NHM
Analysis and control on networks: Trends and perspectives
Fabio Ancona Laura Caravenna Annalisa Cesaroni Giuseppe M. Coclite Claudio Marchi Andrea Marson
Networks & Heterogeneous Media 2017, 12(3): i-ii doi: 10.3934/nhm.201703i
keywords: Conservation laws balance laws
NHM
On the convergence rate in multiscale homogenization of fully nonlinear elliptic problems
Fabio Camilli Claudio Marchi
Networks & Heterogeneous Media 2011, 6(1): 61-75 doi: 10.3934/nhm.2011.6.61
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
CPAA
On the convergence of singular perturbations of Hamilton-Jacobi equations
Claudio Marchi
Communications on Pure & Applied Analysis 2010, 9(5): 1363-1377 doi: 10.3934/cpaa.2010.9.1363
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
Analysis and control on networks: Trends and perspectives
Fabio Ancona Laura Caravenna Annalisa Cesaroni Giuseppe M. Coclite Claudio Marchi Andrea Marson
Networks & Heterogeneous Media 2017, 12(2): i-ii doi: 10.3934/nhm.201702i
keywords: Conservation laws balance laws
DCDS-S
Asymptotic behaviour for operators of Grushin type: Invariant measure and singular perturbations
Paola Mannucci Claudio Marchi Nicoletta Tchou
Discrete & Continuous Dynamical Systems - S 2019, 12(1): 119-128 doi: 10.3934/dcdss.2019008

This paper concerns singular perturbation problems where the dynamics of the fast variable evolve in the whole space according to an operator whose infinitesimal generator is formed by a Grushin type second order part and a Ornstein-Uhlenbeck first order part.

We prove that the dynamics of the fast variables admits an invariant measure and that the associated ergodic problem has a viscosity solution which is also regular and with logarithmic growth at infinity. These properties play a crucial role in the main theorem which establishes that the value functions of the starting perturbation problems converge to the solution of an effective problem whose operator and initial datum are given in terms of the associated invariant measure.

keywords: Subelliptic equations Grushin vector fields invariant measure singular perturbations viscosity solutions degenerate elliptic equations
DCDS-S
A flame propagation model on a network with application to a blocking problem
Fabio Camilli Elisabetta Carlini Claudio Marchi
Discrete & Continuous Dynamical Systems - S 2018, 11(5): 825-843 doi: 10.3934/dcdss.2018051
We consider the Cauchy problem
$\left\{ \begin{array}{*{35}{l}} {{\partial }_{t}}u+H(x,Du) = 0&(x,t)\in \Gamma \times (0,T) \\ u(x,0) = {{u}_{0}}(x)&x\in \Gamma \\\end{array} \right.$
where
$\Gamma$
is a network and
$H$
is a positive homogeneous Hamiltonian which may change from edge to edge. In the first part of the paper, we prove that the Hopf-Lax type formula gives the (unique) viscosity solution of the problem. In the latter part of the paper we study a flame propagation model in a network and an optimal strategy to block a fire breaking up in some part of a pipeline; some numerical simulations are provided.
keywords: Evolutive Hamilton-Jacobi equation viscosity solution network Hopf-Lax formula approximation
DCDS
A model problem for Mean Field Games on networks
Fabio Camilli Elisabetta Carlini Claudio Marchi
Discrete & Continuous Dynamical Systems - A 2015, 35(9): 4173-4192 doi: 10.3934/dcds.2015.35.4173
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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