CPAA
On the convergence of singular perturbations of Hamilton-Jacobi equations
Claudio Marchi
This paper is devoted to singular perturbation problems for first order equations. Under some coercivity and periodicity assumptions, we establish the uniform convergence and we provide an estimate of the rate of convergence, which we consider the main result of the paper.
    We shall also show that our results apply to the homogenization problem for coercive and periodic equations. Finally, some examples arising in optimal control and differential games theory will be discussed.
keywords: Singular perturbations Hamilton-Jacobi equations rate of convergence. viscosity solutions ergodicity homogenization
NHM
On the convergence rate in multiscale homogenization of fully nonlinear elliptic problems
Fabio Camilli Claudio Marchi
This paper concerns periodic multiscale homogenization for fully nonlinear equations of the form $u^\epsilon+H^\epsilon (x,\frac{x}{\epsilon},\ldots,\frac{x}{epsilon^k},Du^\epsilon,D^2u^\epsilon)=0$. The operators $H^\epsilon$ are a regular perturbations of some uniformly elliptic, convex operator $H$. As $\epsilon\to 0^+$, the solutions $u^\epsilon$ converge locally uniformly to the solution $u$ of a suitably defined effective problem. The purpose of this paper is to obtain an estimate of the corresponding rate of convergence. Finally, some examples are discussed.
keywords: Nonlinear elliptic equations rate of convergence. multiscale homogenization Bellman equations
NHM
Analysis and control on networks: Trends and perspectives
Fabio Ancona Laura Caravenna Annalisa Cesaroni Giuseppe M. Coclite Claudio Marchi Andrea Marson
keywords: Conservation laws balance laws
NHM
Analysis and control on networks: Trends and perspectives
Fabio Ancona Laura Caravenna Annalisa Cesaroni Giuseppe M. Coclite Claudio Marchi Andrea Marson
keywords: Conservation laws balance laws
DCDS
A model problem for Mean Field Games on networks
Fabio Camilli Elisabetta Carlini Claudio Marchi
In [14], Guéant, Lasry and Lions considered the model problem ``What time does meeting start?'' as a prototype for a general class of optimization problems with a continuum of players, called Mean Field Games problems. In this paper we consider a similar model, but with the dynamics of the agents defined on a network. We discuss appropriate transition conditions at the vertices which give a well posed problem and we present some numerical results.
keywords: Mean Field Games numerical methods. stochastic optimal control Networks

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