Eder Marinho Martins

Computing the First Eigenpair of the p-Laplacian via Inverse Iteration of Sublinear Supersolutions

We introduce an iterative method for computing the first eigenpair (λp,ep) for the p-Laplacian operator with homogeneous Dirichlet data as the limit of (μq,uq) as q→p−, where uq is the positive solution of the sublinear Lane-Emden equation

Eigenvalues and eigenfunctions of the Laplacian via inverse iteration with shift

-\Delta_{p}u_{q}=\mu_{q}u_{q}^{q-1}

Computing the best constant in the Sobolev inequality for a ball

with the same boundary data. The method is shown to work for any smooth, bounded domain. Solutions to the Lane-Emden problem are obtained through inverse iteration of a super-solution which is derived from the solution to the torsional creep problem. Convergence of uq to ep is in the C1-norm and the rate of convergence of μq to λp is at least O(p−q). Numerical evidence is presented.

Positive solution for a class of coupled (p, q)-Laplacian nonlinear systems

In this paper we present an iterative method, inspired by the inverse iteration with shift technique of finite linear algebra, designed to find the eigenvalues and eigenfunctions of the Laplacian with homogeneous Dirichlet boundary condition for arbitrary bounded domains ? ? R N . This method, which has a direct functional analysis approach, does not approximate the eigenvalues of the Laplacian as those of a finite linear operator. It is based on the uniform convergence away from nodal surfaces and can produce a simple and fast algorithm for computing the eigenvalues with minimal computational requirements, instead of using the ubiquitous Rayleigh quotient of finite linear algebra. Also, an alternative expression for the Rayleigh quotient in the associated infinite dimensional Sobolev space which avoids the integration of gradients is introduced and shown to be more efficient. The method can also be used in order to produce the spectral decomposition of any given function u ? L 2 ( ? ) .

Computing the first eigenpair of the p-Laplacian in annuli

ABSTRACT Let be the unit ball of and let if and if For each let be the positive function such that and In this paper, we develop an iterative method for obtaining the pair , starting from . Since is explicitly known, the method is computationally practical, as our numerical tests show.