On length-preserving and area-preserving inverse curvature flow of planar curves with singularities

This paper aims to investigate the evolution problem for planar curves with singularities. Motivated by the inverse curvature flow introduced by Li and Wang (Calc. Var. Partial Differ. Equ. 62 (2023), No. 135), we intend to consider the area-preserving and length-preserving inverse curvature flow with nonlocal term for $\ell$-convex Legendre curves. For the area-preserving flow, an $\ell$-convex Legendre curve %of with initial algebraic area $A_0>0$ evolves to a circle of radius $\sqrt{\frac{A_0}{\pi}}$. For the length-preserving flow, an $\ell$-convex Legendre curve %of with initial algebraic length $L_0$ evolves to a circle of radius $\frac{L_0}{2\pi}$. As the by-product, we obtain some geometric inequalities for $\ell$-convex Legendre curves through the length-preserving flow.