Computing the first eigenvalue of the \(p\)-Laplacian via the inverse power method. (English) Zbl 1172.35047

Authors’ abstract: The authors discuss a new method for computing the first Dirichlet eigenvalue of the \(p\)-Laplacian inspired by the inverse power method in finite dimensional linear algebra. The iterative technique is independent of the particular method used in solving the \(p\)-Laplacian equation and therefore can be made as efficient as the latter. The method is validated theoretically for any ball in \({\mathbb R}^n\) if \(p > 1\) and for any bounded domain in the particular case \(p=2\). For \(p > 2\) the method is validated numerically for the square.


35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
47J10 Nonlinear spectral theory, nonlinear eigenvalue problems
Full Text: DOI


[1] Allegretto, W.; Huang, Y.X., A Picone’s identity for the p-Laplacian and applications, Nonlinear anal., 32, 819-830, (1998) · Zbl 0930.35053
[2] Anderson, G.D.; Vamanamurthy, M.K.; Vuorinen, M., Inequalities for quasiconformal mappings in space, Pacific J. math., 160, 1-18, (1993) · Zbl 0793.30014
[3] Andreianov, B.; Boyer, F.; Hubert, F., Finite volume schemes for the p-Laplacian on Cartesian meshes, ESAIM math. model. numer. anal., 38, 6, 931-959, (2004) · Zbl 1081.65105
[4] Damascelli, L., Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. sc. norm. super. Pisa cl. sci., 26, 4, 689-707, (1998)
[5] Demmel, J.W., Applied numerical linear algebra, (1997), SIAM · Zbl 0879.65017
[6] Feng, X.; He, Y., High order iterative methods without derivatives for solving nonlinear equations, Appl. math. comput., 186, 1617-1623, (2007) · Zbl 1119.65036
[7] Kawohl, B.; Fridman, V., Isoperimetric estimates for the first eigenvalue of the p-Laplace operator and the Cheeger constant, Comment. math. univ. carolin., 44, 659-667, (2003) · Zbl 1105.35029
[8] Lefton, L.; Wei, D., Numerical approximation of the first eigenpair of the p-Laplacian using finite elements and the penalty method, Numer. funct. anal. optim., 18, 3-4, 389-399, (1997) · Zbl 0884.65103
[9] Watkins, D.S., Fundamentals of matrix computations, (2002), John Wiley & Sons · Zbl 1005.65027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.