Weak KAM aspects of convex Hamilton-Jacobi equations with Neumann type boundary conditions

Hitoshi Ishii

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    13 Citations (Scopus)

    Abstract

    We study convex Hamilton-Jacobi equations H(x,Du)=0 and ut+H(x,Du)=0 in a bounded domain Ω of Rn with the Neumann type boundary condition Dγu=g in the viewpoint of weak KAM theory, where γ is a vector field on the boundary ∂ Ω pointing a direction oblique to ∂ Ω We establish the stability under the formations of infimum and of convex combinations of subsolutions of convex Hamilton-Jacobi equations, some comparison and existence results for convex and coercive Hamilton-Jacobi equations with the Neumann type boundary condition as well as existence results for the Skorokhod problem. We define the Aubry set associated with the Neumann type boundary problem and establish some properties of the Aubry set including the existence results for the "calibrated" extremals for the corresponding action functional (or variational problem).

    Original languageEnglish
    Pages (from-to)99-135
    Number of pages37
    JournalJournal des Mathematiques Pures et Appliquees
    Volume95
    Issue number1
    DOIs
    Publication statusPublished - 2011 Jan

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    Keywords

    • Aubry-Mather theory
    • Hamilton-Jacobi equations
    • Neumann type boundary conditions
    • Viscosity solutions
    • Weak KAM theory

    ASJC Scopus subject areas

    • Applied Mathematics
    • Mathematics(all)

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