6 — Albero di navigazione del sito 2 — Salta al contenuto

# Università  degli Studi di Udine

A | A | A
Tu sei qui: Portale preprintdimi.2-2010-musina
Title

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

Number2/2010
Deposited by:Roberta Musina
Deposited on:04/03/2010
StatusSubmitted for publication
AuthorsRoberta Musina
Abstract

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

File 2-2010-Musina_loops.pdf

Dipartimento di Matematica e Informatica
via delle Scienze 206 - 33100 UDINE
Tel +39-0432-558400 - Fax +39-0432-558499
email:
pec: dimi@postacert.uniud.it