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Inverse iteration method for higher eigenvalues of the $p$-Laplacian

Vladimir Bobkov, Timur Galimov

TL;DR

This work addresses the computation of higher eigenvalues of the p-Laplacian on bounded domains by introducing Algorithm A, an inverse-iteration scheme that balances the Rayleigh quotients of the positive and negative parts of iterates. Each step solves a p-Poisson problem with a parametrized right-hand side and selects a coefficient to enforce $R^+[u_{k+1}]=R^-[u_{k+1}]$, yielding a sign-changing iterate whose Rayleigh quotient monotonically decreases toward a limit $R^*$ that is a higher eigenvalue; the associated eigenfunctions form the limit set, and the parameter sequence $\alpha_k$ converges to a fixed point $\alpha^*$ of the balancing function. The authors prove well-posedness, monotonicity, and convergence properties, including that subsequences of normalized iterates converge to eigenfunctions of $R^*$ and, under mild isolation assumptions (e.g., for $\lambda_2(\Omega,p)$), convergence to the second eigenpair. Numerical experiments on the unit square validate the method, show convergence to symmetry-consistent eigenvalues, and agree with existing symmetry-based estimates in the literature. The results provide a robust framework for approximating higher nonlinear eigenvalues of the p-Laplacian and demonstrate practical computational viability.

Abstract

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.

Inverse iteration method for higher eigenvalues of the $p$-Laplacian

TL;DR

This work addresses the computation of higher eigenvalues of the p-Laplacian on bounded domains by introducing Algorithm A, an inverse-iteration scheme that balances the Rayleigh quotients of the positive and negative parts of iterates. Each step solves a p-Poisson problem with a parametrized right-hand side and selects a coefficient to enforce , yielding a sign-changing iterate whose Rayleigh quotient monotonically decreases toward a limit that is a higher eigenvalue; the associated eigenfunctions form the limit set, and the parameter sequence converges to a fixed point of the balancing function. The authors prove well-posedness, monotonicity, and convergence properties, including that subsequences of normalized iterates converge to eigenfunctions of and, under mild isolation assumptions (e.g., for ), convergence to the second eigenpair. Numerical experiments on the unit square validate the method, show convergence to symmetry-consistent eigenvalues, and agree with existing symmetry-based estimates in the literature. The results provide a robust framework for approximating higher nonlinear eigenvalues of the p-Laplacian and demonstrate practical computational viability.

Abstract

We propose a characterization of a -Laplace higher eigenvalue based on the inverse iteration method with balancing the Rayleigh quotients of the positive and negative parts of solutions to consecutive -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 -Poisson problems.

Paper Structure

This paper contains 10 sections, 18 theorems, 128 equations, 5 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Brief overview of the inverse iteration method
  3. Main results
  4. Properties of the equation $-\Delta_p v = \left|f\right|^{p-2}f$
  5. Well-posedness and monotonicity
  6. Convergence
  7. Preliminary results
  8. Proof of Theorem \ref{['thm:main']}
  9. Numerical results
  10. Appendix. Basic properties of the $p$-Poisson problem

Key Result

Theorem 3.1

Let $k \geqslant 0$. Then there exists a root $\alpha_k \in (0;1)$ of the equation eq:R+=R-. As a consequence, $u_{k+1} := \varphi_{k} (\alpha_{k})$ is defined, it is sign-changing, solves the $p$-Poisson problem and satisfies Moreover, we have where $R[u_{k+2}] = R[u_{k+1}]$ if and only if $u_{k+1}$ is an eigenfunction.

Figures (5)

  • Figure 1: The first 7501 elements of the sequence \ref{['countersequence']}.
  • Figure 2: Example \ref{['ex:square rad']}: the graph of the initial guess \ref{['square rad']}.
  • Figure 3: Example \ref{['ex:square mid']}: the detailed graphs for some particular values of $p$. Left column -- the Rayleigh quotients sequences $\left\{R[u_k]\right\}_{k=0}^5$ (logarithmic $y$ scale); middle column -- the $W_0^{1,p}(\Omega)$- and $L^p(\Omega)$-norms (red and green, respectively) of the difference between two consecutive approximations $u_k$ and $u_{k+1}$ (logarithmic $y$ scale); right column -- the graphs of the resulting approximations $u_5$. The rows correspond to (from the top to the bottom) $p = 1.6$, $p = 2$, $p = 3.5$, and $p = 5$.
  • Figure 4: Example \ref{['ex:square diag']}: the detailed graphs for some particular values of $p$. Left column -- the Rayleigh quotients sequences $\left\{R[u_k]\right\}_{k=0}^5$ (logarithmic $y$ scale); middle column -- the $W_0^{1,p}(\Omega)$- and $L^p(\Omega)$-norms (red and green, respectively) of the difference between two consecutive approximations $u_k$ and $u_{k+1}$ (logarithmic $y$ scale); right column -- the graphs of the resulting approximations $u_5$. The rows correspond to (from the top to the bottom) $p = 1.6$, $p = 2$, $p = 3.5$, and $p = 5$.
  • Figure 5: Example \ref{['ex:square rad']}: the results of the numerical experiments for $p = 1.7$ (left column), $p = 2$ (millde column), and $p = 3$ (right column). From the top to the bottom: the graphs of the Rayleigh quotients; the graphs of the functions $u_k$ corresponding to the first, the second, and the third stabilization stages of $R[u_k]$, respectively.

Theorems & Definitions (44)

  • Theorem 3.1: Well-posedness and monotonicity
  • Theorem 3.2: Convergence
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Remark 4.1
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • proof
  • ...and 34 more