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On the planar Lp-Minkowski problem
Journal of Differential Equations ( IF 2.4 ) Pub Date : 2021-03-29 , DOI: 10.1016/j.jde.2021.03.035
Shi-Zhong Du

In this paper, we study the planar Lp-Minkowski problem(0.1)uθθ+u=fup1,θS1 for all pR, which was introduced by Lutwak [21]. A detailed exploration of (0.1) on solvability will be presented. More precisely, we will prove that for p(0,2), there exists a positive function fCα(S1),α(0,1) such that (0.1) admits a nonnegative solution vanishes somewhere on S1. In case p(1,0], a surprising a-priori upper/lower bound for solution was established, which implies the existence of positive classical solution to each positive function fCα(S1). When p(2,1], the existence of some special positive classical solution has already been known using the Blaschke-Santalo inequality [7]. Upon the final case p2, we show that there exist some positive functions fCα(S1) such that (0.1) admits no solution. Our results clarify and improve largely the planar version of Chou-Wang's existence theorem [7] for p<2. At the end of this paper, some new uniqueness results will also be shown.



中文翻译:

关于平面L p -Minkowski问题

在本文中,我们研究平面 大号p-Minkowski问题(0.1)üθθ+ü=Füp-1个θ小号1个 对全部 p[R,这是由Lutwak [21]提出的。将介绍(0.1)关于可溶性的详细说明。更确切地说,我们将证明p02个,有一个积极的作用 FCα小号1个α01个 使得(0.1)承认非负解消失在 小号1个。以防万一p-1个0],建立了令人惊讶的先验上/下界解,这意味着每个正函数都存在正经典解FCα小号1个。什么时候p-2个-1个],已经使用Blaschke-Santalo不等式[7]知道了一些特殊的正经典解的存在。在最后的情况下p-2个,我们发现这里存在一些积极的功能 FCα小号1个因此(0.1)不接受任何解决方案。我们的结果在很大程度上澄清和改进了Chou-Wang存在定理[7]的平面形式。p<2个。在本文的结尾,还将显示一些新的唯一性结果。

更新日期:2021-03-29
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