For an arbitrary open, nonempty, bounded set
Ω⊂Rn Ω ⊂ R n ,
n∈N n ∈ N , and sufficiently smooth coefficients
a,b,q a , b , q , we consider the closed, strictly positive, higher-order differential operator
AΩ,2m (a,b,q) A Ω , 2 m ( a , b , q ) in
a44d21c895447b153ac28" title="Click to view the MathML source">L2 (Ω) L 2 ( Ω ) defined on
mage" height="20" width="71" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870816305412-si6.gif"> mage/1-s2.0-S0001870816305412-si6.gif">W 0 2 m , 2 ( Ω ) , associated with the differential expression
and its Krein–von Neumann extension
AK,Ω,2m (a,b,q) A K , Ω , 2 m ( a , b , q ) in
a44d21c895447b153ac28" title="Click to view the MathML source">L2 (Ω) L 2 ( Ω ) . Denoting by
N(λ;AK,Ω,2m (a,b,q)) N ( λ ; A K , Ω , 2 m ( a , b , q ) ) ,
λ>0 λ > 0 , the eigenvalue counting function corresponding to the strictly positive eigenvalues of
AK,Ω,2m (a,b,q) A K , Ω , 2 m ( a , b , q ) , we derive the bound
where
C=C(a,b,q,Ω)>0 C = C ( a , b , q , Ω ) > 0 (with
C(In ,0,0,Ω)=|Ω| C ( I n , 0 , 0 , Ω ) = | Ω | ) is connected to the eigenfunction expansion of the self-adjoint operator
mage" height="20" width="82" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870816305412-si15.gif"> mage/1-s2.0-S0001870816305412-si15.gif">A ˜ 2 m ( a , b , q ) in
L2 (Rn ) L 2 ( R n ) defined on
a4866bc01139f4d46b" title="Click to view the MathML source">W2m,2 (Rn ) W 2 m , 2 ( R n ) , corresponding to
τ2m (a,b,q) τ 2 m ( a , b , q ) . Here
vn :=πn/2 /Γ((n+2)/2) v n : = π n / 2 / Γ ( ( n + 2 ) / 2 ) denotes the (Euclidean) volume of the unit ball in
Rn R n .
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 mage" height="20" width="70" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870816305412-si21.gif"> mage/1-s2.0-S0001870816305412-si21.gif">A ˜ 2 ( a , b , q ) in L2 (Rn ) L 2 ( R n ) .
We also consider the analogous bound for the eigenvalue counting function for the Friedrichs extension AF,Ω,2m (a,b,q) A F , Ω , 2 m ( a , b , q ) in a44d21c895447b153ac28" title="Click to view the MathML source">L2 (Ω) L 2 ( Ω ) of AΩ,2m (a,b,q) A Ω , 2 m ( a , b , q ) .
No assumptions on the boundary ∂Ω of Ω are made.