Existence of Self-similar Solutions to the Anisotropic Affine Curve-shortening Flow

Abstract

Abstract In this paper the existence of positive $2\pi $-periodic solutions to the ordinary differential equation $$\begin{equation*} u^{\prime\prime}+u=\frac{f}{u^3} \ \textrm{ in } \mathbb{R} \end{equation*}$$is studied, where $f$ is a positive $2\pi $-periodic smooth function. By virtue of a new generalized Blaschke–Santaló inequality, we obtain a new existence result of solutions.