Spectral stability of general non-negative self-adjoint operators with applications to Neumann-type operators

Abstract

We prove a stability theorem for the eigenvalues of general non-negative self-adjoint linear operators with compact resolvents and by applying it we prove a sharp stability result for the dependence of the eigenvalues of second order uniformly elliptic linear operators with homogeneous Neumann boundary conditions upon domain perturbation

Spectral stability of Dirichlet second order uniformly elliptic operators

“…However, in this case a simpler transition operator can be used: just the restriction from Ω 1 to Ω 2 (see [3,4]). …”

“…where, for μ = 0, 1, W μ,∞ (Ω) denotes the Sobolev space with fractional order of smoothness (see [3]). …”

“…Some results of such a type have been proved in Burenkov and Davies [1], Burenkov and Lamberti [2,3], Davies [6,8], Kozlov [14], Pang [19].…”

“…In order to prove these results, we apply the general spectral stability theorem proved in [3], which is based on the notion of a 'transition operator.' In this paper we define suitable transition operators which enable us to prove our sharp estimates.…”

“…Since φ (x) ∈ (V j ) ρ 2 for all j ∈ J (x), 0 < < ρ 4 , then by (7.10) and (7.12) it follows that (7.13) for all j ∈ J (x), 0 < < ρ 4 . Finally, since j ∈J (x) ψ j (x) = s j =1 ψ j (x) = 1 because supp ψ j ⊂ (V j ) 3 4 ρ for all j = 1, . .…”