Planar closed curves with prescribed curvature

TL;DR

This paper proves the existence of planar closed curves with specific curvature functions using variational methods and a novel approach to ensure bounded sequences.

Contribution

It introduces a new variational approach combined with a parameter and the monotonicity trick to establish existence results for prescribed curvature curves.

Findings

Existence of closed curves with prescribed curvature for certain functions

Application of the monotonicity trick to geometric variational problems

Development of a method to obtain bounded Palais-Smale sequences

Abstract

By variational methods, we prove existence of planar closed curves with prescribed curvature for some classes of curvature functions. The main difficulty is to obtain bounded Palais-Smale sequences. This is achieved by adding a parameter in the problem and using a version of the "monotonicity trick" introduced by M. Struwe in 1988.