For an arbitrary open, nonempty, bounded set e949dd851938f0fc4478d11304a01" title="Click to view the MathML source">Ω⊂Rn, n∈N, and sufficiently smooth coefficients a,b,q, we consider the closed, strictly positive, higher-order differential operator e552fc970147260c5d" title="Click to view the MathML source">AΩ,2m(a,b,q) in ba44d21c895447b153ac28" title="Click to view the MathML source">L2(Ω) defined on a0e6590734e640a190aac9f6c74">, associated with the differential expression
and its Krein–von Neumann extension a0d791f" title="Click to view the MathML source">AK,Ω,2m(a,b,q) in ba44d21c895447b153ac28" title="Click to view the MathML source">L2(Ω). Denoting by N(λ;AK,Ω,2m(a,b,q)), e5887571a78720" title="Click to view the MathML source">λ>0, the eigenvalue counting function corresponding to the strictly positive eigenvalues of a0d791f" title="Click to view the MathML source">AK,Ω,2m(a,b,q), we derive the bound
where e5" title="Click to view the MathML source">C=C(a,b,q,Ω)>0 (with bc08b8e48a3a" 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 L2(Rn) defined on a08ca5c2aa4866bc01139f4d46b" title="Click to view the MathML source">W2m,2(Rn), corresponding to τ2m(a,b,q). Here a1a8e064040393224564798b3223ed71" title="Click to view the MathML source">vn:=πn/2/Γ((n+2)/2) denotes the (Euclidean) volume of the unit ball in e5ee4c2a0c93f5c7dd08d47" title="Click to view the MathML source">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 in L2(Rn).
We also consider the analogous bound for the eigenvalue counting function for the Friedrichs extension a1004e0773e4317b0b20c7f7e4b3" title="Click to view the MathML source">AF,Ω,2m(a,b,q) in ba44d21c895447b153ac28" title="Click to view the MathML source">L2(Ω) of e552fc970147260c5d" title="Click to view the MathML source">AΩ,2m(a,b,q).