Hadamard's Formula Inside and Out

Abstract

Our goal is to explore boundary variations of spectral problems from the calculus of moving surfaces point of view. Hadamard's famous formula for simple Laplace eigenvalues under Dirichlet boundary conditions is generalized in a number of significant ways, including Neumann and mixed boundary conditions, multiple eigenvalues, and second order variations. Some of these formulas appear for the first time here. Furthermore, we present an analytical framework for deriving general formulas of the Hadamard type. The presented analysis finds direct applications in shape optimization and other variational problems. As a specific application, we discuss equilibrium and stable shapes of electron bubbles. © 2010 Springer Science+Business Media, LLC.