A note on singular perturbation problems via Aubry-Mather theory
Fabio Camilli Annalisa Cesaroni
Discrete & Continuous Dynamical Systems - A 2007, 17(4): 807-819 doi: 10.3934/dcds.2007.17.807
Exploiting the metric approach to Hamilton-Jacobi equation recently introduced by Fathi and Siconolfi [13], we prove a singular perturbation result for a general class of Hamilton-Jacobi equations. Considered in the framework of small random perturbations of dynamical systems, it extends a result due to Kamin [19] to the case of a dynamical system having several attracting points inside the domain.
keywords: PDE Aubry-Mather theory. Singular perturbations viscosity solutions
A semi-discrete approximation for a first order mean field game problem
Fabio Camilli Francisco Silva
Networks & Heterogeneous Media 2012, 7(2): 263-277 doi: 10.3934/nhm.2012.7.263
In this article we consider a model first order mean field game problem, introduced by J.M. Lasry and P.L. Lions in [18]. Its solution $(v,m)$ can be obtained as the limit of the solutions of the second order mean field game problems, when the noise parameter tends to zero (see [18]). We propose a semi-discrete in time approximation of the system and, under natural assumptions, we prove that it is well posed and that it converges to $(v,m)$ when the discretization parameter tends to zero.
keywords: optimal control convex duality Mean field games numerical methods.
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
Characterizing attraction probabilities via the stochastic Zubov equation
Fabio Camilli Lars Grüne
Discrete & Continuous Dynamical Systems - B 2003, 3(3): 457-468 doi: 10.3934/dcdsb.2003.3.457
A stochastic differential equation with an a.s. locally stable compact set is considered. The attraction probabilities to the set are characterized by the sublevel sets of the limit of a sequence of solutions to $2^{nd}$ order partial differential equations. Two numerical examples illustrating the method are presented.
keywords: Stochastic differential equation Zubov's method vicosity solution.
A discrete Hughes model for pedestrian flow on graphs
Fabio Camilli Adriano Festa Silvia Tozza
Networks & Heterogeneous Media 2017, 12(1): 93-112 doi: 10.3934/nhm.2017004

In this paper, we introduce a discrete time-finite state model for pedestrian flow on a graph in the spirit of the Hughes dynamic continuum model. The pedestrians, represented by a density function, move on the graph choosing a route to minimize the instantaneous travel cost to the destination. The density is governed by a conservation law whereas the minimization principle is described by a graph eikonal equation. We show that the discrete model is well-posed and the numerical examples reported confirm the validity of the proposed model and its applicability to describe real situations.

keywords: Pedestrian flow Hughes model conservation law graph eikonal equation well-posed discrete model
Systems of convex Hamilton-Jacobi equations with implicit obstacles and the obstacle problem
Fabio Camilli Paola Loreti Naoki Yamada
Communications on Pure & Applied Analysis 2009, 8(4): 1291-1302 doi: 10.3934/cpaa.2009.8.1291
Aim of this paper is to show that some of the results in the weak KAM theory for $1^{s t}$ order convex Hamilton-Jacobi equations (see [11], [13]) can be extended to systems of convex Hamilton-Jacobi equations with implicit obstacles and to the obstacle problem. We obtain two results: a comparison theorem for systems lacking strict monotonicity; a representation formula for the obstacle problem involving the distance function associated to the Hamiltonian of the equation.
keywords: viscosity solutions obstacle problem Systems of Hamilton-Jacobi equations large deviation Hamiltonians.
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
$\left\{ \begin{array}{*{35}{l}} {{\partial }_{t}}u+H(x,Du) = 0&(x,t)\in \Gamma \times (0,T) \\ u(x,0) = {{u}_{0}}(x)&x\in \Gamma \\\end{array} \right.$
is a network and
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
Fabio Camilli Italo Capuzzo Dolcetta Maurizio Falcone
Networks & Heterogeneous Media 2012, 7(2): i-ii doi: 10.3934/nhm.2012.7.2i
The theory of Mean Field Games (MFG, in short) is a branch of the theory of Differential Games which aims at modeling and analyzing complex decision processes involving a large number of indistinguishable rational agents who have individually a very small influence on the overall system and are, on the other hand, influenced by the mass of the other agents. The name comes from particle physics where it is common to consider interactions among particles as an external mean field which influences the particles. In spite of the optimization made by rational agents, playing the role of particles in such models, appropriate mean field equations can be derived to replace the many particles interactions by a single problem with an appropriately chosen external mean field which takes into account the global behavior of the individuals.

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On numerical approximation of the Hamilton-Jacobi-transport system arising in high frequency approximations
Yves Achdou Fabio Camilli Lucilla Corrias
Discrete & Continuous Dynamical Systems - B 2014, 19(3): 629-650 doi: 10.3934/dcdsb.2014.19.629
In the present article, we study the numerical approximation of a system of Hamilton-Jacobi and transport equations arising in geometrical optics. We consider a semi-Lagrangian scheme. We prove the well posedness of the discrete problem and the convergence of the approximated solution toward the viscosity-measure valued solution of the exact problem.
keywords: numerical approximations Hamilton-Jacobi equation transport equation semiconcavity OSL condition. measure solutions viscosity solutions
Transport of measures on networks
Fabio Camilli Raul De Maio Andrea Tosin
Networks & Heterogeneous Media 2017, 12(2): 191-215 doi: 10.3934/nhm.2017008

In this paper we formulate a theory of measure-valued linear transport equations on networks. The building block of our approach is the initial and boundary-value problem for the measure-valued linear transport equation on a bounded interval, which is the prototype of an arc of the network. For this problem we give an explicit representation formula of the solution, which also considers the total mass flowing out of the interval. Then we construct the global solution on the network by gluing all the measure-valued solutions on the arcs by means of appropriate distribution rules at the vertexes. The measure-valued approach makes our framework suitable to deal with multiscale flows on networks, with the microscopic and macroscopic phases represented by Lebesgue-singular and Lebesgue-absolutely continuous measures, respectively, in time and space.

keywords: Network transport equation measure-valued solutions distribution conditions

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