The effect of perturbations on the first eigenvalue of the \(p\)-Laplacian (Q1854699)

The \(p\)-Laplacian on a compact Riemannian manifold \((M,g)\) is defined by \(\Delta_p f=\delta(|df |^{p-2}df)\), where \(p>1\) and \(\delta= -\text{div}_g\). The real constants \(\lambda\) for which the equation \(\Delta_p f=\lambda|f|^{p-2}f\) has nontrivial solution are called the eigenvalues of \(\Delta_p\) and the associated solutions are the eigenfunctions. It is known that the set of the nonzero eigenvalues is nonempty, bounded subset of \([0,\infty]\). Let \(\Omega\) be a domain of \(M\) equipped with Lipschitzian boundary: it is shown here that one can make the volume of \(M\) arbitrarily close to the volume of \((\Omega,g)\) while the first eigenvalue of the \(p\)-Laplacian on \(M\) remains uniformly bounded from below in terms of the first eigenvalue of the Neumann problem [\textit{L. Veron}, Colloq. Math. Soc., János Bolyai 62, 317-352 (1991; Zbl 0822.58052)] for the \(p\)-Laplacian on \((\Omega,g)\).