Hung V. Tran

Homogenization of Weakly Coupled Systems of Hamilton–Jacobi Equations with Fast Switching Rates

We consider homogenization for weakly coupled systems of Hamilton–Jacobi equations with fast switching rates. The fast switching rate terms force the solutions to converge to the same limit, which is a solution of the effective equation. We discover the appearance of the initial layers, which appear naturally when we consider the systems with different initial data and analyze them rigorously. In particular, we obtain matched asymptotic solutions of the systems and the rate of convergence. We also investigate properties of the effective Hamiltonian of weakly coupled systems and show some examples which do not appear in the context of single equations.

Adjoint methods for static Hamilton–Jacobi equations

We use the adjoint methods to study the static Hamilton–Jacobi equations and to prove the speed of convergence for those equations. The main new ideas are to introduce adjoint equations corresponding to the formal linearizations of regularized equations of vanishing viscosity type, and from the solutions σε of those we can get the properties of the solutions u of the Hamilton–Jacobi equations. We classify the static equations into two types and present two new ways to deal with each type. The methods can be applied to various static problems and point out the new ways to look at those PDE.

Remarks on the large time behavior of viscosity solutions of quasi-monotone weakly coupled systems of Hamilton–Jacobi equations

We investigate the large-time behavior of viscosity solutions of quasi-monotone weakly coupled systems of Hamilton-Jacobi equations on the n-dimensional torus. We establish a convergence result to asymptotic solutions as time goes to infinity under rather restricted assumptions.

A Lagrangian Approach to Weakly Coupled Hamilton--Jacobi Systems

We study a class of weakly coupled Hamilton-Jacobi systems with a specific aim to perform a qualitative analysis in the spirit of weak KAM theory. Our main achievement is the definition of a family of related action functionals containing the Lagrangians obtained by duality from the Hamiltonians of the system. We use them to characterize, by means of a suitable estimate, all the subsolutions of the system, and to explicitly represent some subsolutions enjoying an additional maximality property. A crucial step for our analysis is to put the problem in a suitable random frame. Only some basic knowledge of measure theory is required, and the presentation is accessible to readers without background in probability.

Some Inverse Problems in Periodic Homogenization of Hamilton-Jacobi Equations

We look at the effective Hamiltonian

The vanishing discount problem and viscosity Mather measures. Part 2: Boundary value problems

{\overline{H}}

Stochastic homogenization of viscous superquadratic Hamilton–Jacobi equations in dynamic random environment

H¯ associated with the Hamiltonian