Two remarks on periodic solutions of Hamilton-Jacobi equations

Hitoshi Ishii, Hiroyoshi Mitake

    Research output: Chapter in Book/Report/Conference proceedingChapter

    Abstract

    We show firstly the equivalence between existence of a periodic solution of the Hamilton-Jacobi equation 'formula presented' is a bounded domain of 'formula presented', with the Dirichlet boundary condition 'formula presented' and that of a subsolution of the stationary problem 'formula presented' under the assumptions that the function 'formula presented' is periodic in t and H is coercive. Here 'formula presented' denotes the average of f over the period. This proposition is a variant of a recent result for 'formula presented' due to Bostan-Namah, and we give a different and simpler approach to such an equivalence. Secondly, we establish that any periodic solution u(x, t) of the problem, ut + H(x, Du) = 0 in 'formula presented' and 'formula presented', is constant in t on the Aubry set for H. Here H is assumed to be convex, coercive and strictly convex in a sense.

    Original languageEnglish
    Title of host publicationRecent Progress on Reaction-Diffusion Systems and Viscosity Solutions
    PublisherWorld Scientific Publishing Co.
    Pages97-119
    Number of pages23
    ISBN (Print)9789812834744, 9812834737, 9789812834737
    DOIs
    Publication statusPublished - 2009 Jan 1

    Fingerprint

    Hamilton-Jacobi Equation
    Periodic Solution
    Equivalence
    Subsolution
    Strictly Convex
    Proposition
    Dirichlet Boundary Conditions
    Bounded Domain
    Denote

    Keywords

    • Aubry sets
    • Hamilton-Jacobi equations
    • Periodic solutions

    ASJC Scopus subject areas

    • Mathematics(all)

    Cite this

    Ishii, H., & Mitake, H. (2009). Two remarks on periodic solutions of Hamilton-Jacobi equations. In Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions (pp. 97-119). World Scientific Publishing Co.. https://doi.org/10.1142/9789812834744_0005

    Two remarks on periodic solutions of Hamilton-Jacobi equations. / Ishii, Hitoshi; Mitake, Hiroyoshi.

    Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions. World Scientific Publishing Co., 2009. p. 97-119.

    Research output: Chapter in Book/Report/Conference proceedingChapter

    Ishii, H & Mitake, H 2009, Two remarks on periodic solutions of Hamilton-Jacobi equations. in Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions. World Scientific Publishing Co., pp. 97-119. https://doi.org/10.1142/9789812834744_0005
    Ishii H, Mitake H. Two remarks on periodic solutions of Hamilton-Jacobi equations. In Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions. World Scientific Publishing Co. 2009. p. 97-119 https://doi.org/10.1142/9789812834744_0005
    Ishii, Hitoshi ; Mitake, Hiroyoshi. / Two remarks on periodic solutions of Hamilton-Jacobi equations. Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions. World Scientific Publishing Co., 2009. pp. 97-119
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