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Gradient flow for the Helfrich functional

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Abstract

The author studies the L 2 gradient flow of the Helfrich functional, which is a functional describing the shapes of human red blood cells. For any λ i ≥ 0 and c 0, the author obtains a lower bound on the lifespan of the smooth solution, which depends only on the concentration of curvature for the initial surface.

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References

  1. Helfrich, W., Elastic properties of lipid bilayers: Theory and possible experiments, Z. Naturforsch, 28, 1973, 693–703.

    MathSciNet  Google Scholar 

  2. Kohsaka, Y. and Nagasawa, T., On the existence for the Helfrich flow and its center manifold near spheres, Diff. Intergral Eq., 19(2), 2006, 121–142.

    MathSciNet  MATH  Google Scholar 

  3. Kuwert, E. and Schätzle, R., The Willmore flow with small initial energy, J. Diff. Geom., 57(3), 2001, 409–441.

    MATH  Google Scholar 

  4. Kuwert, E. and Schätzle, R., Gradient flow for the Willmore functional, Comm. Anal. Geom., 10(2), 2002, 307–339.

    MathSciNet  MATH  Google Scholar 

  5. Simonett, G., The Willmore flow near spheres, Diff. Integral Eq., 14(8), 2001, 1005–1014.

    MathSciNet  MATH  Google Scholar 

  6. Liu, Y. N. and Jian, H. Y., A curve flow evolved by a fourth order parabolic equation, Sci. China, Ser. A, 2009, 52(9), 2177–2184.

    Article  MathSciNet  MATH  Google Scholar 

  7. Jian, H. Y. and Liu, Y. N., Ginzburg-Landau vortex and mean curvature flow with external force field, Acta Math. Sin., Engl. Ser., 22(6), 2006, 1831–1842.

    Article  MathSciNet  MATH  Google Scholar 

  8. Jian, H. Y. and Liu, Y. N., Long-time existence of mean curvature flow with external force fields, Pacific J. Math., 234(2), 2008, 311–315.

    Article  MathSciNet  MATH  Google Scholar 

  9. Liu, Y. N. and Jian, H. Y., Evolution of hypersurfaces by mean curvature minus external force field, Sci. China, Ser. A, 50(2), 2007, 231–239.

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Yannan Liu.

Additional information

Project supported by the National Natural Science Foundation of China (No. 11026121) and the Training Programme Foundation for the Excellent Talents of Beijing (No. 2012D005003000004).

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Liu, Y. Gradient flow for the Helfrich functional. Chin. Ann. Math. Ser. B 33, 931–940 (2012). https://doi.org/10.1007/s11401-012-0741-0

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  • DOI: https://doi.org/10.1007/s11401-012-0741-0

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