Eigenvalue problem for fully nonlinear second-order elliptic PDE on balls, II

Norihisa Ikoma, Hitoshi Ishii

    Research output: Contribution to journalArticle

    4 Citations (Scopus)

    Abstract

    This is a continuation of Ikoma and Ishii (Ann Inst H Poincaré Anal Non Linéaire 29:783–812, 2012) and we study the eigenvalue problem for fully nonlinear elliptic operators, positively homogeneous of degree one, on finite intervals or balls. In the multi-dimensional case, we consider only radial eigenpairs. Our eigenvalue problem has a general first-order boundary condition which includes, as a special case, the Dirichlet, Neumann and Robin boundary conditions. Given a nonnegative integer n, we prove the existence and uniqueness, modulo multiplication of the eigenfunction by a positive constant, of an eigenpair whose eigenfunction, as a radial function in the multi-dimensional case, has exactly n zeroes. When an eigenfunction has n zeroes, we call the corresponding eigenvalue of nth order. Furthermore, we establish results concerning comparison of two eigenvalues, characterizations of nth order eigenvalues via differential inequalities, the maximum principle for the boundary value problem in connection with the principal eigenvalue, and existence of a solution having n zeroes, as a radial function in the multi-dimensional case, of the boundary value problem with an inhomogeneous term.

    Original languageEnglish
    Pages (from-to)451-510
    Number of pages60
    JournalBulletin of Mathematical Sciences
    Volume5
    Issue number3
    DOIs
    Publication statusPublished - 2015 Oct 1

    Fingerprint

    Elliptic PDE
    Fully Nonlinear
    Eigenvalue Problem
    Eigenfunctions
    Ball
    Eigenvalue
    Radial Functions
    Zero
    Boundary Value Problem
    Principal Eigenvalue
    Robin Boundary Conditions
    Differential Inequalities
    Nonlinear Operator
    Comparison Result
    Neumann Boundary Conditions
    Maximum Principle
    Elliptic Operator
    Dirichlet Boundary Conditions
    Continuation
    Modulo

    Keywords

    • Eigenvalue problem
    • Fully nonlinear equation
    • General boundary conditions
    • Higher order eigenvalues
    • Principal eigenvalues

    ASJC Scopus subject areas

    • Mathematics(all)

    Cite this

    Eigenvalue problem for fully nonlinear second-order elliptic PDE on balls, II. / Ikoma, Norihisa; Ishii, Hitoshi.

    In: Bulletin of Mathematical Sciences, Vol. 5, No. 3, 01.10.2015, p. 451-510.

    Research output: Contribution to journalArticle

    Ikoma, Norihisa ; Ishii, Hitoshi. / Eigenvalue problem for fully nonlinear second-order elliptic PDE on balls, II. In: Bulletin of Mathematical Sciences. 2015 ; Vol. 5, No. 3. pp. 451-510.
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