Let k: ℂ → ℝ be a smooth given function. A k-loop is a closed curve u in ℂ having prescribed curvature k(p) at every point p Ie ∈ 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 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.

Planar loops with prescribed curvature: Existence, multiplicity and uniqueness results / Musina, Roberta. - In: PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY. - ISSN 0002-9939. - 139:12(2011), pp. 4445-4459. [10.1090/S0002-9939-2011-10915-8]

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

#### Abstract

Let k: ℂ → ℝ be a smooth given function. A k-loop is a closed curve u in ℂ having prescribed curvature k(p) at every point p Ie ∈ 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 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.
##### Scheda breve Scheda completa Scheda completa (DC)
2011
139
12
4445
4459
PII S0002-9939(2011)10915-8
http://www.ams.org/journals/proc/2011-139-12/home.html
Musina, Roberta
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/20.500.11767/32309`
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