For an arbitrary open, nonempty, bounded set e949dd851938f0fc4478d11304a01" title="Click to view the MathML source">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 AΩ,2m(a,b,q) in L2(Ω) defined on e6590734e640a190aac9f6c74">, associated with the differential expression
and its Krein–von Neumann extension ae4f9caee6dda0d791f" 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 ae4f9caee6dda0d791f" title="Click to view the MathML source">AK,Ω,2m(a,b,q), we derive the bound
where a37e118520b7bf93583079aad096e5" title="Click to view the MathML source">C=C(a,b,q,Ω)>0 (with a3a" 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 b045fb410d215a3c8a761dd6" 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 in b045fb410d215a3c8a761dd6" title="Click to view the MathML source">L2(Rn).
We also consider the analogous bound for the eigenvalue counting function for the Friedrichs extension b0b20c7f7e4b3" title="Click to view the MathML source">AF,Ω,2m(a,b,q) in L2(Ω) of AΩ,2m(a,b,q).