Numerical approximation of the first eigenpair of the p-Laplacian using finite elements and the penalty method

Lew Lefton, Dongming Wei

Research output: Contribution to journalArticle

33 Citations (Scopus)


For 1 < p < ∞, consider the eigenvalue problem for the p-Laplacian - △pu = λ|u|p-2u, u|∂Ω = 0 where △pu = div(|▽u|p-2▽u). The first eigenvalue λ1 can be obtained by minimizing the functional ∫Ω|▽u|p/ ∫Ω |u|p over W1,P0(Ω). A method for computing λ1 numerically is presented. The technique uses a finite element approximation to the first eigenfunction and a penalty function to enforce the constraint. Convergence is proved and numerical results are presented. The numerical results are compared with exact values when known. A lower bound for p-Laplacian eigenvalues is also presented. In particular, this work provides a computational framework for obtaining precise approximations of the best constant for the Sobolev imbedding W1,p0(Ω) (Rightwards arrow with hook sign) Lp(Ω).

Original languageEnglish
Pages (from-to)389-399
Number of pages11
JournalNumerical Functional Analysis and Optimization
Issue number3-4
Publication statusPublished - Jan 1 1997



  • Finite elements
  • Nonlinear eigenvalue problems
  • Penalty method
  • p-Laplacian

ASJC Scopus subject areas

  • Analysis
  • Signal Processing
  • Computer Science Applications
  • Control and Optimization

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