0904.35041

[11] G. Bellettini, V. Caselles, A. Chambolle, M. Novaga, Crystalline mean curvature flow of convex sets, Arch. Ration. Mech. Anal. 179 no. 1 (2006), 109-152 | MR 2208291 | Zbl 1148.53049

[12] L. Caffarelli, The obstacle problem revisited, J. Fourier Anal. Appl. 4 (1998), 383-402 | MR 1658612 | Zbl 0928.49030

[13] P. Cardaliaguet, P.-L. Lions, P. Souganidis, A discussion about the homogenization of moving interfaces, J. Math. Pures Appl. 91 no. 4 (2009), 339-363 | MR 2518002 | Zbl 1180.35070

[14] V. Caselles, A. Chambolle, Anisotropic curvature-driven flow of convex sets, Nonlinear Anal. 65 no. 8 (2006), 1547-1577 | MR 2248685 | Zbl 1107.35069

[15] A. Chambolle, An algorithm for mean curvature motion, Interfaces Free Bound. 6 no. 2 (2004), 195-218 | MR 2079603 | Zbl 1061.35147

[16] A. Chambolle, V. Caselles, D. Cremers, M. Novaga, T. Pock, An introduction to total variation for image analysis, Theoretical Foundations and Numerical Methods for Sparse Recovery, Radon Ser. Comp. Appl. Math. vol. 9, De Gruyter (2010), 263-340 | MR 2731599 | Zbl 1209.94004

[17] A. Chambolle, M. Novaga, Implicit time discretization of the mean curvature flow with a discontinuous forcing term, Interfaces Free Bound. 10 (2008), 283-300 | MR 2453133 | Zbl 1170.65054

[18] B. Craciun, K. Bhattacharya, Effective motion of a curvature-sensitive interface through a heterogeneous medium, Interfaces Free Bound. 6 (2004), 151-173 | MR 2079601 | Zbl 1061.35148

[19] M. Crandall, H. Ishii, P.-L. Lions, Userʼs guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. 27 no. 1 (1992), 1-67

[20] F. Da Lio, Comparison results for quasilinear equations in annular domains and applications, Comm. Partial Differential Equations 27 no. 1–2 (2002), 283-323 | MR 1886961 | Zbl 0994.35014

[21] N. Dirr, G. Karali, N.K. Yip, Pulsating wave for mean curvature flow in inhomogeneous medium, European J. Appl. Math. 19 (2008), 661-699 | MR 2463225 | Zbl 1185.53076

[22] M.S. Hutson, Y. Tokutake, M. Chang, J. Bloor, S. Venakides, D.P. Kiehart, G. Edwards, Forces for morphogenesis investigated with laser microsurgery and quantitative modeling, Science 300 no. 5616 (2003), 145-149

[23] R.V. Kohn, S. Serfaty, A deterministic-control based approach to fully nonlinear parabolic and elliptic equations, Comm. Pure Appl. Math. 63 (2010), 1298-1350 | MR 2681474 | Zbl 1204.35070

[24] S. Luckhaus, T. Sturzenhecker, Implicit time discretization for the mean curvature flow equation, Calc. Var. Partial Differential Equations 3 no. 2 (1995), 253-271 | MR 1386964 | Zbl 0821.35003

[25] Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, Univ. Lecture Ser. vol. 22, Amer. Math. Soc., Providence, RI (2001) | MR 1852741

[26] M. Miranda, Frontiere minimali con ostacoli, Ann. Univ. Ferrara 16 no. 1 (1971), 29-37 | MR 301617 | Zbl 0266.49036

[27] R. Phillips, Crystals, Defects and Microstructures, Cambridge University Press (2001)

[28] J.A. Sethian, Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision and Materials Science, Cambridge University Press (1999) | MR 1700751 | Zbl 0973.76003