Unstable Willmore surfaces of revolution subject to natural boundary conditions

Calculus of Variations and Partial Differential Equations

Volumn 48, Issue 3-4, 2013, Pages 293-313

  Dall'Acqua, Anna ^a   Deckelnick, Klaus ^a   Wheeler, Glen ^a  

^a OTTO VON GUERICKE UNIVERSITY   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84885622111     PISSN: 09442669     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00526-012-0551-y     Document Type: Article

References (28)

1. Bauer M., Kuwert E.: Existence of minimizing Willmore surfaces of prescribed genus. Int. Math. Res. Not. 2003(10), 553-576 (2003).
2. Bergner M., Dall'Acqua A., Fröhlich S.: Symmetric Willmore surfaces of revolution satisfying natural boundary conditions. Calc. Var. Partial Differ. Equ. 39(3-4), 361-378 (2010).
3. Bergner, M., Dall'Acqua, A., Fröhlich, S.: Willmore surfaces of revolution with two prescribed boundary circles. J. Geom. Anal. (2012), On-line first.
4. Bernard Y., Riviere T.: Local Palais-Smale sequences for the Willmore functional. Comm. Anal. Geom. 19(3), 563-600 (2011).
5. Blaschke, W.: Vorlesungen über Differentialgeometrie, vol. I-III, (1929).
6. Bryant R. L.: A duality theorem for Willmore surfaces. J. Differ. Geom. 20(1), 23-53 (1984).
7. Bryant R., Griffiths P.: Reduction for constrained variational problems and, Am. J. Math. 108(3), 525-570 (1986).
8. Dacorogna B.: Introduction to the Calculus of Variations. Imperial College Press, London (2004).
9. Dall'Acqua, A.: Uniqueness for the homogeneous Dirichlet Willmore boundary value problem. Ann. Glob. Anal. Geom. On-line first.
10. Dall'Acqua A., Deckelnick K., Grunau H.-C.: Classical solutions to the Dirichlet problem for Willmore surfaces of revolution. Adv. Calc. Var. 1(4), 379-397 (2008).
11. Dall'Acqua A., Fröhlich S., Grunau H.-C., Schieweck F.: Symmetric Willmore surfaces of revolution satisfying arbitrary Dirichlet boundary data. Adv. Calc. Var. 4(1), 1-81 (2011).
12. Deckelnick K., Grunau H.-C.: A Navier boundary value problem for Willmore surfaces of revolution. Analysis 29(3), 229-258 (2009).
13. Grunau, H.-C.: The asymptotic shape of a boundary layer of symmetric willmore surfaces of revolution. In: Bandle, C. et al. (eds.) Inequalities and Applications 2010, Volume 161 of International Series of Numerical Mathematics, pp. 19-29. Springer, Basel (2012).
14. Hertrich-Jeromin U., Pinkall U.: Ein Beweis der Willmoreschen Vermutung für Kanaltori. J. Reine Angew. Math. 430, 21-34 (1992).
15. Kuwert, E.: The Willmore functional. Lectures held at ETH in 2007.
16. n. Arvxiv peprint arXiv: 1007. 3967 (2010).
17. Kuwert, E., Schätzle, R.: Closed surfaces with bounds on their Willmore energy. To appear in Annali Sc. Norm. Sup. Pisa.
18. Langer J., Singer D. A.: Curve straightening and a minimax argument for closed elastic curves. Topology 24(1), 75-88 (1985).
19. Nitsche J. C. C.: Boundary value problems for variational integrals involving surface curvatures. Q. Appl. Math. 51(2), 363-387 (1993).
20. Palmer B.: The conformal Gauss map and the stability of Willmore surfaces. Ann. Glob. Anal. Geom. 9(3), 305-317 (1991).
21. Palmer B.: Uniqueness theorems for Willmore surfaces with fixed and free boundaries. Indiana Univ. Math. J. 49(4), 1581-1602 (2000).
22. Rivière T.: Analysis aspects of Willmore surfaces. Invent. Math. 174(1), 1-45 (2008).
23. Schätzle R.: The Willmore boundary problem. Calc. Var. Partial Differ. Equ. 37(3-4), 275-302 (2010).
24. Simon L.: Existence of surfaces minimizing the Willmore functional. Comm. Anal. Geom. 1(2), 281-326 (1993).
25. Struwe, M.: Plateau's Problem and the Calculus Of Variations. Mathematical Notes, vol. 35. Princeton University Press, Princeton.
26. Thomsen G.: Über konforme Geometrie I: Grundlagen der konformen Flächentheorie. (German). Abh. Math. Sem. Hamburg. 3, 31-56 (1923).
27. Weiner J. L.: On a problem of Chen, Willmore, et al. Indiana Univ. Math. J. 27(1), 19-35 (1978).
28. Willmore T. J.: Riemannian Geometry. Oxford Science Publications. The Clarendon Press Oxford University Press, New York (1993).