Computing the first eigenpair of the p-Laplacian via inverse iteration of sublinear supersolutions

We introduce an iterative method for computing the first eigenpair $({\lambda }_p,e_p)$ for the $p$-Laplacian operator with homogeneous Dirichlet data as the limit of $({\mu }_{q,}{u}_q)$ as $q\to p^{-}$, where $u_q$ is the positive solution of the sublinear Lane-Emden equation $-{\mathrm{\Delta }}_p{u}_q={\mu }_q{u}_q^{q-1}$ with 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 $u_q$ to $e_p$ is in the $C^1$-norm and the rate of convergence of ${\mu }_q$ to ${\lambda }_p$ is at least $O(p-q)$. Numerical evidence is presented.