Searcharxiv⌕ Search

arXiv subjects

Hiroyoshi Mitake

Publications and source records attributed to Hiroyoshi Mitake.

At least 37 records · Page 2Linked to original sources
arXiv2020

The large time profile for Hamilton--Jacobi--Bellman equations

Here, we study the large-time limit of viscosity solutions of the Cauchy problem for second-order Hamilton--Jacobi--Bellman equations with convex Hamiltonians in the torus. This large-time limit solves the corresponding stationary problem, sometimes called the ergodic problem. This problem, however, has multiple viscosity solutions and, thus, a key question is which of these solutions is selected by the limit. Here, we provide a representation for the viscosity solution to the Cauchy problem in terms of generalized holonomic measures. Then, we use this representation to characterize the large-time limit in terms of the initial data and generalized Mather measures. In addition, we establish various results on generalized Mather measures and duality theorems that are of independent interest.

math.AP↗
arXiv2019

The selection problem for some first-order stationary mean-field games

Here, we study the existence and the convergence of solutions for the vanishing discount MFG problem with a quadratic Hamiltonian. We give conditions under which the discounted problem has a unique classical solution and prove convergence of the vanishing-discount limit to a unique solution up to constants. Then, we establish refined asymptotics for the limit. When those conditions do not hold, the limit problem may not have a unique solution and its solutions may not be smooth, as we illustrate in an elementary example. Finally, we investigate the stability of regular weak solutions and address the selection problem. Using ideas from Aubry-Mather theory, we establish a selection criterion for the limit.

math.AP↗
arXiv2019

Remarks on large time behavior of level-set mean curvature flow equations with driving and source terms

We study a level-set mean curvature flow equation with driving and source terms, and establish convergence results on the asymptotic behavior of solutions as time goes to infinity under some additional assumptions. We also study the associated stationary problem in details in a particular case, and establish Alexandrov's theorem in two dimensions in the viscosity sense, which is of independent interest.

math.AP↗
arXiv2019

Rate of convergence in periodic homogenization of Hamilton-Jacobi equations: the convex setting

We study the rate of convergence of $u^ε$, as $ε\to 0+$, to $u$ in periodic homogenization of Hamilton-Jacobi equations. Here, $u^ε$ and $u$ are viscosity solutions to the oscillatory Hamilton-Jacobi equation and its effective equation \begin{equation*} {\rm (C)_ε} \qquad \begin{cases} u_t^ε+H\left(\frac{x}ε,Du^ε\right)=0 \qquad &\text{in} \ \mathbb{R}^n \times (0,\infty), u^ε(x,0)=g(x) \qquad &\text{on} \ \mathbb{R}^n, \end{cases} \end{equation*} and \begin{equation*} {\rm (C)} \qquad \begin{cases} u_t+\overline{H}\left(Du\right)=0 \qquad &\text{in} \ \mathbb{R}^n \times (0,\infty), u(x,0)=g(x) \qquad &\text{on} \ \mathbb{R}^n, \end{cases} \end{equation*} respectively. We assume that the Hamiltonian $H=H(y,p)$ is coercive and convex in the $p$ variable and is $\mathbb{Z}^n$-periodic in the $y$ variable, and the initial data $g$ is bounded and Lipschitz continuous.

math.AP↗
arXiv2019

Generalized ergodic problems: existence and uniqueness structures of solutions

We study a generalized ergodic problem (E), which is a Hamilton-Jacobi equation of contact type, in the flat $n$-dimensional torus. We first obtain existence of solutions to this problem under quite general assumptions. Various examples are presented and analyzed to show that (E) does not have unique solutions in general. We then study uniqueness structures of solutions to (E) in the convex setting by using the nonlinear adjoint method.

math.AP↗
arXiv2018

Existence of asymptotic speed of solutions to birth and spread type nonlinear partial differential equations

In this paper, we prove the existence of asymptotic speed of solutions to fully nonlinear, possibly degenerate parabolic partial differential equations in a general setting. We then give some explicit examples of equations in this setting and study further properties of the asymptotic speed for each equation. Some numerical results concerning the asymptotic speed are presented.

math.AP↗
arXiv2018

On uniqueness sets of additive eigenvalue problems and applications

In this paper, we provide a simple way to find uniqueness sets for additive eigenvalue problems of first and second order Hamilton--Jacobi equations by using a PDE approach. An application in finding the limiting profiles for large time behaviors of first order Hamilton--Jacobi equations is also obtained.

math.AP↗
arXiv2016

Weak KAM theory for discount Hamilton-Jacobi equations and its application

Weak KAM theory for discount Hamilton-Jacobi equations and corresponding discount Lagrangian/Hamiltonian dynamics is developed. Then it is applied to error estimates for viscosity solutions in the vanishing discount process. The main feature is to introduce and investigate the family of $α$-limit points of minimizing curves, with some details in terms of minimizing measures. In error estimates, the family of $α$-limit points is effectively exploited with properties of the corresponding dynamical systems.

math.AP↗
arXiv2016

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

In arXiv:1603.01051 (Part 1 of this series), we have introduced a variational approach to studying the vanishing discount problem for fully nonlinear, degenerate elliptic, partial differential equations in a torus. We develop this approach further here to handle boundary value problems. In particular, we establish new representation formulas for solutions of discount problems, critical values, and use them to prove convergence results for the vanishing discount problems.

math.AP↗
arXiv2016

The Selection problem for discounted Hamilton-Jacobi equations: some non-convex cases

Here, we study the selection problem for the vanishing discount approximation of non-convex, first-order Hamilton-Jacobi equations. While the selection problem is well understood for convex Hamiltonians, the selection problem for non-convex Hamiltonians has thus far not been studied. We begin our study by examining a generalized discounted Hamilton-Jacobi equation. Next, using an exponential transformation, we apply our methods to strictly quasi-convex and to some non-convex Hamilton-Jacobi equations. Finally, we examine a non-convex Hamiltonian with flat parts to which our results do not directly apply. In this case, we establish the convergence by a direct approach.

math.AP↗
arXiv2016

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 $v^λ$ of the discount problem with the factor $λ>0$ converges to a solution of the ergodic problem as $λ\to 0$.

math.AP↗
arXiv2015

On asymptotic speed of solutions to level-set mean curvature flow equations with driving and source terms

We investigate a model equation in the crystal growth, which is described by a level-set mean curvature flow equation with driving and source terms. We establish the well-posedness of solutions, and study the asymptotic speed. Interestingly, a new type of nonlinear phenomena in terms of asymptotic speed of solutions appears, which is very sensitive to the shapes of source terms.

math.AP↗
arXiv2014

Selection problems for a discounted degenerate viscous Hamilton--Jacobi equation

We prove that the solution of the discounted approximation of a degenerate viscous Hamilton--Jacobi equation with convex Hamiltonians converges to that of the associated ergodic problem. We characterize the limit in terms of stochastic Mather measures by naturally using the nonlinear adjoint method, and deriving a commutation lemma. This convergence result was first achieved by Davini, Fathi, Iturriaga, and Zavidovique for the first order Hamilton--Jacobi equation.

math.AP↗
arXiv2014

Existence for stationary mean field games with quadratic Hamiltonians with congestion

In this paper, we investigate the existence and uniqueness of solutions to a stationary mean field game model introduced by J.-M. Lasry and P.-L. Lions. This model features a quadratic Hamiltonian with possibly singular congestion effects. Thanks to a new class of a-priori bounds, combined with the continuation method, we prove the existence of smooth solutions in arbitrary dimensions.

math.AP↗
arXiv2014

Weakly coupled systems of the infinity Laplace equations

We derive the weakly coupled systems of the infinity Laplace equations via a tug-of-war game introduced by Peres, Schramm, Sheffield, and Wilson (2009). We establish existence, uniqueness results of the solutions, and introduce a new notion of "generalized cones" for systems. By using "generalized cones" we analyze blow-up limits of solutions.

math.AP↗
arXiv2013

A new method for large time behavior of degenerate viscous Hamilton--Jacobi equations with convex Hamiltonians

We introduce a new machinery to study the large time behavior for general classes of Hamilton--Jacobi type equations, which include degenerate parabolic equations and weakly coupled systems. We establish the convergence results by using the nonlinear adjoint method and identifying new long time averaging effects. These methods are robust and can easily be adapted to study the large time behavior of related problems.

math.AP↗
arXiv2013

Large-time behavior for obstacle problems for degenerate viscous Hamilton--Jacobi equations

Cagnetti, Gomes, Mitake and Tran (2013) introduced a new idea to study the large time behavior for degenerate viscous Hamilton--Jacobi equations. In this paper, we apply the method to study the large-time behavior of the solution to the obstacle problem for degenerate viscous Hamilton--Jacobi equations. We establish the convergence result under rather general assumptions.

math.AP↗