A heat flow for the mean field equation on a finite graph

Inspired by works of Casteras (Pacific J. Math., 2015), Li-Zhu (Calc. Var., 2019) and Sun-Zhu (Calc. Var., 2020), we propose a heat flow for the mean field equation on a connected finite graph $G=(V,E)$. Namely $$ left{begin{array}{lll} partial_tphi(u)=Delta u-Q+rho frac{e^u}{int_Ve^udmu}[1.5ex] u(cdot,0)=u_0, end{array}right. $$ where $Delta$ is the standard graph Laplacian, $rho$ is a real number, $Q:Vrightarrowmathbb{R}$ is a function satisfying $int_VQdmu=rho$, and $phi:mathbb{R}rightarrowmathbb{R}$ is one of certain smooth functions including $phi(s)=e^s$. We prove that for any initial data $u_0$ and any $rhoinmathbb{R}$, there exists a unique solution $u:Vtimes[0,+infty)rightarrowmathbb{R}$ of the above heat flow; moreover, $u(x,t)$ converges to some function $u_infty:Vrightarrowmathbb{R}$ uniformly in $xin V$ as $trightarrow+infty$, and $u_infty$ is a solution of the mean field equation $$Delta u_infty-Q+rhofrac{e^{u_infty}}{int_Ve^{u_infty}dmu}=0.$$ Though $G$ is a finite graph, this result is still unexpected, even in the special case $Qequiv 0$. Our approach reads as follows: the short time existence of the heat flow follows from the ODE theory; various integral estimates give its long time existence; moreover we establish a Lojasiewicz-Simon type inequality and use it to conclude the convergence of the heat flow.

Calculus of variations on locally finite graphs

Let $G=(V,E)$ be a locally finite graph. Firstly, using calculus of variations, including a direct method of variation and the mountain-pass theory, we get sequences of solutions to several local equations on $G$ (the Schrodinger equation, the mean field equation, and the Yamabe equation). Secondly, we derive uniform estimates for those local solution sequences. Finally, we obtain global solutions by extracting convergent sequence of solutions. Our method can be described as a variational method from local to global.

Existence results for a generalized mean field equation on a closed Riemann surface

Let $Sigma$ be a closed Riemann surface, $h$ a positive smooth function on $Sigma$, $rho$ and $alpha$ real numbers. In this paper, we study a generalized mean field equation begin{align*} -Delta u=rholeft(dfrac{he^u}{int_Sigma he^u}-dfrac{1}{mathrm{Area}left(Sigmaright)}right)+alphaleft(u-fint_{Sigma}uright), end{align*} where $Delta$ denotes the Laplace-Beltrami operator. We first derive a uniform bound for solutions when $rhoin (8kpi, 8(k+1)pi)$ for some non-negative integer number $kin mathbb{N}$ and $alpha otinmathrm{Spec}left(-Deltaright)setminusset{0}$. Then we obtain existence results for $alpha<lambda_1left(Sigmaright)$ by using the Leray-Schauder degree theory and the minimax method, where $lambda_1left(Sigmaright)$ is the first positive eigenvalue for $-Delta$.

Multiple solutions of Kazdan-Warner equation on graphs in the negative case

Let $G=(V,E)$ be a finite connected graph, and let $kappa: Vrightarrow mathbb{R}$ be a function such that $int_Vkappa dmu<0$. We consider the following Kazdan-Warner equation on $G$:[Delta u+kappa-K_lambda e^{2u}=0,] where $K_lambda=K+lambda$ and $K: Vrightarrow mathbb{R}$ is a non-constant function satisfying $max_{xin V}K(x)=0$ and $lambdain mathbb{R}$. By a variational method, we prove that there exists a $lambda^*>0$ such that when $lambdain(-infty,lambda^*]$ the above equation has solutions, and has no solution when $lambdageq lambda^ast$. In particular, it has only one solution if $lambdaleq 0$; at least two distinct solutions if $0<lambda<lambda^*$; at least one solution if $lambda=lambda^ast$. This result complements earlier work of Grigoryan-Lin-Yang cite{GLY16}, and is viewed as a discrete analog of that of Ding-Liu cite{DL95} and Yang-Zhu cite{YZ19} on manifolds.

Blow-up analysis involving isothermal coordinates on the boundary of compact Riemann surface

Using the method of blow-up analysis, we obtain two sharp Trudinger-Moser inequalities on a compact Riemann surface with smooth boundary, as well as the existence of the corresponding extremals. This generalizes early results of Chang-Yang [7] and the first named author [32], and complements Fontanas inequality of two dimensions [15]. The blow-up analysis in the current paper is far more elaborate than that of [32], and particularly clarifies several ambiguous points there. In precise, we prove the existence of isothermal coordinate systems near the boundary, the existence and uniform estimates of the Green function with the Neumann boundary condition. Also our analysis can be applied to the Kazdan-Warner problem and the Chern-Simons Higgs problem on compact Riemman surfaces with smooth boundaries.

Nonexistence of extremals for an inequality of Adimurthi-Druet on a closed Riemann surface

Based on a recent work of Mancini-Thizy [28], we obtain the nonexistence of extremals for an inequality of Adimurthi-Druet [1] on a closed Riemann surface $(Sigma,g)$. Precisely, if $lambda_1(Sigma)$ is the first eigenvalue of the Laplace-Beltrami operator with respect to the zero mean value condition, then there exists a positive real number $alpha^ast<lambda_1(Sigma)$ such that for all $alphain (alpha^ast,lambda_1(Sigma))$, the supremum $$sup_{uin W^{1,2}(Sigma,g),,int_Sigma udv_g=0,,| abla_gu|_2leq 1}int_Sigma exp(4pi u^2(1+alpha|u|_2^2))dv_g$$ can not be attained by any $uin W^{1,2}(Sigma,g)$ with $int_Sigma udv_g=0$ and $| abla_gu|_2leq 1$, where $W^{1,2}(Sigma,g)$ denotes the usual Sobolev space and $|cdot|_2=(int_Sigma|cdot|^2dv_g)^{1/2}$ denotes the $L^2(Sigma,g)$-norm. This complements our earlier result in [39].

Mean field equations on a closed Riemannian surface with the action of an isometric group

Let $(Sigma,g)$ be a closed Riemannian surface, $textbf{G}={sigma_1,cdots,sigma_N}$ be an isometric group acting on it. Denote a positive integer $ell=inf_{xinSigma}I(x)$, where $I(x)$ is the number of all distinct points of the set ${sigma_1(x),cdots,sigma_N(x)}$. A sufficient condition for existence of solutions to the mean field equation $$Delta_g u=8piellleft(frac{he^u}{int_Sigma he^udv_g}-frac{1}{{rm Vol}_g(Sigma)}right)$$ is given. This recovers results of Ding-Jost-Li-Wang (Asian J Math 1997) when $ell=1$ or equivalently $textbf{G}={Id}$, where $Id$ is the identity map.

A Trudinger-Moser inequality for conical metric in the unit ball

In this note, we prove a Trudinger-Moser inequality for conical metric in the unit ball. Precisely, let $mathbb{B}$ be the unit ball in $mathbb{R}^N$ $(Ngeq 2)$, $p>1$, $g=|x|^{frac{2p}{N}beta}(dx_1^2+cdots+dx_N^2)$ be a conical metric on $mathbb{B}$, and $lambda_p(mathbb{B})=infleft{int_mathbb{B}| abla u|^Ndx: uin W_0^{1,N}(mathbb{B}),,int_mathbb{B}|u|^pdx=1right}$. We prove that for any $betageq 0$ and $alpha<(1+frac{p}{N}beta)^{N-1+frac{N}{p}}lambda_p(mathbb{B})$, there exists a constant $C$ such that for all radially symmetric functions $uin W_0^{1,N}(mathbb{B})$ with $int_mathbb{B}| abla u|^Ndx-alpha(int_mathbb{B}|u|^p|x|^{pbeta}dx)^{N/p}leq 1$, there holds $$int_mathbb{B}e^{alpha_N(1+frac{p}{N}beta)|u|^{frac{N}{N-1}}}|x|^{pbeta}dxleq C,$$ where $|x|^{pbeta}dx=dv_g$, $alpha_N=Nomega_{N-1}^{1/(N-1)}$, $omega_{N-1}$ is the area of the unit sphere in $mathbb{R}^N$; moreover, extremal functions for such inequalities exist. The case $p=N$, $-1<beta<0$ and $alpha=0$ was considered by Adimurthi-Sandeep cite{A-S}, while the case $p=N=2$, $betageq 0$ and $alpha=0$ was studied by de Figueiredo-do O-dos Santos cite{F-do-dos}.

Trudinger-Moser inequalities on a closed Riemannian surface with the action of a finite isometric group

Let $(Sigma,g)$ be a closed Riemannian surface, $W^{1,2}(Sigma,g)$ be the usual Sobolev space, $textbf{G}$ be a finite isometric group acting on $(Sigma,g)$, and $mathscr{H}_textbf{G}$ be a function space including all functions $uin W^{1,2}(Sigma,g)$ with $int_Sigma udv_g=0$ and $u(sigma(x))=u(x)$ for all $sigmain textbf{G}$ and all $xinSigma$. Denote the number of distinct points of the set ${sigma(x): sigmain textbf{G}}$ by $I(x)$ and $ell=inf_{xin Sigma}I(x)$. Let $lambda_1^textbf{G}$ be the first eigenvalue of the Laplace-Beltrami operator on the space $mathscr{H}_textbf{G}$. Using blow-up analysis, we prove that if $alpha<lambda_1^textbf{G}$ and $betaleq 4piell$, then there holds $$sup_{uinmathscr{H}_textbf{G},,int_Sigma| abla_gu|^2dv_g-alpha int_Sigma u^2dv_gleq 1}int_Sigma e^{beta u^2}dv_g<infty;$$ if $alpha<lambda_1^textbf{G}$ and $beta>4piell$, or $alphageq lambda_1^textbf{G}$ and $beta>0$, then the above supremum is infinity; if $alpha<lambda_1^textbf{G}$ and $betaleq 4piell$, then the above supremum can be attained. Moreover, similar inequalities involving higher order eigenvalues are obtained. Our results partially improve original inequalities of J. Moser cite{Moser}, L. Fontana cite{Fontana} and W. Chen cite{Chen-90}.

A remark on energy estimates concerning extremals for Trudinger-Moser inequalities on a disc

In this short note, we generalized an energy estimate due to Malchiodi-Martinazzi (J. Eur. Math. Soc. 16 (2014) 893-908) and Mancini-Martinazzi (Calc. Var. (2017) 56:94). As an application, we used it to reprove existence of extremals for Trudinger-Moser inequalities of Adimurthi-Druet type on the unit disc. Such existence problems in general cases had been considered by Yang (Trans. Amer. Math. Soc. 359 (2007) 5761-5776; J. Differential Equations 258 (2015) 3161-3193) and Lu-Yang (Discrete Contin. Dyn. Syst. 25 (2009) 963-979) by using another method.