For an arbitrary open, nonempty, bounded set a01" 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 970147260c5d" title="Click to view the MathML source">AΩ,2m(a,b,q) in L2(Ω) defined on a0e6590734e640a190aac9f6c74">, 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 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 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 976c0bdc5bc08b8e48a3a" 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 975b045fb410d215a3c8a761dd6" title="Click to view the MathML source">L2(Rn) defined on a08ca5c2aa4866bc01139f4d46b" 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 vn:=πn/2/Γ((n+2)/2) denotes the (Euclidean) volume of the unit ball in a0c93f5c7dd08d47" 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 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 AF,Ω,2m(a,b,q) in L2(Ω) of 970147260c5d" title="Click to view the MathML source">AΩ,2m(a,b,q).