Estimates on the Neumann and Steklov principal eigenvalues of collapsing domains

TL;DR

This paper explores how the principal eigenvalues of Neumann and Steklov problems relate in collapsing convex domains, providing partial proofs for conjectures on their ratio in specific geometric settings.

Contribution

It establishes bounds on the eigenvalue ratio for collapsing convex domains, identifying extremal shapes like triangles and rectangles under certain conditions.

Findings

Thinning triangles maximize the eigenvalue ratio.

Thinning rectangles minimize the eigenvalue ratio with symmetry.

Partial proof of a conjecture relating Neumann and Steklov eigenvalues.

Abstract

We investigate the relationship between the Neumann and Steklov principal eigenvalues emerging from the study of collapsing convex domains in $\mathbb{R}^2$. Such a relationship allows us to give a partial proof of a conjecture concerning estimates of the ratio of the former to the latter: we show that thinning triangles maximize the ratio among convex thinning sets, while thinning rectangles minimize the ratio among convex thinning with some symmetry property.