A bound for the eigenvalue counting function for Krein-von Neumann and Friedrichs extensions

作者：Ashbaugh Mark S; Gesztesy Fritz^*; Laptev Ari; Mitrea Marius; Sukhtaiev Selim

来源：Advances in Mathematics, 2017, 304: 1108-1155.

DOI：10.1016/j.aim.2016.09.11

• Krein and Friedrichs extensions of powers of second-order uniformly elliptic partial differential operators
• Bounds on eigenvalue counting functions Spectral analysis
• Buckling problem

For an arbitrary open, nonempty, bounded set Omega subset of R-n, n epsilon N, and sufficiently smooth coefficients a, b, q, we consider the closed, strictly positive, higher-order differential operator A(Omega,2m)(a, b, q) in L-2(Omega) defined on W-0(2m 2)(Omega), associated with the differential expression
T2m (a, b, q) := ((GRAPHICS) (-i partial derivative(j) - b(j))a(J,k)(-i partial derivative(k) - b(k)) +q)(m),
m epsilon N,
and its Krein-von Neumann extension A(K,Omega,2m) (a, b, q) in L-2(Omega). Denoting by N(lambda; A(K,Omega,2m)(a, b, q)), lambda > 0, the eigen-value counting function corresponding to the strictly positive eigenvalues of A(K,Omega,2m) (a, b, q), we derive the bound
N(lambda; A(K,Omega,2m) (a, b, q))
<= Cv(n) (2 pi)(-n) (1+2m/2m+n)(n/(2m)) lambda n/(2m),
lambda > 0,
where C = C (a, b, q, Omega) > 0 (with C(In, 0, 0, Omega) = vertical bar Omega vertical bar) is connected to the eigenfunction expansion of the self-adjoint operator (A) over tilde (2m)(a, b, q) in L-2 (R-n) defined on W-2m,W-2(R-n), corresponding to T2m (a, b, q). Here v(n) := pi(n/2) /Gamma((n+2)/2) denotes the (Euclidean) volume of the unit ball in 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 (A) over tilde (2) (a, b, q) in L-2(R-n).
We also consider the analogous bound for the eigenvalue counting function for the Friedrichs extension A(F,Omega,2m)(a, b, q) in L-2(Omega) of A(Omega,2m)(a, b, q).
No assumptions on the boundary partial derivative Omega of Omega are made.

• 出版日期2017-1-2