Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Planar loops with prescribed curvature: Existence, multiplicity and uniqueness results


Author: Roberta Musina
Journal: Proc. Amer. Math. Soc. 139 (2011), 4445-4459
MSC (2010): Primary 51M25, 53A04, 49J10
DOI: https://doi.org/10.1090/S0002-9939-2011-10915-8
Published electronically: April 5, 2011
MathSciNet review: 2823090
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ k:\mathbb{C}\to \mathbb{R}$ be a smooth given function. A $ k$-loop is a closed curve $ u$ in $ \mathbb{C}$ having prescribed curvature $ k(p)$ at every point $ p\in u$. We use variational methods to provide sufficient conditions for the existence of $ k$-loops. Then we show that a breaking symmetry phenomenon may produce multiple $ k$-loops, in particular when $ k$ is radially symmetric and somewhere increasing. If $ k>0$ is radially symmetric and non-increasing, we prove that any embedded $ k$-loop is a circle; that is, round circles are the only convex loops in $ \mathbb{C}$ whose curvature is a non-increasing function of the Euclidean distance from a fixed point. The result is sharp, as there exist radially increasing curvatures $ k>0$ which have embedded $ k$-loops that are not circles.


References [Enhancements On Off] (What's this?)

  • 1. Aeppli, A., On the uniqueness of compact solutions for certain elliptic differential equations, Proc. Amer. Math. Soc., 11, 826-832 (1960). MR 0121567 (22:12304)
  • 2. Alexandrov, A.D., Uniqueness theorems for surfaces in the large. I, Vestink Leningrad Univ., 11, 5-17 (1956). Amer. Math. Soc. Transl. Ser. 2, 21, 341-354 (1962). MR 0150706 (27:698a)
  • 3. Brezis, H., Coron, J. M., Convergence of solutions of H-systems or how to blow bubbles, Arch. Rat. Mech. Anal., 89, 21-56 (1985). MR 784102 (86g:53007)
  • 4. Caldiroli, P., Guida, M., Closed curves in $ \mathbb{R}\sp 3$ with prescribed curvature and torsion in perturbative cases. I. Necessary condition and study of the unperturbed problem, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 17, 227-242 (2006). MR 2254070 (2007f:53004)
  • 5. Caldiroli, P., Guida, M., Helicoidal trajectories of a charge in a nonconstant magnetic field, Adv. Differential Equations, 12, 601-622 (2007). MR 2319450 (2008j:58013)
  • 6. Caldiroli, P., Musina, R., Existence of minimal H-bubbles, Commun. Contemp. Math., 4, 177-209 (2002). MR 1901145 (2004b:53013)
  • 7. Caldiroli, P., Musina, R., Bubbles with prescribed mean curvature: the variational approach, Nonlinear Analysis TMA, to appear, DOI:10.1016/j.na.2011.01.019
  • 8. Guida, M., Perturbative-type results for some problems of geometric analysis in low dimension, Ph.D. Thesis, Università di Torino (2004).
  • 9. Guida, M., Rolando, S., Symmetric $ k$-loops, Differential Integral Equations, 23, 861-898 (2010). MR 2675586
  • 10. Osserman, R., The four or more vertex theorem, Amer. Math. Monthly, 92, 332-337 (1985). MR 790188 (87e:53001)
  • 11. Pucci, P., Serrin, J., The strong maximum principle revisited, J. Diff. Equations, 196, 1-66 (2004). MR 2025185 (2004k:35033)
  • 12. Schneider, M., Multiple solutions for the planar Plateau problem, preprint, arXiv:0903.1132 (2009).
  • 13. Treibergs, A.E., Wei, W., Embedded hyperspheres with prescribed mean curvature, J. Differential Geom., 18, 513-521 (1983). MR 723815 (85e:53082)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 51M25, 53A04, 49J10

Retrieve articles in all journals with MSC (2010): 51M25, 53A04, 49J10


Additional Information

Roberta Musina
Affiliation: Dipartimento di Matematica ed Informatica, Università di Udine, via delle Scienze, 206-33100 Udine, Italy
Email: roberta.musina@uniud.it

DOI: https://doi.org/10.1090/S0002-9939-2011-10915-8
Received by editor(s): May 3, 2010
Received by editor(s) in revised form: October 14, 2010
Published electronically: April 5, 2011
Communicated by: Matthew J. Gursky
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society