We consider an evolution problem associated to the Kazdan-Warner equation on a closed Riemann surface $(Σ,g)$
\begin{align*}
-Δ_{g}u=8π\left(\frac{he^{u}}{\int_Σhe^{u}{\rm d}μ_{g}}-\frac{1}{\int_Σ{\rm d}μ_{g}}\right)
\end{align*} where the prescribed function $h\geq0$ and $\max_Σh>0$. We prove the global existence and convergence under additional assumptions such as
\begin{align*}
Δ_{g}\ln h(p_0)+8π-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 $Σ$. 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.
In this paper, we study the sign-changing Kazdan-Warner's problem on two dimensional closed Riemannian manifold with negative Euler number $χ(M)<0$. We show that once, the direct method on convex sets is used to find a minimizer of the corresponding functional, then there is another solution via a use of the variational method of mountain pass. In conclusion, we show that there are at least two solutions to the Kazdan-Warner's problem on two dimensional Kazdan-Warner equation provided the prescribed function changes signs and with this average negative.
We concern in this paper the graph Kazdan-Warner equation \begin{equation*} Δf=g-he^f \end{equation*} on an infinite graph, the prototype of which comes from the smooth Kazdan-Warner equation on an open manifold. Different from the variational methods often used in the finite graph case, we use a heat flow method to study the graph Kazdan-Warner equation. We prove the existence of a solution to the graph Kazdan-Warner equation under the assumption that $h\leq0$ and some other integrability conditions or constrictions about the underlying infinite graphs.
We investigate the Kazdan-Warner equation on a network. In this case, the differential equation is defined on each edge, while appropriate transition conditions of Kirchhoff type are prescribed at the vertices. We show that the Kazdan-Warner theory extends to the present setting and we study also the critical case.
We study Kazdan-Warner equations on a connected finite graph via the method of the degree theory. Firstly, we prove that all solutions to the Kazdan-Warner equation with nonzero prescribed function are uniformly bounded and the Brouwer degree is well defined. Secondly, we compute the Brouwer degree case by case. As consequences, we give new proofs of some known existence results for the Kazdan-Warner equation on a connected finite graph.
Let $G=(V,E)$ be a connected finite graph. In this short paper, we reinvestigate the Kazdan-Warner equation $$Δu=c-he^u$$ with $c<0$ on $G$, where $h$ defined on $V$ is a known function. Grigor'yan, Lin and Yang \cite{GLY} showed that if the Kazdan-Warner equation has a solution, then $\overline{h}$, the average value of $h$, is negative. Conversely, if $\overline{h}<0$, then there exists a number $c_-(h)<0$, such that the Kazdan-Warner equation is solvable for every $0>c>c_-(h)$ and it is not solvable for $c<c_-(h)$. Moreover, if $h\leq0$ and $h\not\equiv0$, then $c_-(h)=-\infty$. Inspired by…
Let $G=(V,E)$ be a finite graph and $Δ$ be the usual graph Laplacian. Using the calculus of variations and a method of upper and lower solutions, we give various conditions such that the Kazdan-Warner equation $Δu=c-he^u$ has a solution on $V$, where $c$ is a constant, and $h:V\rightarrow\mathbb{R}$ is a function. We also consider similar equations involving higher order derivatives on graph. Our results can be compared with the original manifold case of Kazdan-Warner (Ann. Math., 1974).
We study the Kazdan-Warner equation on canonically compactifiable graphs. These graphs are distinguished as analytic properties of Laplacians on these graphs carry a strong resemblance to Laplacians on open pre-compact manifolds.
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences（保加利亚科学院数学与信息学研究所）；Departamento de Matemática – Universidade Federal da Paraíba（帕拉伊巴联邦大学数学系）；Instituto de Ciência e Tecnologia – Unifesp（圣保罗联邦大学科学与技术学院）
Let $(M,g)$ be a compact Riemann surface with unit area, $h\in C^{\infty}(M)$ a function which is positive somewhere, $ρ>0$, $p_i\in M$ and $α_i\in(-1,+\infty)$ for $i=1,\cdots,\ell$, we consider the mean field equation \begin{align*} Δv + 4π\sum_{i=1}^{\ell}α_i\left(1-δ_{p_i}\right) = ρ\left(1-\frac{he^v}{\int_Mhe^vdμ}\right), \end{align*} on $M$, where $Δ$ and $dμ$ are the Laplace-Beltrami operator and the area element of $(M,g)$ respectively. Using variational method and blowup analysis, we prove some existence results in the critical case $ρ=8π(1+\min\{0,\min_{1\leq i\leq\ell}α_i\})$. Thes…
Let $(Σ,g)$ be a compact Riemannian surface without boundary and $λ_1(Σ)$ be the first eigenvalue of the Laplace-Beltrami operator $Δ_g$. Let $h$ be a positive smooth function on $Σ$. Define a functional $$J_{α,β}(u)=\frac{1}{2}\int_Σ(|\nabla_gu|^2-αu^2)dv_g-β\log\int_Σhe^udv_g$$ on a function space $\mathcal{H}=\left\{u\in W^{1,2}(Σ): \int_Σudv_g=0\right\}$. If $α<λ_1(Σ)$ and $J_{α,8π}$ has no minimizer on $\mathcal{H}$, then we calculate the infimum of $J_{α,8π}$ on $\mathcal{H}$ by using the method of blow-up analysis. As a consequence, we give a sufficient condition under which a Kazdan-Wa…
Let $G=(V,E)$ be a finite connected graph, and let $κ: V\rightarrow \mathbb{R}$ be a function such that $\int_Vκdμ<0$. We consider the following Kazdan-Warner equation on $G$:\[Δu+κ-K_λe^{2u}=0,\] where $K_λ=K+λ$ and $K: V\rightarrow \mathbb{R}$ is a non-constant function satisfying $\max_{x\in V}K(x)=0$ and $λ\in \mathbb{R}$. By a variational method, we prove that there exists a $λ^*>0$ such that when $λ\in(-\infty,λ^*]$ the above equation has solutions, and has no solution when $λ\geq λ^\ast$. In particular, it has only one solution if $λ\leq 0$; at least two distinct solutions if $0<λ<λ^*$;…
In this paper we introduce and study a geometric heat flow to find Killing vector fields on closed Riemannian manifolds with positive sectional curvature. We study its various properties, prove the global existence of the solution of this flow, discuss its convergence and possible applications, and its relation to the Navier-Stokes equations on manifolds and Kazdan-Warner-Bourguignon-Ezin identity for conformal Killing vector fields. We also provide two new criterions on the existence of Killing vector fields. The similar flow to finding holomorphic vector fields on Kähler manifolds will be st…
We study a prescribing functions problem of a conformally invariant integral equation involving Poisson kernel on the unit ball. This integral equation is not the dual of any standard type of PDE. As in Nirenberg problem, there exists a Kazdan-Warner type obstruction to existence of solutions. We prove existence in the antipodal symmetry functions class.
This paper is devoted to the existence of contact forms of prescribed Webster scalar curvature on a $3-$dimensional CR compact manifold locally conformally CR equivalent to the unit sphere $\mathbb{S}^{3}$ of $\mathbb{C}^{2}$. Due to Kazdan-Warner type obstructions, conditions on the function $H$ to be realized as a Webster scalar curvature have to be given. We prove new existence results based on a new type of Euler-Hopf type formula. Our argument gives an upper bound on the Morse index of the obtained solution. We also give a lower bound on the number of conformal contact forms having the sa…
In this article, we give results of prescribing scalar and mean curvature functions for metrics either pointwise conformal or conformally equivalent to a Riemannian metric that is equipped on a compact manifold with boundary, with dimensions at least $ 3 $. The results are classified by the sign of the first eigenvalue of the conformal Laplacian. This leads to a "Trichotomy Theorem" in terms of both scalar and mean curvature functions, which is a full extension of the "Trichotomy Theorem" given by Kazdan and Warner. We also discuss prescribing Gauss and geodesic curvature problems on compact R…
Analog to the classical result of Kazdan-Warner for the existence of solutions to the prescribed Gaussian curvature equation on compact 2-manifolds without boundary, it is widely known that if $(M,g_0)$ is a closed 4-manifold with zero $Q$-curvature and if $f$ is any non-constant, smooth, sign-changing function with $\int_M f dμ_{{\it g}_0} <0$, then there exists at least one solution $u$ to the prescribed $Q$-curvature equation \[ \mathbf{P}_{g_0} u = f e^{4u}, \] where $\mathbf{P}_{g_0}$ is the Paneitz operator which is positive with kernel consisting of constant functions. In this paper, we…
