Inverse iteration method for higher eigenvalues of the $p$-Laplacian
This work addresses a specific mathematical challenge in partial differential equations for researchers in numerical analysis and applied mathematics, representing an incremental advancement.
The authors tackled the problem of computing higher eigenvalues of the p-Laplacian by proposing an inverse iteration method that balances Rayleigh quotients, proving its well-posedness and convergence, and providing numerical computations.
We propose a characterization of a $p$-Laplace higher eigenvalue based on the inverse iteration method with balancing the Rayleigh quotients of the positive and negative parts of solutions to consecutive $p$-Poisson equations. The approach relies on the second eigenvalue's minimax properties, but the actual limiting eigenvalue depends on the choice of initial function. The well-posedness and convergence of the iterative scheme are proved. Moreover, we provide the corresponding numerical computations. As auxiliary results, which also have an independent interest, we provide several properties of certain $p$-Poisson problems.