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).
- Finite elements
- Nonlinear eigenvalue problems
- Penalty method
ASJC Scopus subject areas
- Signal Processing
- Computer Science Applications
- Control and Optimization