Remarks on the second Neumann eigenvalue

Abstract

his work reviews some basic features on the second (first nontrivial) eigenvalue \(\lambda_2\) to the Neumann problem $$\displaylines{ -\Delta_p u = \lambda |u|^{p-2}u \quad x\in \Omega\cr |\nabla u|^{p-2}\frac{\partial u}{\partial \nu}=0 \quad x\in \partial\Omega, }$$ where \(\Omega\) is a bounded Lipschitz domain of \(\mathbb{R}^N\), \(\nu\) is the outer unit normal, and \(\Delta_p u = \text{div}(|\nabla u|^{p-2}\nabla u)\) is the p-Laplacian operator. We are mainly concerned with the variational characterization of \(\lambda_2\) and place emphasis on the range \(1 < p < 2\), where the nonlinearity \(|u|^{p-2}u\) becomes non smooth. We also address the corresponding result for the p-Laplacian in graphs.