Abstract
This contribution describes recent results on a variational approach for the geometric gradient flow of perimeter-like functionals, which include a class of non-local perimeters. In particular, the consistency of the variational approach with viscosity solutions of an appropriate level set equation is established.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
F. Almgren, J. Taylor and L. Wang, Curvature-driven flows: a variational approach, SIAM J. Control Optim 31 (1993), 387–438.
M. Barchiesi, S. H. Kang, M. L. Triet, M. Morini and M. Ponsiglione, A variational model for infinite perimeter segmentations based on Lipschitz level set functions: denoising while keeping finely oscillatory boundaries, Multiscale Model. Simul. 8 (2010).
L. A. Caffarelli, J.-M. Roquejoffre and O. Savin, Non-local minimal surfaces, Comm. Pure Appl. Math. 63 (2010), 1111–1144.
L. A. Caffarelli and P. E. Souganidis, Convergence of nonlocal threshold dynamics approximations to front propagation, ARMA 195 (2010).
P. Cardaliaguet, Front propagation problems with nonlocal terms, II, J. Math. Anal. Appl. 260 (2001), 572–601.
A. Chambolle, An algorithm for mean curvature motion, Interfaces Free Bound. 6 (2004), 195–218.
A. Chambolle, A. Giacomini and L. Lussardi, Continuous limits of discrete perimeters, M2AN Math. Model. Numer. Anal. 44 (2010), 207–230.
A. Chambolle and J. Darbon, On total variation minimization and surface evolution using parametric maximum flows, International Journal of Computer Vision 84 (2009), 288–307.
A. Chambolle, M. Morini and M. Ponsiglione, A non-local mean curvature flow and its semi-implicit time-discrete approximation, SIAM J. Math. Anal. 44 (2012), 4048–4077.
A. Chambolle, M. Morini and M. Ponsiglione, Nonlocal variational curvature flows, Work in progress.
Chen, Y.-G., Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Di. Geom. 33 (1991).
L. C. Evans and J. Spruck, Motion of level sets by mean curvature, J. Di Geom. 33 (1991).
C. Imbert, Level set approach for fractional mean curvature flows, Interfaces Free Bound 11 (2009).
H. Ishii and P. Souganidis, Generalized motion of noncompact hypersurfaces with velocity having arbitrary growth on the curvature tensor, Tohoku Math. J. (2) 47 (1995), 227–250.
S. Luckhaus and T. Sturzenhecker, Implicit time discretization for the mean curvature flow equation, Calc. Var. Partial Differential Equations 3 (1995), 253–271.
D. Slepčev, Approximation schemes for propagation of fronts with nonlocal velocities and Neumann boundary conditions, Nonlinear Anal. 52 (2003), 79–115.
E. Valdinoci, A fractional framework for perimeters and phase transitions, preprint 2012.
A. Visintin, Nonconvex functionals related to multiphase systems, SIAM J. Math. Anal. 21 (1990), 1281–1304.
A. Visintin, Generalized coarea formula and fractal sets, Japan J. Indust. Appl. Math. 8 (1991), 175–201.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Scuola Normale Superiore Pisa
About this paper
Cite this paper
Chambolle, A., Morini, M., Ponsiglione, M. (2013). Minimizing movements and level set approaches to nonlocal variational geometric flows. In: Chambolle, A., Novaga, M., Valdinoci, E. (eds) Geometric Partial Differential Equations proceedings. CRM Series, vol 15. Edizioni della Normale, Pisa. https://doi.org/10.1007/978-88-7642-473-1_4
Download citation
DOI: https://doi.org/10.1007/978-88-7642-473-1_4
Publisher Name: Edizioni della Normale, Pisa
Print ISBN: 978-88-7642-472-4
Online ISBN: 978-88-7642-473-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)