For an arbitrary open, nonempty, bounded set Ω⊂Rn, b2e8213826" title="Click to view the MathML source">n∈N, and sufficiently smooth coefficients e7e787bf1213b235b01262b51" title="Click to view the MathML source">a,b,q, we consider the closed, strictly positive, higher-order differential operator a775cb1ee552fc970147260c5d" title="Click to view the MathML source">AΩ,2m(a,b,q) in 89aba44d21c895447b153ac28" title="Click to view the MathML source">L2(Ω) defined on e6590734e640a190aac9f6c74">, associated with the differential expression
and its Krein–von Neumann extension 97399b5ae4f9caee6dda0d791f" title="Click to view the MathML source">AK,Ω,2m(a,b,q) in 89aba44d21c895447b153ac28" title="Click to view the MathML source">L2(Ω). Denoting by 89b5ac8934320" title="Click to view the MathML source">N(λ;AK,Ω,2m(a,b,q)), a78720" title="Click to view the MathML source">λ>0, the eigenvalue counting function corresponding to the strictly positive eigenvalues of 97399b5ae4f9caee6dda0d791f" title="Click to view the MathML source">AK,Ω,2m(a,b,q), we derive the bound
where C=C(a,b,q,Ω)>0 (with a79976c0bdc5bc08b8e48a3a" title="Click to view the MathML source">C(In,0,0,Ω)=|Ω|) is connected to the eigenfunction expansion of the self-adjoint operator af80"> in 975b045fb410d215a3c8a761dd6" title="Click to view the MathML source">L2(Rn) defined on W2m,2(Rn), corresponding to e600e305f4eb19886d204c2c0a318fe" 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.
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 89af1b5a5fb"> in 975b045fb410d215a3c8a761dd6" title="Click to view the MathML source">L2(Rn).
We also consider the analogous bound for the eigenvalue counting function for the Friedrichs extension b20c7f7e4b3" title="Click to view the MathML source">AF,Ω,2m(a,b,q) in 89aba44d21c895447b153ac28" title="Click to view the MathML source">L2(Ω) of a775cb1ee552fc970147260c5d" title="Click to view the MathML source">AΩ,2m(a,b,q).