Shape differentiability of the eigenvalues of elliptic systems

TL;DR

This paper investigates how the eigenvalues of elliptic systems change with domain shape, proving their analytic dependence, deriving formulas for their variation, and characterizing critical shapes like balls under volume constraints.

Contribution

It establishes the analyticity of eigenvalue functions, derives Hadamard-type formulas, and characterizes critical domains for elliptic systems, especially highlighting the role of symmetry.

Findings

Eigenvalues' elementary symmetric functions are analytic with respect to domain shape.

Hadamard-type formulas for eigenvalue functions are derived.

Balls are identified as critical domains for rotation-invariant systems under volume constraints.

Abstract

We consider second order elliptic systems of partial differential equations subject to Dirichlet and Neumann boundary conditions. We prove analyticity of the elementary symmetric functions of the eigenvalues, and compute Hadamard-type formulas for such functions. Then we provide a characterization of criticality of the domain under volume constraint, and prove that if the system is rotation invariant, then balls are critical domains for all those functions.