A bound for the eigenvalue counting function for Krein--von Neumann and Friedrichs extensions

الأصل · EN

For an arbitrary open, nonempty, bounded set Ω⊂ Rⁿ, n ∈ N, and sufficiently smooth coefficients a,b,q, we consider the closed, strictly positive, higher-order differential operator AΩ, ₂ₘ (a,b,q) in L²(Ω) defined on W₀²ᵐ,²(Ω), associated with the higher-order differential expression τ₂ₘ (a,b,q):= (∑ⱼ,ₖ₌₁ⁿ (-i ∂ⱼ - bⱼ) aⱼ,ₖ (-i ∂ₖ - bₖ)+q)ᵐ, m ∈ N, and its Krein--von Neumann extension Aₖ, Ω, ₂ₘ (a,b,q) in L²(Ω). Denoting by N(λ; Aₖ, Ω, ₂ₘ (a,b,q)), λ> 0, the eigenvalue counting function corresponding to the strictly positive eigenvalues of Aₖ, Ω, ₂ₘ (a,b,q), we derive the bound N(λ; Aₖ, Ω, ₂ₘ (a,b,q)) ≤ C vₙ (2π)⁻ⁿ (1+2m/2m+n)ⁿ/⁽²ᵐ⁾ λⁿ/⁽²ᵐ⁾, λ> 0, where C = C(a,b,q,Ω)>0 (with C(Iₙ,0,0,Ω) = |Ω|) is connected to the eigenfunction expansion of the self-adjoint operator A₂ₘ (a,b,q) in L²(Rⁿ) defined on W²ᵐ,²(Rⁿ), corresponding to τ₂ₘ (a,b,q). Here vₙ:= πⁿ/²/Γ((n+2)/2) denotes the (Euclidean) volume of the unit ball in Rⁿ. Our method of proof relies on variational considerations exploiting the fundamental link between the Krein--von Neumann extension and an underlying abstract buckling problem, and on the distorted Fourier transform defined in terms of the eigenfunction transform of A₂ (a,b,q) in L²(Rⁿ). We also consider the analogous bound for the eigenvalue counting function for the Friedrichs extension AF,Ω, ₂ₘ (a,b,q) in L²(Ω) of AΩ, ₂ₘ (a,b,q). No assumptions on the boundary ∂ Ω of Ω are made.