Eigenvalue maximization for surfaces of revolution with prescribed boundary

TL;DR

This paper proves the existence and uniqueness of a surface of revolution that maximizes the first Dirichlet eigenvalue given two fixed boundary circles, with the maximum surface depending on the proximity of the circles.

Contribution

It establishes the existence and uniqueness of the eigenvalue-maximizing surface of revolution with prescribed boundary circles in three-dimensional space.

Findings

Existence of a maximizer surface for sufficiently close boundary circles.

Uniqueness of the maximizer surface when the boundary circles are close.

Characterization of the maximizing surface as a surface of revolution.

Abstract

Fix two parallel circles in $\mathbb{R}^3$ centered about a common axis. Among surfaces of revolution immersed in $\mathbb{R}^3$ whose boundary is given by these circles, there is one which maximizes the first Dirichlet eigenvalue. If the circles are sufficiently close together, then this surface is unique.