Analytic solutions of the Weiss mean field equation

Abstract We study the following mean field equation on a flat torus $T:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau )$: $$\begin{equation*} \varDelta u + \rho \left(\frac{e^{u}}{\int_{T}e^u}-\frac{1}{|T|}\right)=0, \end{equation*}$$where $ \tau \in \mathbb{C}, \mbox{Im}\ \tau>0$, and $|T|$ denotes the total area of the torus. We first prove that the solutions are evenly symmetric about any critical point of $u$ provided that $\rho \leq 8\pi $. Based on this crucial symmetry result, we are able to establish further the uniqueness of the solution if $\rho \leq \min{\{8\pi ,\lambda _1(T)|T|\}}$. Furthermore, we also classify all one-dimensional solutions by showing that the level sets must be closed geodesics.

Download Full-text

A mean field equation involving positively supported probability measures: blow-up phenomena and variational aspects

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/prm.2018.30 ◽

2018 ◽

Vol 149 (2) ◽

pp. 325-352 ◽

Cited By ~ 1

Author(s):

Aleks Jevnikar ◽

Wen Yang

Keyword(s):

Field Equation ◽

Elliptic Problem ◽

Blow Up ◽

Existence Of Solutions ◽

Mean Field ◽

Probability Measures ◽

Trudinger Inequality ◽

Mean Field Equation

AbstractWe are concerned with an elliptic problem which describes a mean field equation of the equilibrium turbulence of vortices with variable intensities. In the first part of the paper, we describe the blow-up picture and highlight the differences from the standard mean field equation as we observe non-quantization phenomenon. In the second part, we discuss the Moser–Trudinger inequality in terms of the blow-up masses and get the existence of solutions in a non-coercive regime by means of a variational argument, which is based on some improved Moser–Trudinger inequalities.

Download Full-text

The nonlocal mean-field equation on an interval

Communications in Contemporary Mathematics ◽

10.1142/s0219199719500287 ◽

2019 ◽

Vol 22 (05) ◽

pp. 1950028

Author(s):

Azahara DelaTorre ◽

Ali Hyder ◽

Luca Martinazzi ◽

Yannick Sire

Keyword(s):

Field Equation ◽

Boundary Conditions ◽

Dirichlet Boundary ◽

Mean Field ◽

Higher Order ◽

Dirichlet Boundary Conditions ◽

Field Equations ◽

Mean Field Equations ◽

Blowing Up ◽

Mean Field Equation

We consider the fractional mean-field equation on the interval [Formula: see text] [Formula: see text] subject to Dirichlet boundary conditions, and prove that existence holds if and only if [Formula: see text]. This requires the study of blowing-up sequences of solutions. We provide a series of tools in particular which can be used (and extended) to higher-order mean field equations of nonlocal type.

Download Full-text

An existence result for the mean-field equation on compact surfaces in a doubly supercritical regime

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s030821051200042x ◽

2013 ◽

Vol 143 (5) ◽

pp. 1021-1045 ◽

Cited By ~ 12

Author(s):

Aleks Jevnikar

Keyword(s):

Field Equation ◽

Existence Result ◽

Mean Field ◽

Type Inequality ◽

Riemannian Surface ◽

Supercritical Regime ◽

Supercritical Parameters ◽

The Mean ◽

First Time ◽

Mean Field Equation

We consider a class of variational equations with exponential nonlinearities on a compact Riemannian surface, describing the mean-field equation of the equilibrium turbulence with arbitrarily signed vortices. For the first time, we consider the problem with both supercritical parameters and we give an existence result by using variational methods. In doing so, we present a new Moser–Trudinger-type inequality under suitable conditions on the centre of mass and the scale of concentration of both eu and e−u, where u is the unknown function in the equation.

Download Full-text