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
aba44d21c895447b153ac28" title="Click to view the MathML source">L2 (Ω) L 2 ( Ω ) defined on
ge" height="20" width="71" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimge id="1-s2.0-S0001870816305412-si6.gif"> ge/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
aba44d21c895447b153ac28" 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 ei
ge nvalue counting function corresponding to the strictly positive ei
ge nvalues 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 ei
ge nfunction expansion of the self-adjoint operator
ge" height="20" width="82" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimge id="1-s2.0-S0001870816305412-si15.gif"> ge/1-s2.0-S0001870816305412-si15.gif">A ˜ 2 m ( a , b , q ) in
L2 (Rn ) L 2 ( R n ) defined on
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 eige nfunction transform of ge" height="20" width="70" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimge id="1-s2.0-S0001870816305412-si21.gif"> ge/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 eige nvalue counting function for the Friedrichs extension AF,Ω,2m (a,b,q) A F , Ω , 2 m ( a , b , q ) in aba44d21c895447b153ac28" 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.