Spectral stability of higher order uniformly elliptic operators

TL;DR

This paper establishes spectral stability estimates for higher order elliptic operators under domain variations, providing bounds based on natural geometric distances and boundary deviations, applicable even with boundary degenerations.

Contribution

It introduces new spectral stability estimates for arbitrary even order elliptic operators on domains with possibly degenerate boundaries, extending previous results to more general settings.

Findings

Derived estimates using a natural distance between open sets

Provided bounds involving the lower Hausdorff-Pompeiu deviation

Achieved boundary-independent estimates for diffeomorphic domains

Abstract

We prove estimates for the variation of the eigenvalues of uniformly elliptic operators with homogeneous Dirichlet or Neumann boundary conditions upon variation of the open set on which an operator is defined. We consider operators of arbitrary even order and open sets admitting arbitrary strong degeneration. The main estimate is expressed via a natural and easily computable distance between open sets with continuous boundaries. Another estimate is obtained via the lower Hausdorff-Pompeiu deviation of the boundaries, which in general may be much smaller than the usual Hausdorff-Pompeiu distance. Finally, in the case of diffeomorphic open sets we obtain an estimate even without the assumption of continuity of the boundaries.