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We consider the two-dimensional mean field equation of the equilibrium turbulence with variable intensities and Dirichlet boundary condition on a pierced domain $$left{ begin{array}{ll} -Delta u=lambda_1dfrac{V_1 e^{u}}{ int_{Omega_{boldsymbolepsilon}} V_1 e^{u} dx } - lambda_2tau dfrac{ V_2 e^{-tau u}}{ int_{Omega_{boldsymbolepsilon}}V_2 e^{ - tau u} dx}&text{in $Omega_{boldsymbolepsilon}=Omegasetminus displaystyle bigcup_{i=1}^m overline{B(xi_i,epsilon_i)}$} u=0 &text{on $partial Omega_{boldsymbolepsilon}$}, end{array} right. $$ where $B(xi_i,epsilon_i)$ is a ball centered at $xi_iinOmega$ with radius $epsilon_i$, $tau$ is a positive parameter and $V_1,V_2>0$ are smooth potentials. When $lambda_1>8pi m_1$ and $lambda_2 tau^2>8pi (m-m_1)$ 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.
In this note, we study symmetry of solutions of the elliptic equation begin{equation*} -Delta _{mathbb{S}^{2}}u+3=e^{2u} hbox{on} mathbb{S}^{2}, end{equation*} that arises in the study of rigidity problem of Hawking mass in general relativity. We p
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(
We study the long time behaviour of the kinetic Fokker-Planck equation with mean field interaction, whose limit is often called Vlasov-Fkker-Planck equation. We prove a uniform (in the number of particles) exponential convergence to equilibrium for t
We investigate regularity properties of the $overline{partial}$-equation on domains in a complex euclidean space that depend on a parameter. Both the interior regularity and the regularity in the parameter are obtained for a continuous family of pseu
In the present work, we study deterministic mean field games (MFGs) with finite time horizon in which the dynamics of a generic agent is controlled by the acceleration. They are described by a system of PDEs coupling a continuity equation for the den