The first eigenvalue of the \(p\)-Laplacian on a compact Riemannian manifold.
Let \(M\) be a compact Riemannian manifold of nonnegative Ricci curvature with or without boundary. In the paper under review, the authors give lower bound estimates of the first (nonlinear) eigenvalue \(\mu_{1,p}\), \(p\geq 2\), of the \(p\)-Laplacian on \(M\). More precisely, they prove that: \[ \mu_{1,p}\geq\frac{1}{p-1}\left(\frac{\pi}{4}\right)^{p} \frac{1}{d^{p}} \] where \(d\) denotes the diameter of \(M\). Moreover, if \(M\) has a boundary, they also prove similar estimates, in terms of the inscribed radius, for some nonlinear boundary value problem.
scientific article; zbMATH DE number 2003968
scientific article