## Differential and Integral Equations

- Differential Integral Equations
- Volume 23, Number 9/10 (2010), 861-898.

### Symmetric $\kappa $-loops

Michela Guida and Sergio Rolando

#### Abstract

We prove the existence of planar closed curves with prescribed curvature $
\kappa $ ($\kappa $*-loops*) for classes of symmetric curvature functions
$\kappa :\mathbb{C}\rightarrow \mathbb{R}$ of any sign, either exhibiting some
homogeneity, or satisfying a uniform condition on the growth along radial
directions. The problem of $\kappa $-loops is equivalent to the problem of
$1$-periodic solutions $u\in C^{2}(\mathbb{R},\mathbb{C})$ to a nonlinear ODE,
namely $u^{\prime \prime }=i\left\| u\right\| _{L^{2}(\left[ 0,1\right] )}\kappa
\left( u\right) u^{\prime }$, which also bears different physical and
geometrical interpretations. Such a problem is variational in nature and, thanks
to low dimension, the main difficulty is the existence of bounded Palais-Smale
sequences, which cannot be granted by standard arguments of critical point
theory.

#### Article information

**Source**

Differential Integral Equations, Volume 23, Number 9/10 (2010), 861-898.

**Dates**

First available in Project Euclid: 20 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356019116

**Mathematical Reviews number (MathSciNet)**

MR2675586

**Zentralblatt MATH identifier**

1240.58008

**Subjects**

Primary: 53A04: Curves in Euclidean space 34B15: Nonlinear boundary value problems 58E99: None of the above, but in this section

#### Citation

Guida, Michela; Rolando, Sergio. Symmetric $\kappa $-loops. Differential Integral Equations 23 (2010), no. 9/10, 861--898. https://projecteuclid.org/euclid.die/1356019116