A note on domain monotonicity for the Neumann eigenvalues of the Laplacian

TL;DR

This paper proves a variant of domain monotonicity for Neumann Laplacian eigenvalues in convex domains and derives an upper bound for these eigenvalues, advancing understanding of spectral properties in convex geometry.

Contribution

It introduces a new variant of domain monotonicity for Neumann eigenvalues and provides an upper bound for convex domains, which was not previously established.

Findings

Proved a variant of domain monotonicity for Neumann eigenvalues.

Derived an upper bound for Neumann eigenvalues of convex domains.

Enhanced understanding of spectral properties in convex geometry.

Abstract

Given a convex domain and its convex sub-domain we prove a variant of domain monotonicity for the Neumann eigenvalues of the Laplacian. As an application of our method we also obtain an upper bound for Neumann eigenvalues of the Laplacian of a convex domain.