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

TL;DR

This paper extends Reilly's inequality to the $p$-Laplacian on submanifolds in space forms, providing an upper bound for the first eigenvalue based on mean curvature and ambient curvature.

Contribution

It generalizes Reilly's inequality from the Laplacian to the $p$-Laplacian for certain $p$, broadening the scope of eigenvalue estimates for submanifolds.

Findings

Derived an upper bound for the first nonzero eigenvalue of the $p$-Laplacian.

Extended classical inequalities to the $p$-Laplacian setting.

Connected eigenvalue bounds with geometric properties of submanifolds.

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.