Hiroyoshi Mitake

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.

A PDE Approach to Large-Time Asymptotics for Boundary-Value Problems for Nonconvex Hamilton–Jacobi Equations

We investigate the large-time behavior of three types of initial-boundary value problems for Hamilton–Jacobi Equations with nonconvex Hamiltonians. We consider the Neumann or oblique boundary condition, the state constraint boundary condition and Dirichlet boundary condition. We establish general convergence results for viscosity solutions to asymptotic solutions as time goes to infinity via an approach based on PDE techniques. These results are obtained not only under general conditions on the Hamiltonians but also under weak conditions on the domain and the oblique direction of reflection in the Neumann case.

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.

Singular Neumann problems and large-time behavior of solutions of noncoercive Hamilton-Jacobi equations

We investigate the large-time behavior of viscosity solutions of Hamilton-Jacobi equations with noncoercive Hamiltonian in a multidimensional Euclidean space. Our motivation comes from a model describing growing faceted crystals recently discussed by E. Yokoyama, Y. Giga and P. Rybka. Surprisingly, growth rates of viscosity solutions of these equations depend on the x-variable. In a part of the space called the effective domain, growth rates are constant, but outside of this domain, they seem to be unstable. Moreover, on the boundary of the effective domain, the gradient with respect to the x-variable of solutions blows up as time goes to infinity. Therefore, we are naturally led to study singular Neumann problems for stationary HamiltonJacobi equations. We establish the existence, stability and comparison results for singular Neumann problems and apply the results for a large-time asymptotic profile on the effective domain of viscosity solutions of Hamilton-Jacobi equations with noncoercive Hamiltonian.

On the Large Time Behavior of Solutions of Hamilton-Jacobi Equations Associated with Nonlinear Boundary Conditions

In this article, we study the large time behavior of solutions of first-order Hamilton–Jacobi Equations set in a bounded domain with nonlinear Neumann boundary conditions, including the case of dynamical boundary conditions. We establish general convergence results for viscosity solutions of these Cauchy–Neumann problems by using two fairly different methods: the first one relies only on partial differential equations methods, which provides results even when the Hamiltonians are not convex, and the second one is an optimal control/dynamical system approach, named the “weak KAM approach”, which requires the convexity of Hamiltonians and gives formulas for asymptotic solutions based on Aubry–Mather sets.

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.

The vanishing discount problem and viscosity Mather measures. Part 1: The problem on a torus

We develop a variational approach to the vanishing discount problem for fully nonlinear, degenerate elliptic, partial differential equations. Under mild assumptions, we introduce viscosity Mather measures for such partial differential equations, which are natural extensions of the Mather measures. Using the viscosity Mather measures, we prove that the whole family of solutions

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

v^\lambda

On convergence rates for solutions of approximate mean curvature equations

of the discount problem with the factor

The Linearized Monge-Ampère Equation

\lambda>0