Abstract
We study the asymptotic analysis of a singularly perturbed weakly parabolic system of m-equations of anisotropic reaction-diffusion type. Our main result formally shows that solutions to the system approximate a geometric motion of a hypersurface by anisotropic mean curvature. The anisotropy, supposed to be uniformly convex, is explicit and turns out to be the dual of the star-shaped combination of the m original anisotropies.
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.
References
L. Ambrosio, P. Colli Franzone and G. Savaré, On the asymptotic behaviour of anisotropic energies arising in the cardiac bidomain model, Interface Free Bound, 2 (2000), 213–266.
D. Bao, S.-S. Chern and Z. Shen, “An Introduction to Riemann-Finsler Geometry”, Graduate Texts in Mathematics, Vol. 200, Springer-Verlag, 2000.
G. Bellettini, “Lecture Notes on Mean Curvature Flow: Barriers and Singular Perturbations”, Edizioni Sc. Norm. Sup. Pisa, to appear.
G. Bellettini, P. Colli Franzone and M. Paolini, Convergence of front propagation for anisotropic bistable reaction-diffusion equations, Asymptotic Anal. 15 (1997), 325–358.
G. Bellettini and M. Paolini, Quasi—optimal error estimates for the mean curvature flow with a forcing term, Differential Integral Equations 8 (1995), 735–752.
G. Bellettini and M. Paolini, Anisotropic motion by mean curvature in the context of Finsler geometry, Hokkaido Math. J. 25 (1996), 537–566.
G. Bellettini, M. Paolini and F. Pasquarelli, Non convex mean curvature flow as a formal singular limit of the non linear bidomain model, Advances in Differential Equations, to appear.
G. Bellettini, M. Paolini and S. Venturini, Some results on surface measures in Calculus of Variations, Ann. Mat. Pura Appl. 170 (1996), 329–359.
J. Bence, B. Merriman and S. Osher, Diffusion generated motion by mean curvature, Computational Crystal Growers Workshop, J. Taylor (ed.), Selected Lectures in Math., Amer. Math. Soc. (1992), 73–83.
E. Bonnetier, E. Bretin and A. Chambolle, Consistency result for a non monotone scheme for anisotropic mean curvature flow, Interfaces Free Bound. 14 (2012), 1–35.
L. Bronsard and F. Reitich, On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation Arch. Ration. Mech. Anal. 124 (1993), 355–379.
A. Chambolle and M. Novaga, Convergence of an algorithm for the anisotropic and crystalline mean curvature flow, SLAM J. Math. Anal. 37 (2006), 1878–1987.
P. Colli Franzone, M. Pennacchio and G. Savaré, Multiscale modeling for the bioelectric activity of the heart, SLAM J. Math. Anal. 37 (2005), 1333–1370.
P. Colli Franzone and G. Savaré, Degenerate evolution systems modeling the cardiac electric field ad micro and macroscopic level, Evolution equations, semigroups and functional analysis (Milano 2000), Vol. 50 Progress Nonlin. Diff. Equations Appl. Birkhäuser, Basel (2002), 49–78.
J. Coromilas, A. L. Dillon and S. M. Wit, Anisotropy reentry as a cause of ventricular tachyarrhytmias, Cardiac Electrophysiology: From Cell to Bedside, ch.49, W. B. Saunders Co., Philadelphia (1994), 511–526.
R. M. Miura, Accurate computation of the stable solitary wave for the FitzHugh-Nagumo equations, J. Math. Biol. 13 (1981/82), 247–269.
R. T. Rockafellar, “Convex Analysis”, Princeton University Press, Princeton, 1972.
R. Schneider, “Convex Bodies: the Brunn-Minkowski Theory”, Encyclopedia of Mathematics and its Applications, Vol. 44, Cambridge Univ. Press, 1993.
A. C. Thompson, “Minkowski Geometry”, Encyclopedia of Mathematics and its Applications, Vol. 63, Cambridge Univ. Press, 1996.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Scuola Normale Superiore Pisa
About this paper
Cite this paper
Amato, S., Bellettini, G., Paolini, M. (2013). The nonlinear multidomain model: a new formal asymptotic analysis. 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_2
Download citation
DOI: https://doi.org/10.1007/978-88-7642-473-1_2
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)