For an arbitrary open, nonempty, bounded set Ω⊂Rn, n∈N, and sufficiently smooth coefficients 35b01262b51" title="Click to view the MathML source">a,b,q, we consider the closed, strictly positive, higher-order differential operator 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 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
where 3583079aad096e5" title="Click to view the MathML source">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 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 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).