Symmetric $\kappa $-loops

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.