Reilly-type inequalities for $p$-Laplacian on submanifolds in space forms

Abstract

Let $M$ be an $n$-dimensional closed orientable submanifold in an $N$-dimensional space form. When $1<p \le \frac n2 + 1$, we obtain an upper bound for the first nonzero eigenvalue of the $p$-Laplacian in terms of the mean curvature of $M$ and the curvature of the space form. This generalizes the Reilly inequality for the Laplacian [9, 15] to the $p$-Laplacian and extends the work of [8] for the $p$-Laplacian.