Symmetric \(\kappa \)-loops. (English) Zbl 1240.58008

Summary: 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 one-periodic solutions \(u\in C^2(\mathbb {R},\mathbb {C})\) to a nonlinear ODE, namely \(u''=i\| u\| _{L^2([0,1])}\kappa (u)u'\), which also bears different physical and geometrical interpretations. Such a problem is variational in nature and, thanks to the low dimension, the main difficulty is the existence of bounded Palais-Smale sequences, which cannot be granted by standard arguments of critical point theory.


58E10 Variational problems in applications to the theory of geodesics (problems in one independent variable)
34B15 Nonlinear boundary value problems for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
53A04 Curves in Euclidean and related spaces