On domain monotonicity of Neumann eigenvalues of convex domains

TL;DR

This paper establishes sharp bounds and comparisons for Neumann eigenvalues of convex domains, extending domain monotonicity results and exploring eigenvalue behavior across dimensions and convexity conditions.

Contribution

It provides explicit bounds for eigenvalue ratios on convex domains and investigates their behavior beyond convexity, advancing understanding of Neumann eigenvalues.

Findings

Sharp bounds for eigenvalue ratios on convex domains

Explicit multiplicative factors for domain monotonicity

Analysis of eigenvalue behavior with dimension and convexity

Abstract

Inspired by a recent result of Funano's, we provide a sharp quantitative comparison result between the first nontrivial eigenvalues of the Neumann Laplacian on bounded convex domains $\Omega_{1} \subset \Omega_{2}$ in any dimension $d$ greater than or equal to two, recovering domain monotonicity up to an explicit multiplicative factor. We provide upper and lower bounds for such multiplicative factors for higher-order eigenvalues, and study their behaviour with respect to the dimension and order. We further consider different scenarios where convexity is no longer imposed. In a final section we formulate some related open problems.