For an arbitrary open, nonempty, bounded set Ω⊂Rn, 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 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 e6590734e640a190aac9f6c74">, associated with the differential expression
and its Krein–von Neumann extension e6dda0d791f" 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 e6dda0d791f" 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 C(In,0,0,Ω)=|Ω|) is connected to the eigenfunction expansion of the self-adjoint operator in L2(Rn) defined on 9f4d46b" title="Click to view the MathML source">W2m,2(Rn), corresponding to e600e305f4eb19886d204c2c0a318fe" title="Click to view the MathML source">τ2m(a,b,q). Here b3223ed71" 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 b3" 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).