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 ac28" title="Click to view the MathML source">L2(Ω) defined on ac9f6c74">, associated with the differential expression
and its Krein–von Neumann extension AK,Ω,2m(a,b,q) in ac28" title="Click to view the MathML source">L2(Ω). Denoting by ac8934320" title="Click to view the MathML source">N(λ;AK,Ω,2m(a,b,q)), bd15ec619da6e5887571a78720" title="Click to view the MathML source">λ>0, the eigenvalue counting function corresponding to the strictly positive eigenvalues of AK,Ω,2m(a,b,q), we derive the bound
where C=C(a,b,q,Ω)>0 (with bdc5bc08b8e48a3a" title="Click to view the MathML source">C(In,0,0,Ω)=|Ω|) is connected to the eigenfunction expansion of the self-adjoint operator in bd6d975b045fb410d215a3c8a761dd6" title="Click to view the MathML source">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 bddf196643438789af1b5a5fb"> in bd6d975b045fb410d215a3c8a761dd6" title="Click to view the MathML source">L2(Rn).
We also consider the analogous bound for the eigenvalue counting function for the Friedrichs extension AF,Ω,2m(a,b,q) in ac28" title="Click to view the MathML source">L2(Ω) of AΩ,2m(a,b,q).