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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
61/2020

Global existence and convergence of a flow to Kazdan-Warner equation with non-negative prescribed function

Linlin Sun and Jingyong Zhu

Abstract

We consider an evolution problem associated to the Kazdan-Warner equation on a closed Riemann surface $(\Sigma,g) $\begin{align*} -\Delta_{g}u=8\pi\left(\dfrac{he^{u}}{\int_{\Sigma}he^{u}\textnormal{d}\mu_{g}}-\dfrac{1}{\int_{\Sigma}\textnormal{d}\mu_{g}}\right) \end{align*} where the prescribed function $h\geq0$ and $\max_{\Sigma}h>0$. We prove the global existence and convergence under additional assumptions such as \begin{align*} \Delta_{g}\ln h(p_0)+8\pi-2K(p_0)>0\end{align*}for any maximum point $p_0$ of the sum of $2\ln h$ and the regular part of the Green function, where $K$ is the Gaussian curvature of $\Sigma$. In particular, this gives a new proof of the existence result by Yang and Zhu [Proc. Amer. Math. Soc. 145(2017), no. 9, 3953-3959] which generalizes existence result of Ding, Jost, Li and Wang [Asian J. Math. 1(1997), no. 2, 230-248] to the non-negative prescribed function case.

Received:
Jun 1, 2020
Published:
Jun 1, 2020
MSC Codes:
35B33, 58J35
Keywords:
Kazdan-Warner equation, mean field type flow, global existence, global convergence

Related publications

inJournal
2021 Journal Open Access
Linlin Sun and Jingyong Zhu

Global existence and convergence of a flow to Kazdan-Warner equation with non-negative prescribed function

In: Calculus of variations and partial differential equations, 60 (2021) 1, p. 42