Abstract
The n-dimensional Helfrich variational problem is, roughly speaking, as follows: minimize the integral of squared mean curvature among n-dimensional hypersurfaces with the prescribed area and enclosed volume. In this article the n-dimensional Helfrich flow is considered. This is the L 2-gradient flow associated with the n-dimensional Helfrich variational problem. The equation of the flow is described as a projected gradient flow. A local existence result and partial uniqueness results are presented. In particular the latter improves previous results.
The author partly supported by Grant-in-Aid for Scientific Research (A) (No. 22244010-01), Japan Society for the Promotion Science.
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
References
Amann, H.: Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. In: Schmeisser, H.-J., Triebel, H. (eds.) Function Spaces, Differential Operators and Nonlinear Analysis, Friedrichroda, 1992. Teubner-Text Math., vol. 133, pp. 9–126. Trubner, Stuttgart (1993)
Au, T.K.-K., Wan, T.Y.-H.: Analysis on an ODE arisen from studying the shape of a red blood cell. J. Math. Anal. Appl. 282(1), 279–295 (2003)
Brailsfold, J.D.: Mechnoelastic properties of biological membranes. In: Membrane Fluidity in Biology, vol. 1. Concepts of Membrane Structure, pp. 291–319. Academic Press, San Diego (1983)
Chen, B.-y.: On a variational problem on hypersurfaces. J. Lond. Math. Soc. (2) 6, 321–325 (1973)
Helfrich, W.: Elastic properties of lipid bilayers: Theory and possible experiments. Z. Naturforsch. 28c, 693–703 (1973)
Jenkins, J.T.: Static equilibrium configurations of a model red blood cell. J. Math. Biol. 4, 149–169 (1977)
Kohsaka, Y., Nagasawa, T.: On the existence for the Helfrich flow and its center manifold near spheres. Differ. Integral Equ. 19(2), 121–142 (2006)
Kurihara, T., Nagasawa, T.: On the gradient flow for a shape optimization problem of plane curves as a singular limit. Saitama Math. J. 24, 43–75 (2006/2007)
Nagasawa, T., Takagi, I.: Bifurcating critical points of bending energy with constraints related to the shape of red blood cells. Calc. Var. Partial Differ. Equ. 16(1), 63–111 (2003)
Nagasawa, T., Takagi, I.: in preparation
Nagasawa, T., Yi, T.: Local existence and uniqueness for the n-dimensional Helfrich flow as a projected gradient flow. Hokkaido Math. J. 41(2), 209–226 (2012)
Ou-Yang, Z.-c., Helfrich, W.: Bending energy of vesicle membranes: general expressions for the first, second, and third variation of the shape energy and applications to spheres and cylinders. Phys. Rev. A 39(10), 5280–5288 (1989)
Peterson, M.A.: An instability of the red blood cell shape. J. Appl. Phys. 57(5), 1739–1742 (1985)
Shikhman, V., Stein, O.: Constrained optimization: projected gradient flows. J. Optim. Theory Appl. 140(1), 117–130 (2009)
Watanabe, K.: Plane domains which are spectrally determined. Ann. Glob. Anal. Geom. 18(5), 447–475 (2000)
Watanabe, K.: Plane domains which are spectrally determined. II. J. Inequal. Appl. 7(1), 25–47 (2002)
Acknowledgements
The author thanks referees for various useful comments.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Italia
About this chapter
Cite this chapter
Nagasawa, T. (2013). Existence and Uniqueness of the n-Dimensional Helfrich Flow. In: Magnanini, R., Sakaguchi, S., Alvino, A. (eds) Geometric Properties for Parabolic and Elliptic PDE's. Springer INdAM Series, vol 2. Springer, Milano. https://doi.org/10.1007/978-88-470-2841-8_15
Download citation
DOI: https://doi.org/10.1007/978-88-470-2841-8_15
Publisher Name: Springer, Milano
Print ISBN: 978-88-470-2840-1
Online ISBN: 978-88-470-2841-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)