Reilly inequality for Varifolds

TL;DR

This paper extends Reilly's inequality, which bounds the first Laplacian eigenvalue using mean curvature, to the broader setting of varifolds, including polygons, and examines cases of equality.

Contribution

It generalizes Reilly's inequality to H(2) varifolds and polygons, providing new insights into eigenvalue bounds in geometric measure theory.

Findings

Reilly inequality is extended to varifolds.

The inequality is specialized for polygons.

Analysis of conditions for equality cases.

Abstract

The famous Reilly inequality gives an upper bound for the first eigenvalue of the Laplacian defined on compact submanifolds of the Euclidean space in terms of the $L^2$-norm of the mean curvature vector. In this paper, we generalize this inequality in a Varifold context. In particular we generalize it for the class of $H(2)$ varifolds and for polygons and we analyse the equality case.