Claudio Marchi

Rates of convergence in periodic homogenization of fully nonlinear uniformly elliptic PDEs

We consider periodic homogenization of the fully nonlinear uniformly elliptic equation . We give an estimate of the rate of convergence of ue to the solution u of the homogenized problem . Moreover we describe a numerical scheme for the approximation of the effective nonlinearity and we estimate the corresponding rate of convergence.

Stationary Mean Field Games systems defined on networks

We consider a stationary Mean Field Games system defined on a network. In this framework, the transition conditions at the vertices play a crucial role: the ones here considered are based on the optimal control interpretation of the problem. We prove separately the well-posedness for each of the two equations composing the system. Finally, we prove existence and uniqueness of the solution of the Mean Field Games system.

Homogenization and vanishing viscosity in fully nonlinear elliptic equations: rate of convergence estimates

Abstract This paper is devoted to studying the behavior as ε → 0 of the equations uε + H(x, x/ε, Duε, εγD2uε) = 0 with γ > 0. It is known that, under some periodicity and ellipticity or coercivity assumptions, the solution uε converges to the solution u of an effective equation u + H̅(x, Du) = 0, with an effective Hamiltonian H̅ dependent on the value of γ. The main purpose of this paper is to estimate the rate of convergence of uε to u. Moreover we discuss some examples and model problems.

Eikonal equations on ramified spaces

We generalize the results in [16] to higher dimensional ramified spaces. For this purpose we introduce ramified manifolds and, as special cases, locally elementary polygonal ramified spaces (LEP spaces). On LEP spaces we develop a theory of viscosity solutions for Hamilton-Jacobi equations, providing existence and uniqueness results.

On the convergence rate in multiscale homogenization of fully nonlinear elliptic problems

This paper concerns periodic multiscale homogenization for fully nonlinear equations of the form

Homogenization of a mean field game system in the small noise limit

u^\epsilon+H^\epsilon (x,\frac{x}{\epsilon},\ldots,\frac{x}{epsilon^k},Du^\epsilon,D^2u^\epsilon)=0

Stochastic Homogenization for Functionals with Anisotropic Rescaling and Noncoercive Hamilton--Jacobi Equations

. The operators

A flame propagation model on a network with application to a blocking problem

H^\epsilon

Multiscale problems and homogenization for second-order Hamilton-Jacobi equations

are a regular perturbations of some uniformly elliptic, convex operator

A comparison among various notions of viscosity solution for Hamilton–Jacobi equations on networks

H