Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations

Scott N. Armstrong

Research output: Contribution to journalArticlepeer-review

Abstract

We study the fully nonlinear elliptic equation(0.1)F (D2 u, D u, u, x) = f in a smooth bounded domain Ω, under the assumption that the nonlinearity F is uniformly elliptic and positively homogeneous. Recently, it has been shown that such operators have two principal "half" eigenvalues, and that the corresponding Dirichlet problem possesses solutions, if both of the principal eigenvalues are positive. In this paper, we prove the existence of solutions of the Dirichlet problem if both principal eigenvalues are negative, provided the "second" eigenvalue is positive, and generalize the anti-maximum principle of Clément and Peletier [P. Clément, L.A. Peletier, An anti-maximum principle for second-order elliptic operators, J. Differential Equations 34 (2) (1979) 218-229] to homogeneous, fully nonlinear operators.

Original languageEnglish (US)
Pages (from-to)2958-2987
Number of pages30
JournalJournal of Differential Equations
Volume246
Issue number7
DOIs
StatePublished - Apr 1 2009

Keywords

  • Anti-maximum principle
  • Dirichlet problem
  • Fully nonlinear elliptic equation
  • Principal eigenvalue

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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