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 60c5d" title="Click to view the MathML source">AΩ,2m(a,b,q) in L2(Ω) defined on , associated with the differential expression
and its Krein–von Neumann extension AK,Ω,2m(a,b,q) in L2(Ω). Denoting by 02c64da54ea89b5ac8934320" title="Click to view the MathML source">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
020">
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 60fab11ca22431b52517af80"> in L2(Rn) defined on W2m,2(Rn), corresponding to 600e305f4eb19886d204c2c0a318fe" title="Click to view the MathML source">τ2m(a,b,q). Here vn:=πn/2/Γ((n+2)/2) denotes the (Euclidean) volume of the unit ball in Rn.
020">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 AF,Ω,2m(a,b,q) in L2(Ω) of 60c5d" title="Click to view the MathML source">AΩ,2m(a,b,q).