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We provide an approximation scheme for first-order stationary mean field games with a separable Hamiltonian. First, we discretize Hamilton-Jacobi equations by discretizing in time, and then prove the existence of minimizing holonomic measures for mean field games. At last, we obtain two sequences of solutions $\{u_i\}$ of discrete Hamilton-Jacobi equations and minimizing holonomic measures $\{m_i\}$ for mean field games and show that $(u_i,m_i)$ converges to a solution of the stationary mean field games., 21 pages

This paper deals with the generalized ergodic problem \[ H(x,u(x),Du(x))=c, \quad x\in M, \] where the unknown is a pair $(c,u)$ of a constant $c \in \mathbb{R}$ and a function $u$ on $M$ for which $u$ is a viscosity solution. We assume $H=H(x,u,p)$ satisfies Tonelli conditions in the argument $p\in T^*_xM$ and the Lipschitz condition in the argument $u\in\R$. For a given $c\in \R$, we first discuss necessary and sufficient conditions for the existence of viscosity solutions. Let $\mathfrak{C}$ denote the set of all real numbers $c$'s for which the above equation admits viscosity solutions. Then we show $\mathfrak{C}$ is an interval, whose endpoints $\x$, $\y$ with $\x\leqslant\y$ can be characterized by a min-max formula and a max-min formula, respectively. The most significant finding is that we figure out the structure of $\mathfrak{C}$ without monotonicity assumptions on $u$., 37 pages

We are concerned with the existence and multiplicity of nontrivial time-periodic viscosity solutions to \[ \partial_t w(x,t) + H( x,\partial_x w(x,t),w(x,t) )=0,\quad (x,t)\in \mathbb{S} \times [0,+\infty). \] We find that there are infinitely many nontrivial time-periodic viscosity solutions with different periods when $\frac{\partial H}{\partial u}(x,p,u)\leqslant-\delta

This paper deals with the existence of solutions of a class of contact mean field games systems of first order. Cardaliaguet \cite{CAR} found a link between the weak KAM theory for Hamiltonian systems and mean field games systems. We prove that there is still a connection between the weak KAM theory for contact Hamiltonian systems and contact mean field games systems. By the analysis of properties of the Mather set for contact Hamiltonian systems, we prove the main existence result., Comment: 20 pages

We study the Hamilton-Jacobi equations $H(x,Du,u)=0$ in $M$ and $\partial u/\partial t +H(x,D_xu,u)=0$ in $M\times(0,\infty)$, where the Hamiltonian $H=H(x,p,u)$ depends Lipschitz continuously on the variable $u$. In the framework of the semicontinuous viscosity solutions due to Barron-Jensen, we establish the comparison principle, existence theorem, and representation formula as value functions for extended real-valued, lower semicontinuous solutions for the Cauchy problem. We also establish some results on the long-time behavior of solutions for the Cauchy problem and classification of solutions for the stationary problem., Comment: 42 pages

Various combinatorial optimization NP-hard problems can be reduced to finding the minimizer of an Ising model, which is a discrete mathematical model. It is an intellectual challenge to develop some mathematical tools or algorithms for solving the Ising model. Over the past decades, some continuous approaches or algorithms have been proposed from physical, mathematical or computational views for optimizing the Ising model such as quantum annealing, the coherent Ising machine, simulated annealing, adiabatic Hamiltonian systems, etc.. However, the mathematical principle of these algorithms is far from being understood. In this paper, we reveal the mathematical mechanism of dynamical system algorithms for the Ising model by Morse theory and variational methods. We prove that the dynamical system algorithms can be designed to minimize a continuous function whose local minimum points give all the candidates of the Ising model and the global minimum gives the minimizer of Ising problem. Using this mathematical mechanism, we can easily understand several dynamical system algorithms of the Ising model such as the coherent Ising machine, the Kerr-nonlinear parametric oscillators and the simulated bifurcation algorithm. Furthermore, motivated by the works of C. Conley, we study transit and capture properties of the simulated bifurcation algorithm to explain its convergence by the low energy transit and capture in celestial mechanics. A detailed discussion on $2$-spin and $3$-spin Ising models is presented as application., Comment: 39 pages, 2 figures(including 8 sub-figures)

This paper is concerned with the study of Aubry-Mather and weak KAM theories for contact Hamiltonian systems with Hamiltonians $H(x,u,p)$ defined on $T^*M\times\mathbb{R}$, satisfying Tonelli conditions with respect to $p$ and $00$, where $M$ is a connected, closed and smooth manifold. First, we show the uniqueness of the backward weak KAM solutions of the corresponding Hamilton-Jacobi equation. Using the unique backward weak KAM solution $u_-$, we prove the existence of the maximal forward weak KAM solution $u_+$. Next, we analyse Aubry set for the contact Hamiltonian system showing that it is the intersection of two Legendrian pseudographs $G_{u_-}$ and $G_{u_+}$, and that the projection $\pi:T^*M\times \mathbb{R}\to M$ induces a bi-Lipschitz homeomorphism $\pi|_{\tilde{\mathcal{A}}}$ from Aubry set $\tilde{\mathcal{A}}$ onto the projected Aubry set $\mathcal{A}$. At last, we introduce the notion of barrier functions and study their interesting properties along calibrated curves. Our analysis is based on a recent method by [43,44]., Comment: 34 pages