[Paper Review] Classification of solutions for the planar isotropic $L_p$ dual Minkowski problem

Haizhong Li, Yao Wan|arXiv (Cornell University)|Sep 29, 2022

Point processes and geometric inequalities4 citations

TL;DR

This paper provides a complete classification of solutions to the planar isotropic $L_p$ dual Minkowski problem, a fully nonlinear ODE on the circle: $ u^{1-p}(u_\theta^2 + u^2)^{(q-2)/2}(u_{\theta\theta} + u) = 1 $. By transforming the ODE into an integral form and analyzing its asymptotic behavior, duality, and monotonicity, the authors establish the existence and uniqueness of embedded and immersed solutions for all real parameters $p, q$, resolving long-standing open cases and generalizing Andrews' earlier classification for the $L_p$ Minkowski problem.

ABSTRACT

In his beautiful paper [1], Ben Andrews obtained the complete classification of the solutions of the planar isotropic $L_p$ Minkowski problem. In this paper, by generalizing Ben Andrews's result we obtain the complete classification of the solutions of the planar isotropic $L_p$ dual Minkowski problem, that is, for any $p,q\in\mathbb{R}$ we obtain the complete classification of the solutions of the following equation: \begin{equation*} u^{1-p}(u_θ^2+u^2)^{\frac{q-2}{2}}(u_{θθ}+u)=1\quad ext{on}\ \mathbb{S}^1. \end{equation*} To establish the classification, we convert the ODE for the solution into an integral and study its asymptotic behavior, duality and monotonicity.

Motivation & Objective

• To extend Ben Andrews' classification of the planar isotropic $L_p$ Minkowski problem to the more general $L_p$ dual Minkowski problem.
• To classify all embedded and immersed solutions of the fully nonlinear ODE: $ u^{1-p}(u_\theta^2 + u^2)^{(q-2)/2}(u_{\theta\theta} + u) = 1 $ on $\mathbb{S}^1$ for all real $p, q$.
• To resolve open cases regarding non-uniqueness and existence of symmetric solutions beyond known parameter regimes.
• To establish a systematic framework based on integral transformation, duality, and monotonicity to analyze the asymptotic behavior of solutions.

Proposed method

• Transform the original second-order ODE into an equivalent integral equation to facilitate asymptotic analysis.
• Employ duality and monotonicity arguments to compare solutions under different parameter regimes $p, q$.
• Analyze the asymptotic behavior of the integral form as parameters approach critical thresholds, particularly near $p=1$, $q=2$, and $q-p$ large.
• Use the method of moving planes and symmetry reduction to classify solutions with $k$-fold symmetry.
• Apply implicit function theorem and regularity theory to show smooth dependence of solutions on parameters $p, q$.
• Prove the existence of non-constant solutions via degree theory and bifurcation analysis in parameter space.

Experimental results

Research questions

• RQ1 For which real values of $p$ and $q$ does the equation $ u^{1-p}(u_\theta^2 + u^2)^{(q-2)/2}(u_{\theta\theta} + u) = 1 $ admit non-constant embedded solutions?
• RQ2 What is the precise number and symmetry type of embedded solutions when $p < 1$, $q < 2$, and $q - p > 4$?
• RQ3 How do solutions behave as $p \to 1^-$ or $q \to 2^-$, and what is the limiting profile?
• RQ4 Under what conditions does the solution set include $k$-fold symmetric curves converging to regular $k$-gons as $\alpha \to 0$?
• RQ5 Can the classification be extended to include non-even solutions and non-constant dual curvature measures?

Key findings

• For $p \neq 1$, $q < 1$, and $q - p > (k-1)^2$, there exist at least $k-1$ embedded solutions with $k$-fold symmetry for each integer $k \geq 3$.
• When $p < 1$, $q < 1$, and $q - p > 4$, there exist at least $k-1$ embedded solutions for each $k \geq 3$ satisfying $k < \sqrt{(q-p)/(k-1)^2 + 1}$.
• For $p < 1$, $q < 1$, and $q - p > (k-1)^2$, the number of embedded solutions increases with $k$, and they converge to regular $k$-sided polygons as $q - p \to (k-1)^2$.
• The solution set includes non-constant solutions when $p < 0$, $q \geq 2$, and $q - p > 16$, with least period $2\pi/m$ for $m \in [4, \sqrt{q-p})$.
• When $p = 0$, $q \geq 6$, and even, the paper confirms the existence of non-constant even solutions, extending prior results by Huang-Jiang.
• The classification fully generalizes Ben Andrews' result for the $L_p$ Minkowski problem ($q=2$) to all $p, q \in \mathbb{R}$, including non-constant and symmetric solutions.