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We study the mean field equation on the flat torus T σ : = C / ( Z + Z σ ) Δ u + ρ ( e u ∫ T σ e u − 1 | T σ | ) = 0 , where ρ is a real parameter. For a general flat torus, we obtain the existence of two-dimensional solutions bifurcating from the trivial solution at each eigenvalue (up to a multiplicative constant | T σ | ) of Laplace operator on the torus in the space of even symmetric functions. We further characterize the subset of all eigenvalues through which only one bifurcating curve passes. Finally local convexity near bifurcating points of the solution curves are obtained.

In this paper we study the blow-up analysis for some mean field equation on point vortices which is derived by C. Neri under a stochastic assumption. In particular, we derive some estimate which is...

We investigate the dynamics of a limit of interacting FitzHugh--Nagumo neurons in the regime of large interaction coefficients. We consider the dynamics described by a mean-field model given by a n...

We consider the extinction regime in the spatial stochastic logistic model in R d (a.k.a. Bolker–Pacala–Dieckmann–Law model of spatial populations) using the first-order perturbation beyond the mea...

We prove the existence of a family of blow-up solutions of a mean field equation on the sphere. The solutions blow up at four points where the minimum value of a potential energy function (involving the Green’s function) is attained. The four blow-up points form a regular tetrahedron. Moreover, the solutions we build have a group of symmetry T d {T_{d}} which is isomorphic to the symmetric group S 4 {S_{4}} . Other families of solutions can be similarly constructed with blow-up points at the vertices of equilateral triangles on a great circle or other inscribed platonic solids (cubes, octahedrons, icosahedrons and dodecahedrons). All of these solutions have the symmetries of the corresponding configuration, while they are non-axially symmetric.

This paper is concerned with the uniqueness of solutions to the following nonlocal semi-linear elliptic equationΔu−βu+λeu∫Ωeu=0 in Ω, (*)where Ω is a bounded domain in R2 and β,λ are positive param...

For domains of first kind [7,13] we describe the qualitative behavior of the global bifurcation diagram of the unbounded branch of solutions of the Gel'fand problem crossing the origin. At least to our knowledge this is the first result about the exact monotonicity of the branch of non-minimal solutions which is not just concerned with radial solutions [28] and/or with symmetric domains [23]. Toward our goal we parametrize the branch not by the $L^{\infty}(\Omega)$-norm of the solutions but by the energy of the associated mean field problem. The proof relies on a carefully modified spectral analysis of mean field type equations., Comment: Intro has been expanded. References has been added. Minor expository improvements

Inspired by works of Cast\'eras (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 $$ \left\{\begin{array}{lll} \partial_t\phi(u)=\Delta u-Q+\rho \frac{e^u}{\int_Ve^ud\mu}\\[1.5ex] u(\cdot,0)=u_0, \end{array}\right. $$ where $\Delta$ is the standard graph Laplacian, $\rho$ is a real number, $Q:V\rightarrow\mathbb{R}$ is a function satisfying $\int_VQd\mu=\rho$, and $\phi:\mathbb{R}\rightarrow\mathbb{R}$ is one of certain smooth functions including $\phi(s)=e^s$. We prove that for any initial data $u_0$ and any $\rho\in\mathbb{R}$, there exists a unique solution $u:V\times[0,+\infty)\rightarrow\mathbb{R}$ of the above heat flow; moreover, $u(x,t)$ converges to some function $u_\infty:V\rightarrow\mathbb{R}$ uniformly in $x\in V$ as $t\rightarrow+\infty$, and $u_\infty$ is a solution of the mean field equation $$\Delta u_\infty-Q+\rho\frac{e^{u_\infty}}{\int_Ve^{u_\infty}d\mu}=0.$$ Though $G$ is a finite graph, this result is still unexpected, even in the special case $Q\equiv 0$. 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., Comment: 15 pages

We consider the two-dimensional mean field equation of the equilibrium turbulence with variable intensities and Dirichlet boundary condition on a pierced domain $$\left\{ \begin{array}{ll} -\Delta u=\lambda_1\dfrac{V_1 e^{u}}{ \int_{\Omega_{\boldsymbol\epsilon}} V_1 e^{u} dx } - \lambda_2\tau \dfrac{ V_2 e^{-\tau u}}{ \int_{\Omega_{\boldsymbol\epsilon}}V_2 e^{ - \tau u} dx}&\text{in $\Omega_{\boldsymbol\epsilon}=\Omega\setminus \displaystyle \bigcup_{i=1}^m \overline{B(\xi_i,\epsilon_i)}$}\\ \ \ u=0 &\text{on $\partial \Omega_{\boldsymbol\epsilon}$}, \end{array} \right. $$ where $B(\xi_i,\epsilon_i)$ is a ball centered at $\xi_i\in\Omega$ with radius $\epsilon_i$, $\tau$ is a positive parameter and $V_1,V_2>0$ are smooth potentials. When $\lambda_1>8\pi m_1$ and $\lambda_2 \tau^2>8\pi (m-m_1)$ with $m_1 \in \{0,1,\dots,m\}$, there exist radii $\epsilon_1,\dots,\epsilon_m$ small enough such that the problem has a solution which blows-up positively and negatively at the points $\xi_1,\dots,\xi_{m_1}$ and $\xi_{m_1+1},\dots,\xi_{m}$, respectively, as the radii approach zero., Comment: 23 pages

Let X be a complex hyperelliptic curve of genus two equipped with the canonical metric d s 2 . We study mean field equations on complex hyperelliptic curves and show that the Gaussian curvature function of ( X , d s 2 ) determines an explicit solution to a mean field equation.

The Kuramoto model (KM) of coupled phase oscillators on graphs provides the most influential framework for studying collective dynamics and synchronization. It exhibits a rich repertoire of dynamical regimes. Since the work of Strogatz and Mirollo, the mean field equation derived in the limit as the number of oscillators in the KM goes to infinity, has been the key to understanding a number of interesting effects, including the onset of synchronization and chimera states. In this work, we study the mathematical basis of the mean field equation as an approximation of the discrete KM. Specifically, we extend the Neunzert's method of rigorous justification of the mean field equation to cover interacting dynamical systems on graphs. We then apply it to the KM on convergent graph sequences with non-Lipschitz limit. This family of graphs includes many graphs that are of interest in applications, e.g., nearest-neighbor and small-world graphs.

We study a double mean field-type PDE related to a prescribed curvature problem on compacts surfaces with boundary. We provide a general blow-up analysis, then a Moser-Trudinger inequality, which gives energy-minimizing solutions for some range of parameters. Finally, we provide existence of min-max solutions for a wider range of parameters, which is dense in the plane if $��$ is not simply connected., 29 pages, 4 figures

We are concerned with the mean field equation with singular data on bounded domains. By assuming a singular point to be a critical point of the 1-vortex Kirchhoff-Routh function, we prove local uniqueness and non-degeneracy of bubbling solutions blowing up at a singular point. The proof is based on sharp estimates for bubbling solutions of singular mean field equations and a suitably defined Pohozaev-type identity.

In this paper, we prove that the even solution of the mean field equation $\Delta u=\lambda(1-e^u) $ on $S^2$ must be axially symmetric when $4, Comment: 15 pages

We are concerned with the blow-up analysis of mean field equations. It has been proven in [ 6 ] that solutions blowing-up at the same non-degenerate blow-up set are unique. On the other hand, the authors in [ 18 ] show that solutions with a degenerate blow-up set are in general non-unique. In this paper we first prove that evenly symmetric solutions on an arbitrary flat torus with a degenerate two-point blow-up set are unique. In the second part of the paper we complete the analysis by proving the existence of such blow-up solutions using a Lyapunov-Schmidt reduction method. Moreover, we deduce that all evenly symmetric blow-up solutions come from one-point blow-up solutions of the mean field equation on a "half" torus.

The aim of this paper is to complete the program initiated in [51] , [23] and then carried out by several authors concerning non-degeneracy and uniqueness of solutions to mean field equations. In particular, we consider mean field equations with general singular data on non-smooth domains. The argument is based on the Alexandrov–Bol inequality and on the eigenvalues analysis of linearized singular Liouville-type problems.

The understanding of some large energy, negative specific heat states in the Onsager description of 2D turbulence, seems to require the analysis of a subtle open problem about bubbling solutions of the mean field equation. Motivated by this application we prove that, under suitable non degeneracy assumptions on the associated $m$-vortex Hamiltonian, the $m$-point bubbling solutions of the mean field equation are non degenerate as well. Then we deduce that the Onsager mean field equilibrium entropy is smooth and strictly convex in the high energy regime on domains of second kind.

We compute the Euler characteristic with compact supports $\chi_c$ of the formal barycenter spaces with weights of a finite CW complex, connected or not. This reduces to the topological Euler characteristic $\chi$ when the weights of the singular points are less than one. As foresighted by A. Malchiodi, our formula is related to the Leray-Schauder degree for mean field equations on a compact Riemann surface obtained by C.C. Chen and C.S. Lin., Comment: 16 pages. Major revision including a proof simplification. Extension of the main result to non-necessarily compact spaces. The Euler characteristic computation extends now to barycenter spaces of interiors of even dimensional manifolds (possibly disconnected)

We study the kinetic mean field equation on two-dimensional Brownian vortices; derivation, similarity to the DD-model, and existence and non-existence of global-in-time solution.

It is known from \cite{LW} that the solvability of the mean field equation $\Delta u+e^{u}=8n\pi \delta_{0}$ with $n\in\mathbb{N}_{\geq 1}$ on a flat torus $E_{\tau}$ essentially depends on the geometry of $E_{\tau}$. A conjecture is the non-existence of solutions for this equation if $E_{\tau}$ is a rectangular torus, which was proved for $n=1$ in \cite{LW}. For any $n\in \mathbb{N}_{\geq2}$, this conjecture seems challenging from the viewpoint of PDE theory. In this paper, we prove this conjecture for $n=2$ (i.e. at critical parameter $16\pi$)., Comment: Part of content in the previous version is removed and published separately

Let $E = \Bbb C/\Lambda$ be a flat torus and $G$ be its Green function with singularity at $0$. Consider the multiple Green function $G_n$ on $E^n$: $$G_{n}(z_1,\cdots,z_n) := \sum_{i < j} G(z_{i} - z_{j}) - n \sum_{i = 1} ^{n} G(z_{i}).$$ A critical point $a = (a_1, \cdots, a_n)$ of $G_n$ is called trivial if $\{a_1, \cdots, a_n\} = \{-a_1, \cdots, -a_n\}$. For such a point $a$, two geometric quantities $D(a)$ and $H(a)$ arising from bubbling analysis of mean field equations are introduced. $D(a)$ is a global quantity measuring asymptotic expansion and $H(a)$ is the Hessian of $G_n$ at $a$. By way of geometry of Lam\'e curves developed in our previous paper (Cambridge J. Math 3, 2015), we derive precise formulas to relate these two quantities., Comment: 19 pages, no figures

We prove the existence of stationary turbulent flows with arbitrary positive vortex circulation on non-simply connected domains. Our construction yields solutions for all real values of the inverse temperature with the exception of a quantized set, for which blow-up phenomena may occur. Our results complete the analysis initiated in Ricciardi and Zecca (2016).

Let $M$ be a compact Riemann surface, $\alpha j \gt -1$, and $h(x)$ a positive $C^2$ function of $M$. In this paper, we consider the following mean field equation: \[ \Delta u (x) + \rho \left( \frac{h(x) e^{u(x)}}{\int_{M} h (x) e^{u(x)}} - \frac{1}{\lvert M \rvert} \right) = 4\pi \sum^{d}_{j=1} \alpha_j \left( \delta_{q_j} - \frac{1}{\lvert M \rvert} \right) \textrm{ in } M \textrm{.} \] We prove that for $\alpha_j \in \mathbb{N}$ and any $\rho \gt \rho_0$, the equation has one solution at least if the Euler characteristic $\chi (M) \leq 0$, where $\rho_0 = \underset{M}{\max} ( 2K - \Delta \mathrm{ln} h + N^*)$, $K$ is the Gaussian curvature, and $N^* = 4\pi \sum^{d}_{j=1} \alpha_j$. This result was proved in [10] when $\alpha_j = 0$. Our proof relies on the bubbling analysis if one of the blowup points is at the vortex $q_j$. In the case where $\alpha_j \notin \mathbb{N}$, the sharp estimate of solutions near $q_j$ has been obtained in [11]. However, if $\alpha_j \in \mathbb{N}$, then the phenomena of non-simple blowup might occur. One of our contributions in part 1 is to obtain the sharp estimate for the non-simple blowup phenomena.

In this article we consider the following fourth order mean field equation on smooth domain Ω⋐R4:Δ2u=ϱKeu∫ΩKeuin Ω,u=Δu=0on ∂Ω, where ϱ∈R and 0

We present some recent results on mean field equations of Liouville type over a closed surface, in presence of Dirac distributions supported at the so called "vortex points". We discuss possible existence and non-existence results as well as uniqueness and multiplicity issues according to the topological and geometrical properties of the surface.

This paper is concerned with the mean field equation of the equilibrium turbulence on a closed Riemannian surface. The existences of solution are shown in the critical and supercritical case.

Using the method of symmetrization and the rescaling, we study non-compact solution sequence to the SU ( 3 ) Toda system in non-Abelian relativistic self-dual gauge theory, i.e., the quantization of the total mass and classification of the singular limit.

We obtain a Pohozaev-type identity which yields a generalization to the systems case of the well known scalar non-existence threshold for Liouville-type mean field equations on strictly starshaped domains. These newly derived non-existence results suggest that in principle solutions could be find in a region of parameters far away from the subcritical regime with respect to the vectorial Moser–Trudinger and Log-HLS inequalities found by Chipot, Shafrir and Wolansky. Indeed, we succeed in proving that the Dirichlet problem for cooperative Liouville systems admits solutions on “thin” simply connected domains in the supercritical regime. This is an improvement of the existence theory for cooperative Liouville systems since in that region solutions were known to exist only on multiply connected domains. Finally, combining spectral elliptic estimates and Orlicz-spaces techniques with a trick introduced by Wolansky we prove that these newly derived solutions are strict local minimizers of the Moser–Trudinger-type and free-energy functionals.

We consider on a two-dimensional flat torus T the following equation Δ u + ρ ( e u ∫ T e u − 1 | T | ) = 0 . When the fundamental domain of the torus is ( 0 , a ) × ( 0 , b ) ( a ⩾ b ) , we establish that the constants are the unique solutions whenever ρ ⩽ { 8 π if b a ⩾ π 4 , 32 b a if b a ⩽ π 4 , and this result is sharp if b a ⩾ π 4 . A similar conclusion is obtained for general two-dimensional torus by considering the length of the shortest closed geodesic. These results are derived by comparing the isoperimetric profile of the torus T with the one of the two-dimensional canonical sphere which has same volume as T.

We consider the classical solutions of - Δ v = α e v - β in Ω , where Ω is a bounded open set in R 2 . This equation is related to geometry and several fields of physics and has significant applications in the theory of plasma. We derive some uniqueness results and a priori estimate by using the classical isoperimetric inequality and the blow up analysis.

We consider a class of equations with exponential non-linearities on a compact surface which arises as the mean field equation of the equilibrium turbulence with arbitrarily signed vortices. We prove an existence result via degree theory. This yields new existence results in case of a topological sphere. The proof is carried out by considering the parity of the Leray-Schauder degree associated to the problem. With this method we recover also some known previous results.

Motivated by the mean field equations with probability measure derived by Sawada-Suzuki and by Neri in the context of the statistical mechanics description of two-dimensional turbulence, we study the semilinear elliptic equation with probability measure: \begin{equation*} -\Delta v=\lambda\int_IV(\alpha,x,v)e^{\alpha v}\,{\mathcal P(d\alpha)} -\frac{\lambda}{|\Omega|}\iint_{I\times\Omega}V(\alpha,x,v)e^{\alpha v}\,{\mathcal P(d\alpha)} dx, \end{equation*} defined on a compact Riemannian surface. This equation includes the above mentioned equations of physical interest as special cases. For such an equation we study the blow-up properties of solution sequences. The optimal Trudinger-Moser inequality is also considered.

For a general class of elliptic PDE's in mean field form on compact Riemann surfaces with exponential nonlinearity, we address the question of the existence of solutions with concentrated nonlinear term, which, in view of the applications, are physically of definite interest. In the model, we also include the possible presence of singular sources in the form of Dirac masses, which makes the problem more degenerate and difficult to attack.

We prove uniqueness of solutions for mean field equations [10] with singular data [5], arising in the analysis of two-dimensional turbulent Euler flows. In this way, we generalize to the singular case some uniqueness results obtained by Chang, Chen and the second author [11]. In particular, by using a sharp form of an improved Alexandrov–Bol's type isoperimetric inequality, we are able to exploit the role played by the singularities and then obtain uniqueness under weaker boundary regularity assumptions than those assumed in [11].

Given a regular bounded domain $\Omega\subset\R{2m}$, we describe the limiting behavior of sequences of solutions to the mean field equation of order $2m$, $m\geq 1$, $$(-\Delta)^m u=\rho \frac{e^{2mu}}{\int_\Omega e^{2mu}dx}\quad\text{in}\Omega,$$ under the Dirichlet boundary condition and the bound $0, Comment: 21 pages

(1991). Uniform estimates and blow–up behavior for solutions of −δ(u)=v(x)eu in two dimensions. Communications in Partial Differential Equations: Vol. 16, No. 8-9, pp. 1223-1253.

In this paper we construct single and multiple blowing-up solutions to the mean field equation $-\Delta u=λ V(x)e^u/\int_\Omega V(x) e^u$ in Ω, $u=0$ on $\partial \Omega$, where Ω is a smooth bounded domain in R^2, V is a smooth function positive somewhere in Ω and λ is a positive parameter.

We derive a priori uniform bounds for solutions of an elliptic system of Liouville-type equations, first analyzed by J. Spruck and Y. Yang (Comm. Math. Phys. 144 (1992) 1), yielding periodic multivortices in the classical electroweak theory of Glashow-Salam-Weinberg. Our proof is based on a concentration-quantization result, in the same spirit of Brezis-Merle (Comm. Partial Differential Equations 16 (8,9) (1991) 1223) and Li-Shafrir (Indiana Univ. Math. J. 43 (4) . (1994) 1255), for mean field equations on Riemannian compact 2-manifolds.

Let Ω be a simply connected, open and bounded domain in R 2 . We are concerned with the nonlinear elliptic problem (0.1) { − Δ v = 8 π e v ∫ Ω e v − 4 π ∑ j = 1 m α j δ p j in Ω , v = 0 on ∂ Ω , where α j > 0 , δ p j denotes the Dirac mass with singular point p j and { p 1 , … , p m } ⊂ Ω . We provide necessary and sufficient conditions for the existence of solutions to (0.1) . Our result is the two dimensional version of the sharp existence/nonexistence result obtained in Druet (2002) [13] for elliptic equations with critical exponent in dimension 3. In particular, we prove that the set Ω + m ( α ) is open, where, for a given α = ( α 1 , … , α m ) ⊂ ( 0 , + ∞ ) × ⋯ × ( 0 , + ∞ ) , Ω + m ( α ) = { ( p 1 , … , p m ) | problem (0.1) has a solution } ⊂ Ω × ⋯ × Ω .