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Classification of solutions for the planar isotropic $L_p$ dual Minkowski problem
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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_{\theta}^2+u^2)^{\frac{q-2}{2}}(u_{\theta\theta}+u)=1\quad\text{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.
Forward citations
Cited by 2 Pith papers
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Compactness of the $L_p$ dual Minkowski problem in $\mathbb{R}^3$
For convex bodies in R^3, bounded L_p qth dual curvature with p in [0,1) and q>2+p forces a uniform diameter upper bound and volume lower bound.
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Uniqueness of $S_2$-isotropic solutions to the isotropic $L_p$ Minkowski problem
Under a spectral-gap assumption on the Hilbert-Brunn-Minkowski operator, every S2-isotropic solution of the isotropic Lp Minkowski problem in the supercritical range p<-n is the unit ball.
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