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
Externally publishedYes

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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

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