Remarks on a planar conformal curvature problem
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- 作者：Sun Yijing (1)
Cao Xiaoqiang (1)
1. Department of Mathematics ; University of Chinese Academy of Sciences ; Beijing ; 100049 ; People鈥檚 Republic of China - 关键词：Minkowski problem ; Negative powers ; Variational method ; 34B16 ; 34C25 ; 52A40
- 刊名：Monatshefte f篓鹿r Mathematik
- 出版年：2015
- 出版时间：April 2015
- 年：2015
- 卷：176
- 期：4
- 页码：623-636
- 全文大小：186 KB
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- 出版者：Springer Vienna
- ISSN：1436-5081
文摘
Given \(p and a positive \(L^{1}\) function \(g\) , with \(g(\theta +T)=g(\theta )\) for some \(T\le 2\pi \) , it is shown that the functional \({\mathcal {I}}_{T}(u)=\frac{1}{2}\int _{S}u^{2}+u'^{2}d\theta +\frac{1}{-p}\int _{S}gu^{p}d\theta \) has a minimizer in the class of nonnegative T-periodic functions \(u\in H^{1}(S)\) for which \(\int _{S}u^{2}+u'^{2}-gu^{p}d\theta \ge 0\) . This minimizer is used to prove the existence of a positive periodic solution \(u\in H^{1}(S)\) of a one dimensional conformal curvature problem \(-u''+u=\frac{g(\theta )}{u^{1-p}}\) .