Aubry-Mather Theory and Hamilton-Jacobi Equations

Bessi, Ugo

doi:10.1007/s00220-002-0781-5

Cited by 25 publications

(28 citation statements)

References 5 publications

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“…Our result is a generalization from the one dimensional case to the d-dimensional case of the results of Jauslin, Kreiss and Moser [22], as well as of Bessi [4]. Our method is very close to the one in [4]: we use a variational representation formula for φ ε , which is a stochastic version of the "Lax-Oleinik formula" used to represent solutions of the non-viscous equation. Then, using standard techniques in stochastic calculus, we estimate the limiting behavior, which allows us to obtain the same conclusion in higher dimensions.…”

Section: X P)mentioning

confidence: 85%

“…Remark 6. As already mentioned, this result is a generalization to higher dimension (but in the autonomous case) of Theorem 1 in [4]. In that paper, the assumption that the Aubry set consists in a finite union of hyperbolic periodic orbits is expressed in a slightly different form, namely:…”

Section: Preliminaries Andmentioning

confidence: 91%

“…Then, using standard techniques in stochastic calculus, we estimate the limiting behavior, which allows us to obtain the same conclusion in higher dimensions. See Section 2 for a more detailed comparison between our assumptions and those of [4].…”

Section: X P)mentioning

confidence: 99%

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Physical solutions of the Hamilton-Jacobi equation

Anantharaman¹,

Iturriaga²,

Padilla³

et al. 2005

Discrete &Amp; Continuous Dynamical Systems - B

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We consider a Lagrangian system on the d-dimensional torus, and the associated Hamilton-Jacobi equation. Assuming that the Aubry set of the system consists in a finite number of hyperbolic periodic orbits of the Euler-Lagrange flow, we study the vanishing-viscosity limit, from the viscous equation to the inviscid problem. Under suitable assumptions, we show that solutions of the viscous Hamilton-Jacobi equation converge to a unique solution of the inviscid problem.

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Section: X P)mentioning

confidence: 85%

Section: Preliminaries Andmentioning

confidence: 91%

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Physical solutions of the Hamilton-Jacobi equation

Anantharaman¹,

Iturriaga²,

Padilla³

et al. 2005

Discrete &Amp; Continuous Dynamical Systems - B

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“…The selection problem for possibly degenerate, fully nonlinear Hamilton-Jacobi-Bellman equations was considered in [15,16]. A related selection problem was addressed in [5,17] and selection questions motivated by finite-difference schemes were examined in [22]. In all these papers, the convexity of the Hamiltonian was essential and no extensions to non-convex Hamiltonians were offered.…”

Section: Introductionmentioning

confidence: 99%

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

Gomes

Mitake

Tran

2018

J. Math. Soc. Japan

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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.

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“…Some nonconvex cases are studied in [16]. A partial result to the selection problem in the vanishing viscosity method is obtained in [3], [2], and that in a finite difference method is given in [24]. The lecture note [19] states lots about related topics based on the nonlinear adjoint method.…”

Section: Introductionmentioning

confidence: 99%

Weak KAM theory for discounted Hamilton–Jacobi equations and its application

Mitake

Soga

2018

Calc. Var.

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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.

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Aubry-Mather Theory and Hamilton-Jacobi Equations

Cited by 25 publications

References 5 publications

Physical solutions of the Hamilton-Jacobi equation

Physical solutions of the Hamilton-Jacobi equation

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

Weak KAM theory for discounted Hamilton–Jacobi equations and its application

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