In this paper we provide the existence of classical solutions to stationary mean field game systems in the whole space $\mathbb{R}^N$, with coercive potential, aggregating local coupling, and under general conditions on the Hamiltonian, completing the analysis started in the companion paper [6]. The only structural assumption we make is on the growth at infinity of the coupling term in terms of the growth of the Hamiltonian. This result is obtained using a variational approach based on the analysis of the non-convex energy associated to the system. Finally, we show that in the vanishing viscosity limit mass concentrates around the flattest minima of the potential, and that the asymptotic shape of the solutions in a suitable rescaled setting converges to a ground state, i.e. a classical solution to a mean field game system without potential.
The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points
从$Ω$中的拟平稳分布出发，考虑过阻尼Langevin动力学$$d X_t = -\nabla f(X_t) d t + \sqrt{h} \ d B_t$$的随机过程$(X_t)_{t\ge 0}$解的有界域$Ω$的第一出口点分布.在小温区（$h\to 0$）和对$f$相当一般的假设下（特别是$f$在$Ω$中可能有几个临界点），证明了第一出口点分布的支持集中在实现$\partial Ω$上$f$最小的几个点上.证明依赖于工具来研究隧道效应在半经典分析。扩展的结果更一般的初始分布比准平稳分布。
Nonlinear Schr{ö}dinger equation: concentration on circles driven by an external magnetic field
本文研究了定态磁非线性薛定谔方程\begin{align}\label{eq:initialabstract}\left( i \hbar \nabla + A(x) \right)^2 u + V(x) u = |u|^{p-2} u, \quad x\in \mathbb{R}^{3},\end{align}的半经典极限，其中$p\textgreater{}2$，$A$是与给定磁场$B$相关联的矢量势，即$\nabla \times A =B$，$V$是非负的标量（电）势，它可以在原点奇异，在无穷远处或紧集外为零.我们假定$A$和$V$满足圆柱对称性.利用精细罚定理证明了模集中在圆上的上方程的半经典柱对称解的存在性，其模集中在圆上，如$\hbar \to 0$。我们强调的浓度是由磁势和电势驱动的。因此，我们的结果表明，在半经典极限，磁场也影响的位置的解决方案的$(\ref{eq:initialabstract})$，如果他们的浓度发生在一个轨迹，而不是一个单一的点。
Singularity formed by the collision of two collapsing solitons in interaction for the 2D Keller-Segel system
CY Cergy Paris Université（塞吉-巴黎大学）；NYUAD Research Institute, New York University Abu Dhabi（纽约大学阿布扎比分校研究院）；Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）；Department of Mathematics, Institute of Applied Mathematical Sciences, National Taiwan University（台湾大学应用数学研究所数学系）
