Computing the first eigenpair of the \(p\)-Laplacian via inverse iteration of sublinear supersolutions. (English) Zbl 1255.65205

Summary: 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 \(q\rightarrow p ^{ - }\), where \(u _{q }\) is the positive solution of the sublinear Lane-Emden equation \(-\Delta_{p}u_{q}=\mu_{q}u_{q}^{q-1}\) 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 \(u _{q }\) 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.


65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs


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