论文标题
平面各向同性$ L_P $ DUAL MINKOWSKI问题的解决方案分类
Classification of solutions for the planar isotropic $L_p$ dual Minkowski problem
论文作者
论文摘要
本·安德鲁斯（Ben Andrews）在他美丽的论文[1]中获得了平面各向同性$ L_P $ Minkowski问题的完整分类。在本文中，通过概括本·安德鲁斯的结果，我们获得了平面各向同性$ l_p $ dual minkowski问题的完整分类，也就是说，对于任何$ p，q \ in \ mathbb {r} $，我们获得了以下等式的解决方案的完整分类： u^{1-p}（u_θ^2+u^2）^{\ frac {q-2} {2}}}}}（u_ {θθ}+u）= 1 \ quad \ text \ text {on} \ \ \ \ \ \ \ \ \ m马理{s}^1。 \ end {方程*}为了建立分类，我们将溶液的颂歌转换为积分，并研究其渐近行为，二元性和单调性。
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\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.
