A shape variation result via the geometry of eigenfunctions

TL;DR

This paper investigates geometric properties of the first eigenfunctions of the Laplace operator under Zaremba boundary conditions on annular domains, revealing how eigenvalues change with domain shape modifications.

Contribution

It introduces new geometric insights into eigenfunctions and demonstrates how shape calculus can predict eigenvalue behavior as domain geometry varies.

Findings

First eigenfunctions exhibit foliated Schwarz symmetry and monotonicity properties.

The first eigenvalue decreases as the inner ball approaches the outer boundary.

Shape calculus effectively links geometric properties to eigenvalue variations.

Abstract

We discuss some of the geometric properties, such as the foliated Schwarz symmetry, the monotonicity along the axial and the affine-radial directions, of the first eigenfunctions of the Zaremba problem for the Laplace operator on annular domains. These fine geometric properties, together with the shape calculus, help us to prove that the first eigenvalue is strictly decreasing as the inner ball moves towards the boundary of the outer ball.