Classification of solutions for the planar isotropic L_p dual Minkowski problem

2022/09/29 by Haizhong Li, Yao Wan, Li, Haizhong +1 · 2 citations
Mathematics · #Point processes and geometric inequalities #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2209.14630

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∈ℝ we obtain the complete classification of the solutions of the following equation: u^1-p(u_θ²+u²)^(q-2)/(2)(u_θθ+u)=1\quadon \mathbbS¹. To establish the classification, we convert the ODE for the solution into an integral and study its asymptotic behavior, duality and monotonicity.

Cited by

• Uniqueness of S₂-isotropic solutions to the isotropic L_p Minkowski problem
• The L_p Dual Minkowski Problem for Group-Invariant Convex Bodies

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