Computing the first eigenvalue of the p-Laplacian via the inverse power method

Abstract

In this paper, we 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 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.

Inequalities for Eigenfunctions of the P-Laplacian

“…Numerous other authors, see e. g. [2], [3,4], [5], [6], [7] and the bibliographies of these papers, have extended this work in various directions including the study of generalized hyperbolic functions and their inverses. Our goal here to study these p-trigonometric and p-hyperbolic functions and to prove several inequalities for them.…”

Computing the sinp function via the inverse power method

“…It is easy to verify that if λ 1 is the first eigenvalue of 2) and v 1 is the corresponding positive eigenfunction, then λ 1 is also the first eigenvalue for (2.1) with…”

“…In this paper, we present a new method to compute the function sin p , inspired by recent work done by the authors in [2], where an iterative algorithm based on the inverse power method of linear algebra was introduced for the computation of the first eigenvalue and first eigenfunction of the Dirichlet problem for the p-Laplacian in arbitrary domains in R N . The functions sin p , 1 < p < ∞, can be thought of as generalizations of the familiar trigonometric functions.…”

“…Our method depends on the convergence of a sequence of functions whose definition, as in [2], is motivated by an extension of the inverse power method of linear algebra for obtaining the first eigenvalue and first eigenfunction of finite dimensional linear operators. These functions are recursively defined and can be given in integral form, so that they can be obtained by numerical integration.…”

Approximation theorem for principle eigenvalue of discrete p-Laplacian

“…Next, let B k pf q " f k`1´fk , then the weighted p-Laplacian Ω p (p ą 1) can be rewritten as Ω p f pkq " ν k | B k pf q | p´1 sgnpB k pf qq´ν k´1 | B k´1 pf q | p´1 sgnpB k´1 pf qq. (1) Let λ p denote the principal eigenvalue of the weighted p-Laplacian Ω p and set When N ď 8, the approximation procedure for λ p is presented in the following theorem.…”