Differential and Integral Equations

Symmetric $\kappa $-loops

Michela Guida and Sergio Rolando

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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

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

First available in Project Euclid: 20 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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

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