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 $q\rightarrow p^{-}$, where $u_{q}$ is the positive solution of the sublinear Lane-Emden equation $-\Delta_{p}u_{q}=\mu_{q}u_{q}^{q-1}$ with 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.Comment: Section 5 was rewritten. Jed Brown was added as autho

Retinoids cause apoptosis in pancreatic cancer cells via activation of RAR-γ and altered expression of Bcl-2/Bax

All-trans-retinoic acid and 9-cis-retinoic acid have been reported to have inhibitory effects on pancreatic adenocarcinoma cells and we have shown that this is partly due to induction of apoptosis. In this study, the mechanisms whereby 9-cis-retinoic acid induces apoptosis in these cells were investigated. An involvement of the Bcl-2 family of proteins was shown, such that 9-cis-retinoic acid causes a decrease in the Bcl-2/Bax ratio. Overexpression of Bcl-2 also resulted in inhibition of apoptosis induced by 9-cis-retinoic acid. Furthermore, two broad-range caspase inhibitors blocked DNA fragmentation induced by 9-cis-retinoic acid, but had no effect on viability defined by mitochondrial activity. Using synthetic retinoids, which bind selectively to specific retinoic acid receptor subtypes, we further established that activation of retinoic acid receptor-γ is essential for induction of apoptosis. Only pan-retinoic acid receptor and retinoic acid receptor-γ selective agonists reduced viability and a cell line expressing very low levels of retinoic acid receptor-γ is resistant to the effects of 9-cis-retinoic acid. A retinoic acid receptor-β/γ selective antagonist also suppressed the cytotoxic effects of 9-cis-retinoic acid in a dose-dependent manner. This study provides important insight into the mechanisms involved in suppression of pancreatic tumour cell growth by retinoids. Our results encourage further work evaluating the clinical use of receptor subtype selective retinoids in pancreatic carcinoma