Sharp spectral stability estimates via the Lebesgue measure of domains for higher order elliptic operators

TL;DR

This paper establishes precise stability bounds for eigenvalues of higher order elliptic operators based on the Lebesgue measure of domain differences, applicable to both Dirichlet and Neumann conditions.

Contribution

It provides the first sharp spectral stability estimates for higher order elliptic operators in terms of domain measure differences.

Findings

Sharp bounds for eigenvalue variations are derived.

Estimates hold for both Dirichlet and Neumann boundary conditions.

Results improve understanding of spectral stability under domain perturbations.

Abstract

We prove sharp stability estimates for the variation of the eigenvalues of non-negative self-adjoint elliptic operators of arbitrary even order upon variation of the open sets on which they are defined. These estimates are expressed in terms of the Lebesgue measure of the symmetric difference of the open sets. Both Dirichlet and Neumann boundary conditions are considered.