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

Guy Barles, Hitoshi Ishii, Hiroyoshi Mitake

    Research output: Contribution to journalArticle

    9 Citations (Scopus)

    Abstract

    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.

    Original languageEnglish
    Pages (from-to)515-558
    Number of pages44
    JournalArchive for Rational Mechanics and Analysis
    Volume204
    Issue number2
    DOIs
    Publication statusPublished - 2012 May

    Fingerprint

    Hamiltonians
    Large Time Behavior
    Nonlinear Boundary Conditions
    Hamilton-Jacobi Equation
    Behavior of Solutions
    Dynamical Boundary Conditions
    Boundary conditions
    Asymptotic Solution
    Viscosity Solutions
    Neumann Problem
    Neumann Boundary Conditions
    Convergence Results
    Partial differential equations
    Convexity
    Bounded Domain
    Cauchy Problem
    Dynamical systems
    Optimal Control
    Partial differential equation
    Dynamical system

    ASJC Scopus subject areas

    • Analysis
    • Mechanical Engineering
    • Mathematics (miscellaneous)

    Cite this

    On the Large Time Behavior of Solutions of Hamilton-Jacobi Equations Associated with Nonlinear Boundary Conditions. / Barles, Guy; Ishii, Hitoshi; Mitake, Hiroyoshi.

    In: Archive for Rational Mechanics and Analysis, Vol. 204, No. 2, 05.2012, p. 515-558.

    Research output: Contribution to journalArticle

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