Stability and symmetry in the navier problem for the one-dimensional willmore equation

SIAM Journal on Mathematical Analysis

Volumn 40, Issue 5, 2009, Pages 2055-2076

  Deckelnick, Klaus ^a   Grunau, Hans Christoph ^a  

^a OTTO VON GUERICKE UNIVERSITY   (Germany)

Author keywords

Morse index; Navier boundary conditions; Stability; Symmetry; Willmore equation

Indexed keywords

ADMISSIBLE FUNCTIONS; BOUNDARY CURVATURE; MORSE INDEX; NAVIER BOUNDARY CONDITIONS; STABILITY PROBLEM; SUBRANGE; SYMMETRIC FUNCTIONS; SYMMETRIC SOLUTION; SYMMETRY; WILLMORE EQUATION;

BOUNDARY CONDITIONS;

STABILITY;

EID: 70350099504     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/07069033X     Document Type: Article

References (19)

1. M. Bauer and E. Kuwert, Existence of minimizing Willmore surface of prescribed genus, Int. Math. Res. Not., 2003, No.10 (2003), pp. 553-576.
2. A. Dall'Acqua, K. Deckelnick, and H.-Ch. Grunau, Classical solutions to the Dirichlet problem for Willmore surfaces of revolution, Adv. Calc. Var., to appear.
3. K. Deckelnick and G. Dziuk, Error analysis of a finite element method for the Willmore flow of graphs, Interfaces Free Bound., 8 (2006), pp. 21-46.
4. K. Deckelnick and H.-Ch. Grunau, Boundary value problems for the one-dimensional Will-more equation, Calc. Var. Partial Differential Equations, 30 (2007), pp. 293-314.
5. G. Dziuk, E. Kuwert, and R. Schatzle, Evolution of elastic curves in Rn: Existence and computation, SIAM J. Math. Anal., 33 (2002), pp. 1228-1245.
6. L. Euler, Opera Omnia, Ser. 1, 24, Orell Füssli, Zürich, 1952.
7. D. KINDERLEHRER and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Classics in Appl. Math. 31, SIAM, Philadelphia, 2000.
8. E. Kuwert and R. Schätzle, The Willmore flow with small initial energy, J. Differential Geom., 57 (2001), pp. 409-441.
9. E. Kuwert and R. Schatzle, Gradient flow for the Willmore functional, Comm. Anal. Geom., 10 (2002), pp. 307-339.
10. 10] E. Kuwert and R. Schatzle, Removability of point singularities of Willmore surfaces, Ann. of Math. (2), 160 (2004), pp. 315-357.
11. 11] Yu. Latushkin, J. Prüss, and R. Schnaubelt, Stable and unstable manifolds for quasilinear parabolic systems with fully nonlinear boundary conditions, J. Evol. Equ., 6 (2006), pp. 537-576.
12. 12] A. Linnér, Explicit elastic curves, Ann. Global Anal. Geom., 16 (1998), pp. 445-475.
13. U. F. Mayer and G. Simonett, A numerical scheme for axisymmetric solutions of curvature-driven free boundary problems, with applications to the Willmore flow, Interfaces Free Bound., 4 (2002), pp. 89-109.
14. 14] J. C. C. Nitsche, Boundary value problems for variational integrals involving surface curvatures, Quart. Appl. Math., 51 (1993), pp. 363-387.
15. A. Polden, Curves and Surfaces of Least Total Curvature and Fourth-Order Flows, Ph.D. thesis, University of Tubingen, Tubingen, Germany, 1996.
16. R. Schatzle, The Willmore Boundary Value Problem, preprint, University of Tubingen, Tubingen, Germany, 2006.
17. L. Simon, Existence of surfaces minimizing the Willmore functional, Comm. Anal. Geom., 1 (1993), pp. 281-326.
18. G. Simonett, The Willmore flow near spheres, Differential Integral Equations, 14 (2001), pp. 1005-1014.
19. T. J. Willmore, Total Curvature in Riemannian Geometry, Ellis Horwood Ser. Math. Appl., Ellis Horwood Limited, Chichester, Halsted Press, New York, 1982.