Planar loops with prescribed curvature via Hardy's inequality

TL;DR

This paper explores the existence of closed planar loops with specific curvature by employing a variational approach based on Hardy's inequality and related functional analysis techniques.

Contribution

It introduces a novel variational method utilizing Hardy's inequality to establish the existence of planar loops with prescribed curvature.

Findings

Existence of closed planar loops with given curvature proven

Development of a Hardy inequality-based variational framework

New functional space tailored for curvature problems

Abstract

We investigate the existence of closed planar loops with prescribed curvature. Our approach is variational, and relies on a Hardy type inequality and its associated functional space.