Extrinsic upper bound of the eigenvalue for \(p\)-Laplacian (Q1988442)

Let \(M\) be an \(m\)-dimensional closed orientable submanifold in an \(n\)-dimensional Riemannian manifold \(N\). Let \(M\) be an \(n\)-dimensional compact Riemannian manifold. Recall that the \(p\)-Laplacian (\(p>1\)) on \(M\) is defined by \[\Delta_pu=\mbox{div}\left(|\nabla u|^{p-2}\nabla u\right).\] When \(p=2\), it is the usual Laplacian; when \(p\neq2\) it is a second-order quasilinear elliptic operator. In this paper, the author focuses on an \(m\)-dimensional closed submanifold \(M\) of an \(n\)-dimensional Riemannian manifold \(N\). When \(N\) is the Euclidean space, R. Reilly obtained a well-known estimate for the first nonzero eigenvalue of Laplacian. When the sectional curvature of \(N\) is bounded above by \(\delta\), the author obtains an upper bound for the first nonzero eigenvalue of the \(p\)-Laplacian in terms of the second fundamental form of \(M\) and \(\delta\). This generalizes the Reilly-type inequality for the Laplacian [\textit{E. Heintze}, Math. Ann. 280, No. 3, 389--402 (1988; Zbl 0628.53044); \textit{R. C. Reilly}, Comment. Math. Helv. 52, 525--533 (1977; Zbl 0382.53038)] to the \(p\)-Laplacian and extends the work of \textit{H. Chen} and \textit{G. Wei} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 184, 210--217 (2019; Zbl 1419.58009)] and \textit{F. Du} and \textit{J. Mao} [Front. Math. China 10, No. 3, 583--594 (2015; Zbl 1334.58018)] for the \(p\)-Laplacian.