Mean Field Equation of Liouville Type with Singular Data Topological Degree

CHIUN-CHUAN CHEN National Taiwan University Chang-Shou Lin National Taiwan University

Analysis of PDEs mathscidoc:1705.03004

CPAM, 887-947, 2015
We consider the following mean field equation: $$\Delta_{g}v+\rho\left(\frac{h^*e^v}{\int_Mh^*e^v}-1\right)=4\pi\sum_{j=1}^N\alpha_j(\delta_{q_j}-1)$$ on M, where M is a compact Riemann surface with volume 1, $h^*$ is a positive $C^1$ function on M, and  $\alpha_j$ are constants satisfying $\alpha_j>-1$. In this paper, we derive the topological degree counting formula for noncritical values of $\rho$. We also give several applications of this formula, including existence of curvature +1 metric with conic singularities, doubly periodic solutions of electroweak theory, and a special case of self-gravitating strings.
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@inproceedings{chiun-chuan2015mean,
  title={Mean Field Equation of Liouville Type with Singular Data Topological Degree},
  author={CHIUN-CHUAN CHEN, and Chang-Shou Lin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170530175133086541779},
  booktitle={CPAM},
  pages={887-947},
  year={2015},
}
CHIUN-CHUAN CHEN, and Chang-Shou Lin. Mean Field Equation of Liouville Type with Singular Data Topological Degree. 2015. In CPAM. pp.887-947. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170530175133086541779.
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