Hadamard's variational formula for simple eigenvalues

TL;DR

This paper investigates Hadamard's variational formula for simple eigenvalues, demonstrating harmonic convexity of the first Laplacian eigenvalue under certain deformations and deriving new inequalities in 2D domains.

Contribution

It establishes harmonic convexity of the first eigenvalue under dynamical and conformal deformations, leading to novel inequalities for Laplacian eigenvalues in two dimensions.

Findings

Harmonic convexity of the first eigenvalue under mixed boundary conditions

New inequalities for Laplacian eigenvalues in 2D domains

Extension of Hadamard's variational formula to conformal deformations

Abstract

We study Hadamard's variational formula for simple eigenvalues under dynamical and conformal deformations. Particularly, harmonic convexity of the first eigenvalue of the Laplacian under the mixed boundary condition is established for two-dimensional domain, which implies several new inequalities.