zbMATH — the first resource for mathematics

Mean curvature flow with obstacles. (English) Zbl 1252.49072
Summary: 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.

49Q20 Variational problems in a geometric measure-theoretic setting
35R37 Moving boundary problems for PDEs
35R45 Partial differential inequalities and systems of partial differential inequalities
49J40 Variational inequalities
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
Full Text: DOI arXiv
[1] Almeida, L.; Bagnerini, P.; Habbal, A.; Noselli, S.; Serman, F., Tissue repair modeling, () · Zbl 1190.35218
[2] Almeida, L.; Bagnerini, P.; Habbal, A.; Noselli, S.; Serman, F., A mathematical model for dorsal closure, J. theoret. biol., 268, 1, 105-119, (2011) · Zbl 1411.92023
[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. · Zbl 1252.92027
[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] Almgren, F.; Taylor, J.E.; Wang, L.-H., Curvature-driven flows: a variational approach, SIAM J. control optim., 31, 2, 387-438, (1993) · Zbl 0783.35002
[6] Ambrosio, L., Movimenti minimizzanti, Rend. accad. naz. sci. XL mem. mat. appl. (5), 19, 191-246, (1995)
[7] Ambrosio, L.; Fusco, N.; Pallara, D., Functions of bounded variation and free discontinuity problems, Oxford math. monogr., (2000), The Clarendon Press/Oxford University Press New York · Zbl 0957.49001
[8] Barles, G.; Cesaroni, A.; Novaga, M., Homogenization of fronts in highly heterogeneous media, SIAM J. math. anal., 43, 1, 212-227, (2011) · Zbl 1228.35026
[9] Barles, G.; Da Lio, F., Remarks on the Dirichlet and state-constraint problems for quasilinear parabolic equations, Adv. differential equations, 8, 8, 897-922, (2003) · Zbl 1073.35120
[10] Bellettini, G.; Novaga, M., Comparison results between minimal barriers and viscosity solutions for geometric evolutions, Ann. sc. norm. super. Pisa cl. sci., 26, 1, 97-131, (1998) · Zbl 0904.35041
[11] Bellettini, G.; Caselles, V.; Chambolle, A.; Novaga, M., Crystalline Mean curvature flow of convex sets, Arch. ration. mech. anal., 179, 1, 109-152, (2006) · Zbl 1148.53049
[12] Caffarelli, L., The obstacle problem revisited, J. Fourier anal. appl., 4, 383-402, (1998) · Zbl 0928.49030
[13] Cardaliaguet, P.; Lions, P.-L.; Souganidis, P., A discussion about the homogenization of moving interfaces, J. math. pures appl., 91, 4, 339-363, (2009) · Zbl 1180.35070
[14] Caselles, V.; Chambolle, A., Anisotropic curvature-driven flow of convex sets, Nonlinear anal., 65, 8, 1547-1577, (2006) · Zbl 1107.35069
[15] Chambolle, A., An algorithm for Mean curvature motion, Interfaces free bound., 6, 2, 195-218, (2004) · Zbl 1061.35147
[16] Chambolle, A.; Caselles, V.; Cremers, D.; Novaga, M.; Pock, T., An introduction to total variation for image analysis, (), 263-340 · Zbl 1209.94004
[17] Chambolle, A.; Novaga, M., Implicit time discretization of the Mean curvature flow with a discontinuous forcing term, Interfaces free bound., 10, 283-300, (2008) · Zbl 1170.65054
[18] Craciun, B.; Bhattacharya, K., Effective motion of a curvature-sensitive interface through a heterogeneous medium, Interfaces free bound., 6, 151-173, (2004) · Zbl 1061.35148
[19] Crandall, M.; Ishii, H.; Lions, P.-L., Userʼs guide to viscosity solutions of second order partial differential equations, Bull. amer. math. soc., 27, 1, 1-67, (1992) · Zbl 0755.35015
[20] Da Lio, F., Comparison results for quasilinear equations in annular domains and applications, Comm. partial differential equations, 27, 1-2, 283-323, (2002) · Zbl 0994.35014
[21] Dirr, N.; Karali, G.; Yip, N.K., Pulsating wave for Mean curvature flow in inhomogeneous medium, European J. appl. math., 19, 661-699, (2008) · Zbl 1185.53076
[22] Hutson, M.S.; Tokutake, Y.; Chang, M.; Bloor, J.; Venakides, S.; Kiehart, D.P.; Edwards, G., Forces for morphogenesis investigated with laser microsurgery and quantitative modeling, Science, 300, 5616, 145-149, (2003)
[23] Kohn, R.V.; Serfaty, S., A deterministic-control based approach to fully nonlinear parabolic and elliptic equations, Comm. pure appl. math., 63, 1298-1350, (2010) · Zbl 1204.35070
[24] Luckhaus, S.; Sturzenhecker, T., Implicit time discretization for the Mean curvature flow equation, Calc. var. partial differential equations, 3, 2, 253-271, (1995) · Zbl 0821.35003
[25] Meyer, Y., Oscillating patterns in image processing and nonlinear evolution equations, Univ. lecture ser., vol. 22, (2001), Amer. Math. Soc. Providence, RI, the fifteenth Dean Jacqueline B. Lewis memorial lectures
[26] Miranda, M., Frontiere minimali con ostacoli, Ann. univ. ferrara, 16, 1, 29-37, (1971) · Zbl 0266.49036
[27] Phillips, R., Crystals, defects and microstructures, (2001), Cambridge University Press
[28] Sethian, J.A., Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision and materials science, (1999), Cambridge University Press · Zbl 0973.76003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.