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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
Published electronically: April 5, 2011
MathSciNet review: 2823090
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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.

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Additional Information

Roberta Musina
Affiliation: Dipartimento di Matematica ed Informatica, Università di Udine, via delle Scienze, 206-33100 Udine, Italy

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.

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