A numerical scheme for axisymmetric solutions of curvature-driven free boundary problems, with applications to the Willmore flow

Interfaces and Free Boundaries

Volumn 4, Issue 1, 2002, Pages 89-109

  Mayer, Uwe F ^a   Slmonett, Gleri ^b  

^a Vanderbilt University ^*  (United States)

^b VANDERBILT UNIVERSITY   (United States)

Author keywords

Numerical solutions; Singularities; Willmore flow

Indexed keywords

EID: 78651547991     PISSN: 14639963     EISSN: None     Source Type: Journal    
DOI: 10.4171/IFB/54     Document Type: Article

References (27)

1. Angenent, S. On the formation of singularities in the curve shortening flow. J. Diff. Geotru 33, (1991) 601-633.
2. Bertozzi, A. L., Bernoff, A. J., & Witelski, T. P. Axisymmetric surface diffusion: dynamics and stability of self-similar pinch-off. J. Stat. Phys. 93, (1998) 725-776.
3. Bradley, J. XV. 1053 Floyd Terrace, Bryn Mawr, PA, USA (1994).
4. Brakke, K. E. The surface evolver. Exp. Math. 1. (1992) 141-165.
5. Bryant, R. A duality theorem for Willmore surfaces. J. Diff. Ceom. 20. (1984) 23-53.
6. Escher, J., Mayer. U. F., & Simonett, G. The surface diffusion flow for immersed hypersurfaces. SIAM J. Math. Anal. 29.(1998) 1419-1433.
7. Giga, Y. & Ito, K. On pinching of curves moved by surface diffusion. Comm. Appl. Anal. 2, (1998) 393-405.
8. Giga, Y. & Ito, K. Loss of convexity of simple closed curves moved by surface diffusion. In: Escher, J. et al. (ed), Topics in Nonlinear Analysis. The Herbert Amann Anniversary Volume. (Prog. Nonlinear Diff. Equ. Appl. 35). pp. 305-320. Birkhauser. Basel (1999).
9. Hsu, L., Kusner, R. & Sullivan, J. Minimizing the squared mean curvature integral for surfaces in space forms. Exp. Math. 1, (1992) 191-207.
10. Kusner, R. Estimates for the biharmonic energy on unbounded planar domains, and the existence of surfaces of every genus that minimize the squared-mean-curvature integral. In: Peters, A. K. (ed), Elliptic and Parabolic Methods in Geometry (Minneapolis, MN. 1994). pp. 67-72. Wellesley, MA (1996).
11. Kuwert, E. & Schatzle, R. Gradient flow for the Willmore functional. Preprint.
12. Kuwert, E. & Schatzle, R. The Willmore flow with small initial energy. Preprint.
13. Langer, J. & Singer, D. Curves in the hyperbolic plane and mean curvatures of tori in 3-space. Bull. London Math. Soc. 16. (1984) 531-534.
14. Li, P. & Yau, S. T. A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces. Invent. Math. 69, (1982) 269-291.
15. Mayer, U. F. A numerical scheme for free boundary problems that are gradient flows for the area functional. Eur. J. Appl. Math. 11, (2000) 61-80.
16. Mayer, U. F. Numerical solutions for the surface diffusion flow in three space dimensions. Comput. Appl. Math. (2001), accepted.
17. Mayer, U. F. & Simonett, G. Self-intersections for the surface diffusion and the volume preserving mean curvature flow. Diff. Int. Equ. 13, (2000) 1189-1199.
18. Montiel, S. & Ros, A. Minimal immersions of surfaces by the first eigenfunctions and conformal area. Invent. Math. 83. (1985) 153-166.
19. Munzner, T., Levy, S., & Phillips, M. Geomview. The Geometry- Center, University of Minnesota. (1994).
20. Plnkall, U. Hopf tori in S3. Invent. Math. 81. (1985) 379-386.
21. Pinkall, U. & Sterling, I. Willmore surfaces. Math. Intelligencer 9. (1987) 38-43.
22. Simon, L. Existence of surfaces minimizing the Willmore functional. Comm. Anal. Geom. 1. (1993) 281-326.
23. Simonett, G. The Willmore flow near spheres. Diff. Int. Equ. 14. (2001) 1005-1014.
24. Weiner, J. L. On a problem of Chen. Willmore et al. Indiana University Math. J. 27. (1978) 19-35.
25. Willmore, T. J. Note on embedded surfaces. An. $ti. Univ. Al. I. Cuza' la$i Sec(. fa Mat. (NS) B11. (1965) 493-496.
26. Willmore, T. J. Mean curvature of immersed surfaces. An. Sti. Univ. 'Al. I. Cuza' la$i Secf. la Mat. (NS) 14, (1968) 99-103.
27. Willmore, T. J. Riemannian Geometry. Clarendon. Oxford (1993).